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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

## Book.Geometry istorija

2019 balandžio 29 d., 00:34 atliko AndriusKulikauskas -
Pridėtos 184-185 eilutės:

http://drmichaelrobinson.net/sheaftutorial/index.html
2019 sausio 19 d., 11:14 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
[[https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6 | [++++几何++++]]] _ _ _ _ [+++געאָמעטרי+++]
į:
[[https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6 | [++++几何++++]]] _ _ _ _ [[https://yi.wikipedia.org/wiki/%D7%92%D7%A2%D7%90%D7%9E%D7%A2%D7%98%D7%A8%D7%99%D7%A2 | [+++געאָמעטרי+++]]]
2019 sausio 19 d., 11:12 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
[[https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6 | [++++几何++++]]] _ _ _ _ [+++מאטעמאטיק+++]
į:
[[https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6 | [++++几何++++]]] _ _ _ _ [+++געאָמעטרי+++]
2019 sausio 19 d., 11:11 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
[++++几何++++] _ _ _ _ [+++מאטעמאטיק+++]
į:
[[https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6 | [++++几何++++]]] _ _ _ _ [+++מאטעמאטיק+++]
2019 sausio 19 d., 11:09 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

[++++几何++++] _ _ _ _ [+++מאטעמאטיק+++]
2019 sausio 03 d., 14:27 atliko AndriusKulikauskas -
Pakeistos 25-26 eilutės iš
* Defining equivalence. For example, what makes shapes equivalent?
į:
* Geometry is the regularity of choice.
* Geometry is about defining equivalence (of shapes), thus the transformations that maintain equivalence, and the symmetries of those transformations.
2019 sausio 03 d., 11:56 atliko AndriusKulikauskas -
Pakeistos 30-31 eilutės iš
į:
* Unions of spaces.
Pridėta 35 eilutė:
* Linear equations are intersections of hyperplanes.
2019 sausio 03 d., 11:37 atliko AndriusKulikauskas -
Pridėtos 22-25 eilutės:

'''General notions'''

* Defining equivalence. For example, what makes shapes equivalent?
2019 sausio 03 d., 11:23 atliko AndriusKulikauskas -
Pakeista 8 eilutė iš:
* To understand what geometry contributes to the overall map of mathematics.
į:
* To understand what geometry contributes to the overall map of mathematics.http://www.ms.lt/sodas/Book/Geometry?action=diff
Pakeista 25 eilutė iš:
į:
* Affine geometry: Point + Vector = Point. Vector + Vector = Vector. Point - Point = Vector. But we can't add two points because we don't have any origin for them to reference.
2019 sausio 03 d., 11:09 atliko AndriusKulikauskas -
Pridėtos 30-33 eilutės:
* Projective geometry is linear algebra.
* Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
* In projective geometry, vectors are points and [[https://en.wikipedia.org/wiki/Bivector | bivectors]] are lines.
* Projective geometry transforms conics into conics.
Ištrintos 113-115 eilutės:
* Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
* In projective geometry, vectors are points and [[https://en.wikipedia.org/wiki/Bivector | bivectors]] are lines.
* Projective geometry transforms conics into conics.
2019 sausio 03 d., 10:52 atliko AndriusKulikauskas -
Pakeista 29 eilutė iš:
* Map lines to lines. Projective geometry additionally maps zero to zero. And infinity to infinity?
į:
* Map lines to lines. Projective geometry additionally maps zero to zero. And infinity to infinity? And do the lines have an orientation? And is that orientation preserved?
2019 sausio 03 d., 10:51 atliko AndriusKulikauskas -
Pridėta 29 eilutė:
* Map lines to lines. Projective geometry additionally maps zero to zero. And infinity to infinity?
Pakeistos 34-36 eilutės iš
į:
* Projective geometry: Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
Ištrintos 83-84 eilutės:
Books
Ištrinta 120 eilutė:
* Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
2019 sausio 03 d., 10:47 atliko AndriusKulikauskas -
Pakeistos 29-35 eilutės iš
[[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]

Fundamental Theorem of Affine geometry. let {$X,X'$} be two finite dimensional affine spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:X\to X'$} be a bijection that sends collinear points to collinear points, i.e. such that for all {$a,b,c\in X$} that are collinear, {$f(a),f(b),f(c)$} are collinear too. Then {$f$} is a semi-affine isomorphism.

This means that there is a field isomorphism {$\sigma:K\to K'$} such that for any point {$a\in X$} the map induced by {$f_a: X_a\to X'_{f(a)}$} is a {$\sigma$}-semi-linear isomorphism.

Fundamental Theorem of Projective geometry. let {$P(X),P(X')$} be two finite dimensional projective spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:P(X)\to P(X')$} be a bijection that sends collinear points to collinear points. Then {$f$} is a semi-linear isomorphism.
į:
* [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
** Fundamental Theorem of Affine geometry. let {$X,X'$} be two finite dimensional affine spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:X\to X'$} be a bijection that sends collinear points to collinear points, i.e. such that for all {$a,b,c\in X$} that are collinear, {$f(a),f(b),f(c)$} are collinear too. Then {$f$} is a semi-affine isomorphism.
** This means that there is a field isomorphism {$\sigma:K\to K'$} such that for any point {$a\in X$} the map induced by {$f_a: X_a\to X'_{f(a)}$} is a {$\sigma$}-semi-linear isomorphism.
** Fundamental Theorem of Projective geometry. let {$P(X),P(X')$} be two finite dimensional projective spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:P(X)\to P(X')$} be a bijection that sends collinear points to collinear points. Then {$f$} is a semi-linear isomorphism.
2019 sausio 03 d., 10:47 atliko AndriusKulikauskas -
Pakeistos 29-30 eilutės iš
į:
[[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
Ištrinta 75 eilutė:
* [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
2019 sausio 03 d., 10:46 atliko AndriusKulikauskas -
Pridėtos 28-34 eilutės:

Fundamental Theorem of Affine geometry. let {$X,X'$} be two finite dimensional affine spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:X\to X'$} be a bijection that sends collinear points to collinear points, i.e. such that for all {$a,b,c\in X$} that are collinear, {$f(a),f(b),f(c)$} are collinear too. Then {$f$} is a semi-affine isomorphism.

This means that there is a field isomorphism {$\sigma:K\to K'$} such that for any point {$a\in X$} the map induced by {$f_a: X_a\to X'_{f(a)}$} is a {$\sigma$}-semi-linear isomorphism.

Fundamental Theorem of Projective geometry. let {$P(X),P(X')$} be two finite dimensional projective spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:P(X)\to P(X')$} be a bijection that sends collinear points to collinear points. Then {$f$} is a semi-linear isomorphism.
2019 sausio 03 d., 10:40 atliko AndriusKulikauskas -
Pridėtos 20-38 eilutės:

[++Four kinds of geometry++]

'''Path geometry'''

'''Line geometry'''

'''Angle geometry'''

* A metric yields distance, an inner product and angles.

'''Oriented area geometry'''
2018 gruodžio 19 d., 13:42 atliko AndriusKulikauskas -
Pakeistos 82-83 eilutės iš
[[https://www.maths.ed.ac.uk/~v1ranick/papers/beutel.pdf | Projective Geometry: From Foundations to Applications]] Beutelspacher and Rosenbaum
į:
* [[http://morpheo.inrialpes.fr/people/Boyer/Teaching/M2R/geoProj.pdf | Projective Geometry. A Short Introduction]]
*
[[https://www.maths.ed.ac.uk/~v1ranick/papers/beutel.pdf | Projective Geometry: From Foundations to Applications]] Beutelspacher and Rosenbaum
2018 gruodžio 19 d., 13:40 atliko AndriusKulikauskas -
Pridėtos 81-82 eilutės:

[[https://www.maths.ed.ac.uk/~v1ranick/papers/beutel.pdf | Projective Geometry: From Foundations to Applications]] Beutelspacher and Rosenbaum
2018 gruodžio 09 d., 21:31 atliko AndriusKulikauskas -
Pridėta 78 eilutė:
* The link between projective geometry and fractions (as equivalence classes).
2018 rugsėjo 11 d., 08:50 atliko AndriusKulikauskas -
Pakeista 96 eilutė iš:
* Sylvain Poirer: Some key ideas, probably you know, but just in case:
į:
* Sylvain Poirier: Some key ideas, probably you know, but just in case:
Pakeista 105 eilutė iš:
* Sylvain Poirer: Affine representations of that quadric are classified by the choice of
į:
* Sylvain Poirier: Affine representations of that quadric are classified by the choice of
Pakeista 112 eilutė iš:
* Sylvain Poirer: We can understand the stereographic projection as the effect of the
į:
* Sylvain Poirier: We can understand the stereographic projection as the effect of the
Pakeista 257 eilutė iš:
* Sylvain Poirer
į:
* Sylvain Poirier
2018 liepos 14 d., 12:16 atliko AndriusKulikauskas -
Pakeistos 543-544 eilutės iš
''How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is speciWed in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a
Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the n-dimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6).'' Penrose, Road to Reality, page 293, §14.1.
į:
''How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is speciWed in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the n-dimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6).'' Penrose, Road to Reality, page 293, §14.1.
2018 liepos 14 d., 12:15 atliko AndriusKulikauskas -
Pridėtos 542-544 eilutės:

''How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is speciWed in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a
Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the n-dimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6).'' Penrose, Road to Reality, page 293, §14.1.
2018 liepos 12 d., 11:29 atliko AndriusKulikauskas -
Pridėtos 303-305 eilutės:

Videos
* [[https://www.youtube.com/watch?v=pXGTevGJ01o | Symmetric geometry and classical mechanics]], Tobias Osborne
2017 lapkričio 11 d., 19:43 atliko AndriusKulikauskas -
Pakeista 437 eilutė iš:
Demicubes
į:
Understanding the demicubes
Pridėtos 439-445 eilutės:

Defining my own demicubes
* Each vertex is plus or minus. Can we think of that as the center being inside or outside of it? As the vertex being either an inner point or an outer point? With the center being inside or outside? Or does the vertex exist or not? (Defining a subsimplex.) Is it filled or not? (As with the filling of a cycle in homology so that it is a "boundary".)
* For the distinguished point, is it necessarily an outer point, so that the center is on the outside?
* In homology, we have edges defining the vertices on either end as positive and negative. How does that work for vertices? What does it mean for a vertex to be positive or negative? And how does that relate to defining the inside or outside of a cycle?
* The ambiguity 2 may arise upon thinking of the axes of the cube, defined by pairs of opposite vertices.
* Or the ambiguity may come from the orientation of any cycle being ambiguous, and defining the inside or outside.
2017 lapkričio 11 d., 19:21 atliko AndriusKulikauskas -
Ištrintos 18-22 eilutės:
* Think of demihypercubes (coordinates sytems) Dn given by simplexes (like An) but in coordinate system presentation (standard simplexes rather than barycentric). So this requires an extra dimension. But then Dn and An are "dual" to each other in some sense.
* Understand symmetry groups, especially for the polytopes, such as [[https://en.wikipedia.org/wiki/Octahedral_symmetry | octahedral symmetry]]. Try to define an infinite family of "coordinate systems", simplexes with distinguished element, for which Dn is the symmetry group. Figure out how to count the subsimplexes and see what is the analogue for Pascal's triangle. Understand octahedron as composed of pairs of vertices.
* Note that the orientation of the simplexes, positive and negative, distinguishes inside and outside. On common edges they go in opposite directions. Also, this seems to relate the coordinates x1, x2, x3 etc. in terms of their canonical order. What does all this mean for cross polytopes?
* Boundaries distinguish inside and outside. So then how does it follow that boundaries don't have boundaries?
Pridėtos 443-452 eilutės:

* Simplexes consists of cycles with fillings.
* Cross polytopes are cycles without fillings.
* Cubes are fillings without boundaries.
* Demicubes should be without fillings and without boundaries.

* Think of demihypercubes (coordinates sytems) Dn given by simplexes (like An) but in coordinate system presentation (standard simplexes rather than barycentric). So this requires an extra dimension. But then Dn and An are "dual" to each other in some sense.
* Understand symmetry groups, especially for the polytopes, such as [[https://en.wikipedia.org/wiki/Octahedral_symmetry | octahedral symmetry]]. Try to define an infinite family of "coordinate systems", simplexes with distinguished element, for which Dn is the symmetry group. Figure out how to count the subsimplexes and see what is the analogue for Pascal's triangle. Understand octahedron as composed of pairs of vertices.
* Note that the orientation of the simplexes, positive and negative, distinguishes inside and outside. On common edges they go in opposite directions. Also, this seems to relate the coordinates x1, x2, x3 etc. in terms of their canonical order. What does all this mean for cross polytopes?
* Boundaries distinguish inside and outside. So then how does it follow that boundaries don't have boundaries?
2017 lapkričio 09 d., 13:40 atliko AndriusKulikauskas -
Pakeistos 21-22 eilutės iš
* Note that the orientation of the simplexes, positive and negative, distinguishes inside and outside. On common edges they go in opposite directions. Also, this seems to relate the coordinates x1, x2, x3 etc. in terms of their canonical order. What does all this mean for cross polytopes?
į:
* Note that the orientation of the simplexes, positive and negative, distinguishes inside and outside. On common edges they go in opposite directions. Also, this seems to relate the coordinates x1, x2, x3 etc. in terms of their canonical order. What does all this mean for cross polytopes?
* Boundaries distinguish inside and outside. So then how does it follow that boundaries don't have boundaries
?
2017 lapkričio 09 d., 13:31 atliko AndriusKulikauskas -
Pridėta 21 eilutė:
* Note that the orientation of the simplexes, positive and negative, distinguishes inside and outside. On common edges they go in opposite directions. Also, this seems to relate the coordinates x1, x2, x3 etc. in terms of their canonical order. What does all this mean for cross polytopes?
2017 lapkričio 09 d., 13:05 atliko AndriusKulikauskas -
Pakeista 20 eilutė iš:
* Understand symmetry groups, especially for the polytopes, such as [[https://en.wikipedia.org/wiki/Octahedral_symmetry | octahedral symmetry]]. Try to define an infinite family of "coordinate systems", simplexes with distinguished element, for which Dn is the symmetry group. Figure out how to count the subsimplexes and see what is the analogue for Pascal's triangle.
į:
* Understand symmetry groups, especially for the polytopes, such as [[https://en.wikipedia.org/wiki/Octahedral_symmetry | octahedral symmetry]]. Try to define an infinite family of "coordinate systems", simplexes with distinguished element, for which Dn is the symmetry group. Figure out how to count the subsimplexes and see what is the analogue for Pascal's triangle. Understand octahedron as composed of pairs of vertices.
2017 lapkričio 09 d., 11:53 atliko AndriusKulikauskas -
Pridėta 20 eilutė:
* Understand symmetry groups, especially for the polytopes, such as [[https://en.wikipedia.org/wiki/Octahedral_symmetry | octahedral symmetry]]. Try to define an infinite family of "coordinate systems", simplexes with distinguished element, for which Dn is the symmetry group. Figure out how to count the subsimplexes and see what is the analogue for Pascal's triangle.
2017 lapkričio 08 d., 18:31 atliko AndriusKulikauskas -
Pridėtos 18-19 eilutės:

* Think of demihypercubes (coordinates sytems) Dn given by simplexes (like An) but in coordinate system presentation (standard simplexes rather than barycentric). So this requires an extra dimension. But then Dn and An are "dual" to each other in some sense.
2017 lapkričio 08 d., 12:44 atliko AndriusKulikauskas -
Pakeista 137 eilutė iš:
* [[http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf | Algebraic Curves: An Introduction to Algebraic Geometry]], William Fulton
į:
* [[http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf | Algebraic Curves: An Introduction to Algebraic Geometry]], William Fulton - section 2.10 has exercises about exact sequences
2017 lapkričio 08 d., 12:44 atliko AndriusKulikauskas -
Pridėta 137 eilutė:
* [[http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf | Algebraic Curves: An Introduction to Algebraic Geometry]], William Fulton
2017 spalio 28 d., 17:06 atliko AndriusKulikauskas -
Pridėta 43 eilutė:
** [[https://www.youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM | Triangle geometry]]
Pridėta 46 eilutė:
** [[https://www.youtube.com/watch?v=rTw6XbmO8Nc&list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB | Algebraic calculus one]]
2017 spalio 25 d., 11:24 atliko AndriusKulikauskas -
Pridėta 363 eilutė:
* Atiyah speculation: Space + Circle = 4 dimensions (Riemannian). Donaldson theory -> geometric models of matter? Signature of 4-manifold = electric charge. Second Betti number = number of protons + neutrons.
2017 spalio 25 d., 11:18 atliko AndriusKulikauskas -
Pridėtos 359-362 eilutės:

Geometry challenges
* Dimension 3: relate Jones quantum invariants (knots, any manifold) with Perlman-Thurston.
* Dimension 4: understand the structure of simply-connected 4-manifolds and the relation to physics.
2017 spalio 25 d., 11:07 atliko AndriusKulikauskas -
Pakeista 353 eilutė iš:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1. Icosahedron is the fake sphere in 3-dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture.
į:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1. Icosahedron is the fake sphere in 3-dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture. Icosahedron would be in A5 but reality is given by A4 and so A5 is insolvable!
2017 spalio 25 d., 11:05 atliko AndriusKulikauskas -
Pakeista 353 eilutė iš:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1.
į:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1. Icosahedron is the fake sphere in 3-dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture.
2017 spalio 25 d., 11:03 atliko AndriusKulikauskas -
Pakeista 349 eilutė iš:
* Geometry is the study of curvature (Atiyah's video talk on Geometry in 2, 3 and 4 dimensions. Intrinsic and extrinsic curvature. Sphere has constant curvature.
į:
* Geometry is the study of curvature (Atiyah's video talk on Geometry in 2, 3 and 4 dimensions. Intrinsic and extrinsic curvature. Sphere has constant curvature. Sphere - positive - genus 0. Torus (cylinder) - zero curvature - genus 1. Higher genus - negative curvature.
Pridėta 353 eilutė:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1.
2017 spalio 25 d., 10:55 atliko AndriusKulikauskas -
Pridėtos 349-352 eilutės:
* Geometry is the study of curvature (Atiyah's video talk on Geometry in 2, 3 and 4 dimensions. Intrinsic and extrinsic curvature. Sphere has constant curvature.
** 2 dimensions - Scalar curvature R
** 3 dimensions - Ricci curvature Rij
** 4 dimensions - Riemann curvature Rijk
2017 spalio 24 d., 07:47 atliko AndriusKulikauskas -
Pakeistos 188-190 eilutės iš
į:
* Constructiveness - closed sets any intersections and finite unions are open sets constructive
* A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other.
Pridėta 200 eilutė:
* Long exact sequence from short exact sequence: derived functors.
Pakeistos 202-203 eilutės iš
į:
* Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?
Pridėtos 415-418 eilutės:
* Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).
* Cube: all vertices have a genealogy, a combination of +s and -s.
* Half-cube defines + for all, thus defines marked opposites.
Pridėtos 421-429 eilutės:
* Symmetric group action on an octahedron is marked, 1 and -1, the octahedron itself is unmarked.
* Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the cross-polytope. (Or consider Bernstein's polynomials.)

Demicubes
* Is the fusion of vertices in the demicube related to the fusion of edges of a square to create a torus, or of vertices to create a circle, etc.?

Dual:
* Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory).
* Cross-polytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory).
2017 spalio 24 d., 07:43 atliko AndriusKulikauskas -
Pakeistos 187-188 eilutės iš
į:
* If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
Pakeistos 379-380 eilutės iš
į:
* Trikampis - išauga požiūrių skaičius apibudinant: affine-taškai-0, projective-tiesės-1, conformal-kampai-2, symplectic-plotai-3.
Pridėta 457 eilutė:
* Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.
2017 spalio 24 d., 07:40 atliko AndriusKulikauskas -
Pakeistos 117-118 eilutės iš
į:
* square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball: pi-to-the-n/2 over (n/2)!
Ištrinta 324 eilutė:
Pridėtos 375-378 eilutės:
Triangles
* What is the significance of a triangle or a trilateral? They are the fourth row of Pascal's triangle.
* A triangle on a sphere together with its antipodes (defined in terms of the center) defines eight triangles, an octahedron. A triangle in three dimensional space defines a demicube (simplex) in terms of the origin. A triangle with its center defines a simplex. How is a triangle related to a cube?
Pridėtos 409-411 eilutės:
Cross-polytope
* A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
Ištrintos 422-423 eilutės:
Ištrinta 428 eilutė:
Ištrintos 488-513 eilutės:

'''Figuring things out'''

Cross-polytope
* A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.

square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball: pi-to-the-n/2 over (n/2)!

'''Notes'''

Triangles
* What is the significance of a triangle or a trilateral? They are the fourth row of Pascal's triangle.
* A triangle on a sphere together with its antipodes (defined in terms of the center) defines eight triangles, an octahedron. A triangle in three dimensional space defines a demicube (simplex) in terms of the origin. A triangle with its center defines a simplex. How is a triangle related to a cube?

Variables
* A variable is an "atom" of meaning as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.

Apibendrinimas
* Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) turi tam tikras bendras savybes, tada tas apibendrintas savybes gali naujai priskirti naujoms jų apibudintoms sąvokoms.
* Apibendrinimas yra "objekto" kūrimas.

>>bgcolor=#EEEEEE<<
2017 spalio 24 d., 07:36 atliko AndriusKulikauskas -
Pridėta 194 eilutė:
* Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
Ištrintos 507-510 eilutės:

Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
2017 spalio 24 d., 07:34 atliko AndriusKulikauskas -
Pakeistos 451-454 eilutės iš
Reflection introduces the action of Z2. It is the reflection across the boundary of self and world. (We can later also think of reflection across the horizon around us, as inversion.) This is the parity of multisets (element or not an element). And that circle S02 is then referenced by rotations and shear mapping and all work with angles. And then the relationship between two dimensions is given perhaps by Z2 x S02, the relationship between two axes: x vs. x (dilation), x vs. 1/x (squeeze) and x vs. y (translation).

Squeeze specification draws a hyperbola (x vs. 1/x). Dilation draws a line (x vs. x). Are there specifications that draw circles (rotation?), ellipses? parabolas?
į:
* Reflection introduces the action of Z2. It is
the reflection across the boundary of self and world. (We can later also think of reflection across the horizon around us, as inversion.) This is the parity of multisets (element or not an element). And that circle S02 is then referenced by rotations and shear mapping and all work with angles. And then the relationship between two dimensions is given perhaps by Z2 x S02, the relationship between two axes: x vs. x (dilation), x vs. 1/x (squeeze) and x vs. y (translation).
* Squeeze specification draws a hyperbola (x vs. 1/x). Dilation draws a line (x vs. x). Are there specifications that draw circles (rotation?), ellipses? parabolas?
* Transformacijos sieja nepriklausomus matus.
Pakeista 478 eilutė iš:
Other
į:
Other transformations
Ištrintos 509-511 eilutės:
Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia.

Transformacijos sieja nepriklausomus matus.
2017 spalio 24 d., 07:22 atliko AndriusKulikauskas -
Pakeistos 71-72 eilutės iš
į:
* Affine geometry - free monoid - without negative sign (subtraction) - lattice of steps - such as Young tableaux as paths on Pascal's triangle.
Pridėtos 145-147 eilutės:
Ideas
* Fiber is a Zero.
Pakeistos 185-186 eilutės iš
į:
* Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, (kaip kad ramybe - lūkesčių nebuvimu), tai sutapatinama, kaip kad "cross-cap".
Pakeistos 195-196 eilutės iš
į:
* Dievas žmogui yra skylė gyvenime, prasmė - neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje - homologijoje.
Pridėta 471 eilutė:
* Squeeze transformacija trijuose matuose: a b c = 1. Tai simetrinė funkcija.
Pakeistos 496-510 eilutės iš
What is the significance of a triangle or a trilateral? They are the fourth row of Pascal's triangle.

A triangle on a sphere together with its antipodes (defined in terms of the center) defines eight triangles, an octahedron. A triangle in three dimensional space defines a demicube (simplex) in terms of the origin. A triangle with its center defines a simplex. How is a triangle related to a cube?

A variable is an "atom" of meaning as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.

Dievas žmogui yra skylė gyvenime, prasmė - neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje - homologijoje.

Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, tai sutapatinama, kaip kad
"cross-cap".

Fiber is a Zero.

Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) turi tam tikras bendras savybes, tada tas apibendrintas savybes gali naujai priskirti naujoms jų apibudintoms sąvokoms
.
į:
Triangles
*
What is the significance of a triangle or a trilateral? They are the fourth row of Pascal's triangle.
* A triangle on a sphere together with its antipodes (defined in terms of the center) defines eight triangles, an octahedron. A triangle in three dimensional space defines a demicube (simplex) in terms of the origin. A triangle with its center defines a simplex. How is a triangle related to a cube?

Variables
*
A variable is an "atom" of meaning as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.

Apibendrinimas
* Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) turi tam tikras bendras savybes, tada tas apibendrintas savybes gali naujai priskirti naujoms jų apibudintoms sąvokoms.
* Apibendrinimas yra
"objekto" kūrimas.
Ištrintos 508-509 eilutės:
Apibendrinimas yra "objekto" kūrimas.
Ištrintos 510-511 eilutės:
Squeeze transformacija trijuose matuose: a b c = 1. Tai simetrinė funkcija.
Ištrintos 511-512 eilutės:

Affine geometry - free monoid - without negative sign (subtraction) - lattice of steps - such as Young tableaux as paths on Pascal's triangle.
2017 spalio 06 d., 15:12 atliko AndriusKulikauskas -
Pridėta 70 eilutė:
* Affine varieties correspond to prime ideals and as such are irreducible. So they are the building blocks of the closed subsets of the Zariski topology.
2017 spalio 06 d., 13:45 atliko AndriusKulikauskas -
Pridėtos 70-75 eilutės:

>>bgcolor=#FFFFC0<<

* What does it mean that the point at infinity is a zero of a polynomial? Is that establishing and modeling the limiting process?

>><<
2017 spalio 06 d., 13:41 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
* What does it mean that the eigenvalues of a matrix are the zeros of its characteristic polynomial? And that the matrix itself is a zero of its characteristic polynomial? And then what doe the symmetric functions of the eigenvalues of a matrix mean? The coefficients of the polynomial can be expressed in terms of the same eigenvalues that are its solutions. So in what sense are they dual?
į:
* What does it mean that the eigenvalues of a matrix are the zeros of its characteristic polynomial? And that the matrix itself is a zero of its characteristic polynomial? And then what doe the symmetric functions of the eigenvalues of a matrix mean? The coefficients of the polynomial can be expressed in terms of the same eigenvalues that are its solutions. So in what sense are they dual? Ask at Math Overflow.
2017 spalio 06 d., 13:40 atliko AndriusKulikauskas -
Pridėtos 28-29 eilutės:

>><<
2017 spalio 06 d., 13:40 atliko AndriusKulikauskas -
Pridėtos 24-27 eilutės:

>>bgcolor=#FFFFC0<<

* What does it mean that the eigenvalues of a matrix are the zeros of its characteristic polynomial? And that the matrix itself is a zero of its characteristic polynomial? And then what doe the symmetric functions of the eigenvalues of a matrix mean? The coefficients of the polynomial can be expressed in terms of the same eigenvalues that are its solutions. So in what sense are they dual?
2017 spalio 04 d., 11:40 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
* Why are rings important for geometry rather than just groups?
į:
* Why are rings important for geometry rather than just groups? Because want to work with ideals and not subrings, because we are dealing with what is not as well as what is, because we are constructing both top-down and bottom-up.
2017 spalio 04 d., 11:16 atliko AndriusKulikauskas -
Pridėta 116 eilutė:
2017 spalio 04 d., 11:12 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
Why are rings important for geometry rather than just groups?
į:
* Why are rings important for geometry rather than just groups?
2017 spalio 04 d., 11:12 atliko AndriusKulikauskas -
Pridėtos 16-17 eilutės:

Why are rings important for geometry rather than just groups?
2017 spalio 04 d., 11:02 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Attach:geometry.png
2017 spalio 04 d., 10:27 atliko AndriusKulikauskas -
Pakeistos 14-29 eilutės iš
* Study Bezier curves and Bernstein polynomials.

* Relate the first Betti number with my version of the Euler characteristic, C - V + E - F + T.

* Try to use the tetrahedron as a way to model the 4th dimension so as to imagine how a trefoil knot could be untangled.

Bernstein polynomials
* x = 1/2 get simplex
* x = 1/3 or 2/3 get cube and cross-polytope

Generalize this result to n-dimensions (starting with 4-dimensions): [[http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L12.html | Full finite symmetry groups in 3 dimensions]]

į:
Pridėtos 337-348 eilutės:

>>bgcolor=#FFFFC0<<

Center and Totality
* Relate the first Betti number with my version of the Euler characteristic, C - V + E - F + T.
* Study Bezier curves and Bernstein polynomials.
* Bernstein polynomials x = 1/2 get simplex, x = 1/3 or 2/3 get cube and cross-polytope.
* Try to use the tetrahedron as a way to model the 4th dimension so as to imagine how a trefoil knot could be untangled.

Generalize this result to n-dimensions (starting with 4-dimensions): [[http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L12.html | Full finite symmetry groups in 3 dimensions]]

>><<
2017 spalio 04 d., 10:24 atliko AndriusKulikauskas -
Ištrintos 17-22 eilutės:

* In topology product rule d(MxN) = dM x N union MxN addition is union (whereas in the Zariski topology multiplication is union). Why? The product rule is related to the deRham cohomology.
* What happens to the corners of the shapes?
* What is the topological quotient for an equilateral triangle or a simplex?
* Topological product (for a torus) is a list, has an order. In general, a Cartesian product is a list. What if such a product is unordered? How do we get there in the limit to F1?
* How can you cut in half a topological object if you have no metric? How can you be sure whether you will get two or three pieces?
Pakeistos 19-21 eilutės iš
* Try to imagine what a 3-sphere looks like as we pass through it from time t = -1 to 1.
į:
Pridėtos 155-163 eilutės:

>>bgcolor=#FFFFC0<<
* In topology product rule d(MxN) = dM x N union MxN addition is union (whereas in the Zariski topology multiplication is union). Why? The product rule is related to the deRham cohomology.
* What happens to the corners of the shapes?
* What is the topological quotient for an equilateral triangle or a simplex?
* Topological product (for a torus) is a list, has an order. In general, a Cartesian product is a list. What if such a product is unordered? How do we get there in the limit to F1?
* How can you cut in half a topological object if you have no metric? How can you be sure whether you will get two or three pieces?
* Try to imagine what a 3-sphere looks like as we pass through it from time t = -1 to 1.
>><<
2017 spalio 04 d., 10:21 atliko AndriusKulikauskas -
Ištrintos 15-18 eilutės:
* Do the six natural bases of the symmetric functions correspond to the six transformations?
* Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?
* In category theory, where do symmetric functions come up? What are eigenvalues understood as? What would be symmetric functions of eigenvalues?
Pakeistos 17-18 eilutės iš
* Think of how transformations act on 0, 1, infinity, for example, translations can take 0 to 1, but infinity to infinity.
į:
Pakeistos 26-27 eilutės iš
* How is the Zariski topology related to the Binomial theorem?
į:
Pakeistos 34-35 eilutės iš
Relate Cayley's theorem to the field with one element
į:
Pridėtos 125-130 eilutės:
>>bgcolor=#FFFFC0<<

* How is the Zariski topology related to the Binomial theorem?

>><<
Pridėtos 401-408 eilutės:

>>bgcolor=#FFFFC0<<
* Think of how transformations act on 0, 1, infinity, for example, translations can take 0 to 1, but infinity to infinity.

* Do the six natural bases of the symmetric functions correspond to the six transformations?
* Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?
* In category theory, where do symmetric functions come up? What are eigenvalues understood as? What would be symmetric functions of eigenvalues?
>><<
2017 spalio 04 d., 10:17 atliko AndriusKulikauskas -
Pridėtos 101-124 eilutės:
* Sylvain Poirer: Some key ideas, probably you know, but just in case:
The (n+p-1)-dimensional projective space associated with a quadratic
space with signature (n,p), is divided by its (n+p-2)-dimensional
surface (images of null vectors), which is a conformal space with
signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p)
and positive curvature, the other with dimension (n,p-1) and negative
curvature. Just by changing convention, the one with signature (n-1,p) and
positive curvature can also seen as a space with signature (p,n-1) and
negative curvature; and similarly for the other.
* Sylvain Poirer: Affine representations of that quadric are classified by the choice of
the horizon, or equivalently the polar point of that horizon (the
point representing in the projective space the direction orthogonal to
that hyperplane). So there are 3 possibilities.
The null one sees it as a paraboloid and gives it an affine geometry.
The 2 others, with the different signs, see it as a quadric whose
center is the polar point, and give it 2 different curved geometries
* Sylvain Poirer: We can understand the stereographic projection as the effect of the
projective transformation of the space, which changes the sphere into
a paraboloid, itself projected into an affine space.

* (1 + ti)(1 + ti) = (1 - t2) + (2t) i is the rational parametrization of the circle.
* What about the sphere? The stereographic projection of the circle onto the plane in Cartesian coordinates is given by (1 + xi + yj)(1 + xi + yj) where ij + ji = 1, that is, i and j anticommute.
* Note also that infinity is the flip side of zero - they make a pair. They are alternate ways of linking together the positive and negative values.
Pakeistos 180-182 eilutės iš
į:
* Our Father relates a left exact sequence and a right exact sequence.
* Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.
Pakeistos 390-393 eilutės iš
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the rational parametrization of the circle.
* What about the sphere? The stereographic projection of the circle onto the plane in Cartesian coordinates is given by (1 + xi + yj)(1 + xi + yj) where ij + ji = 1, that is, i and j anticommute.
* Note also that infinity is the flip side of zero - they make a pair. They are alternate ways of linking together the positive and negative values.
į:
Pakeistos 452-454 eilutės iš

A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
į:
Cross-polytope
*
A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
Ištrintos 458-482 eilutės:
'''Sylvain Poirer'''

Some key ideas, probably you know, but just in case:
The (n+p-1)-dimensional projective space associated with a quadratic
space with signature (n,p), is divided by its (n+p-2)-dimensional
surface (images of null vectors), which is a conformal space with
signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p)
and positive curvature, the other with dimension (n,p-1) and negative
curvature.
Just by changing convention, the one with signature (n-1,p) and
positive curvature can also seen as a space with signature (p,n-1) and
negative curvature; and similarly for the other.

Affine representations of that quadric are classified by the choice of
the horizon, or equivalently the polar point of that horizon (the
point representing in the projective space the direction orthogonal to
that hyperplane). So there are 3 possibilities.
The null one sees it as a paraboloid and gives it an affine geometry.
The 2 others, with the different signs, see it as a quadric whose
center is the polar point, and give it 2 different curved geometries

We can understand the stereographic projection as the effect of the
projective transformation of the space, which changes the sphere into
a paraboloid, itself projected into an affine space.
Pakeistos 467-470 eilutės iš
Exact sequence
* Our Father relates a left exact sequence and a right exact sequence.
* Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.
į:
Ištrintos 486-491 eilutės:

[+Geometry Intuition+]

2017 spalio 04 d., 10:12 atliko AndriusKulikauskas -
Pridėtos 145-149 eilutės:
Ideas
* Tadashi Tokieda: Basic strategy of topology. When a problem has degeneracies, then deform (or perturb) to a problem without degeneracies, then deform back. We can use the same approach to show some problems are unsolvable.
* Quotient is gluing is equivalence on a boundary. Topology is the creation of a smaller space from a larger space.
* Cross cap introduces contradiction, which breaks the segregation between orientations, whether inside and outside, self and world, or true and false.
Pakeistos 379-395 eilutės iš
I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry. And I imagine that they are as intuitive as the various ways that we interpret multiplication in arithmetic. I suppose that they may include translation, rotation, scaling, homothety, similarity, reflection and shear.

Relate to the six transformations in the anharmonic group of the
[[https://en.wikipedia.org/wiki/Cross-ratio | cross-ratio]]. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?

shear map takes parallelogram to square, preserves area

The 6 specifications can be compared with cinematographic movements of a camera.
* Reflection: a camera in a mirror, a frame within a frame...
* Rotation: a camera swivels from left to right, makes a choice, like turning one's head
* Dilation: a camera zooms for the desired composition
.
* Translation: a camera moves around.
But I don't know how to think of shear or squeeze mappings in terms of
a camera. However, consider what a camera would do to a tiled floor.
* Shear:
* Squeeze:
the camera looks out onto the horizon.

Note that the [[https://en
.wikipedia.org/wiki/SL2(R)#Classification_of_elements | classification of elements of SL2(R)]] includes elliptic (conjugate to a rotation), parabolic (shear) and hyperbolic (squeeze). Similarly, see the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Classification | classification of Moebius transformations]].
į:
Sources to think about
* [[http://settheory.net/geometry#transf | Sylvain Poirer's list of permutations]] which I used.
* Grothendieck's six operations:
** pushforward along a morphism and its left adjoint
** compactly supported pushforward and its right adjoint
** tensor product and its adjoint internal hom
* The various ways that we interpret multiplication in arithmetic.
* [[https://en
.m.wikipedia.org/wiki/Möbius_transformation | Möbius transformation]] combines translation, inversion, reflection, rotation, homothety. See the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Classification | classification of Moebius transformations]]. Note also that the [[https://en.wikipedia.org/wiki/SL2(R)#Classification_of_elements | classification of elements of SL2(R)]] includes elliptic (conjugate to a rotation), parabolic (shear) and hyperbolic (squeeze).
*
The six transformations in the anharmonic group of the [[https://en.wikipedia.org/wiki/Cross-ratio | cross-ratio]]. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?
* The 6 specifications can be compared with cinematographic movements of a camera
. But I don't know how to think of shear or squeeze mappings in terms of a camera. However, consider what a camera would do to a tiled floor. Shear? Squeeze: the camera looks out onto the horizon?
** Reflection: a camera in a mirror, a frame within a frame
...
** Rotation: a camera swivels from left to right, makes
a choice, like turning one's head
** Dilation: a camera zooms for the desired composition
.
** Translation: a camera moves around
.
Pakeistos 398-400 eilutės iš
* Harmonic analysis, periodic functions, circle are rotation.
* Homotopy is translation.
į:
Reflection
Pridėtos 400-401 eilutės:
Shear
* Shear map takes parallelogram to square, preserves area
Pridėtos 403-415 eilutės:
Rotation
* Harmonic analysis, periodic functions, circle are rotation.
* Rotations are multiplicative but not additive. This brings to mind the field with one element.
Dilation
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively.
Complex number dilation (rotating).
* Homothety is related to dilation. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
* https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
* Dilation brings to mind the Cartesian product A x B. There is also the inner (direct) product A + B. How is it related to the disjoint union? And there is the tensor product which I think is like an expansion in terms of A.B and so is like multiplication.
Squeeze
* [[https://en.wikipedia.org/wiki/Squeeze_mapping | Squeeze mapping]]
Translation
* Homotopy is translation.
Ištrintos 416-417 eilutės:

* http://settheory.net/geometry#transf
Pakeistos 418-422 eilutės iš
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively. Complex number dilation (rotating). Rotations are multiplicative but not additive. This brings to mind the field with one element.
* Dilation brings to mind the Cartesian product A x B. There is also the inner (direct) product A + B. How is it related to the disjoint union? And there is the tensor product which I think is like an expansion in terms of A.B and so is like multiplication.
* In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
* [[https://en.wikipedia.org/wiki/Squeeze_mapping | Squeeze mapping]]
* Isometry
į:
Other
Pridėta 421 eilutė:
* Isometry
Pakeistos 425-432 eilutės iš
* https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
* https://en.m.wikipedia.org/wiki/Möbius_transformation combines
translation, inversion, reflection, rotation, homothety
Grothendieck's six operations:
* pushforward along a morphism and its left adjoint
* compactly supported pushforward and its right adjoint
* tensor product and its adjoint internal hom
į:
Pakeistos 429-430 eilutės iš
Tadashi Tokieda: Basic strategy of topology. When a problem has degeneracies, then deform (or perturb) to a problem without degeneracies, then deform back. We can use the same approach to show some problems are unsolvable.
į:
Pakeistos 434-436 eilutės iš
Quotient is gluing is equivalence on a boundary. Topology is the creation of a smaller space from a larger space.

Cross cap introduces contradiction, which breaks the segregation between orientations, whether inside and outside, self and world, or true and false.
į:
2017 spalio 04 d., 09:43 atliko AndriusKulikauskas -
Pridėtos 229-235 eilutės:
Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
* perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
* orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
* most important theorem: Pythagoras q=q1+q2 - q1q2
* second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3
Compare to: Beltrami-Klein model of hyperbolic geometry
Pakeistos 309-315 eilutės iš
Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity. And a different way of relating two dimensions.

Each geometry is the action of a monoid, thus a language. But that monoid may contain an inverse, which distinguishes the projective geometry from the affine geometry.

In a free monoid the theorems are equations and they are determined by what can be done with associativity. This is first order logic. A second order logic or higher order logic would be given by what can be expressed, for example, by counting various possibilities.

Consider a trigon with 3 directed sides A, B, C:
į:
Ideas
*
Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity. And a different way of relating two dimensions.
* Each geometry is the action of a monoid, thus a language. But that monoid may contain an inverse, which distinguishes the projective geometry from the affine geometry.
* In a free monoid the theorems are equations and they are determined by what can be done with associativity. This is first order logic. A second order logic or higher order logic would be given by what can be expressed, for example, by counting various possibilities.
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.

'''Four Basic Geometries'''

Consider a triangle
with 3 directed sides A, B, C:
Pakeistos 325-326 eilutės iš
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.
į:
Pakeista 337 eilutė iš:
* Affine geometry supposes the integers
į:
* Affine geometry supposes the natural numbers
Pridėtos 345-346 eilutės:
Projective geometry adds points at infinity to affine geometry. Conformal geometry or inversive geometry adds a distinguished circle. Symplectic geometry adds an area product. Moebius strip plays with the distinguished circle changing orientation if you go around.
Pakeistos 349-352 eilutės iš
Ordered geometry features the concept of intermediacy. It is a common foundation for affine, Euclidean, absolute geometry and hyperbolic geometry, but not projective geometry. Like projective geometry, it omits the notion of measurement.

[[https://en
.m.wikipedia.org/wiki/Foundations_of_geometry | Absolute geometry]], also known as neutral geometry, is based on the axioms of Euclidean geometry (including the first four of Euclid's axioms) but with the parallel postulate removed.
į:
Special geometries
* Euclidean geometry: empty space + tools: straightedge, compass, area measurer
** most important theorem: Pythagoras q=q1+q2
** (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
* Ordered
geometry features the concept of intermediacy. It is a common foundation for affine, Euclidean, absolute geometry and hyperbolic geometry, but not projective geometry. Like projective geometry, it omits the notion of measurement.
*
[[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Absolute geometry]], also known as neutral geometry, is based on the axioms of Euclidean geometry (including the first four of Euclid's axioms) but with the parallel postulate removed.
Ištrintos 356-370 eilutės:

Projective geometry adds points at infinity to affine geometry. Conformal geometry or inversive geometry adds a distinguished circle. Symplectic geometry adds an area product. Moebius strip plays with the distinguished circle changing orientation if you go around.

Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
* perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
* orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
* most important theorem: Pythagoras q=q1+q2 - q1q2
* second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3

Compare to: Beltrami-Klein model of hyperbolic geometry

Euclidean geometry: empty space + tools: straightedge, compass, area measurer
* most important theorem: Pythagoras q=q1+q2
* (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
2017 rugsėjo 24 d., 15:31 atliko AndriusKulikauskas -
Ištrinta 15 eilutė:
* Look at Wildberger's three binormal forms.
Pakeista 19 eilutė iš:
* How does the geometric product in a Clifford Algebra model angular momentum, the basis for symplectic geometry, which is otherwise typically described by the cross product?
į:
Pakeista 22 eilutė iš:
* Relate sheaves and vector bundles.
į:
Pridėtos 50-55 eilutės:
>>bgcolor=#FFFFC0<<

* Look at Wildberger's three binormal forms.

>><<
Pridėtos 158-163 eilutės:
>>bgcolor=#FFFFC0<<

* Relate sheaves and vector bundles.

>><<
Pridėtos 236-240 eilutės:
>>bgcolor=#FFFFC0<<

* How does the geometric product in a Clifford Algebra model angular momentum, the basis for symplectic geometry, which is otherwise typically described by the cross product?

>><<
2017 rugsėjo 23 d., 13:50 atliko AndriusKulikauskas -
Pridėtos 69-95 eilutės:
Affine geometry
* Allowing only positive "coefficients" is related to positive definiteness, convexity.
* Does not assume Euclid's third and fourth axioms.
* Different coordinate systems don't agree on any origin.
* Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
** In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
** Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
* In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
* https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume
* https://en.m.wikipedia.org/wiki/Motive_(algebraic_geometry) related to the connection between affine and projective space
* Tiesė perkelta į kitą tiesę išsaugoja trijų taškų paprastą santykį (ratio).

Projective geometry
* Projective geometry relates one plane (upon which the projection is made) with another plane (where the "eye" is, the zero where all the lines come from). And thus the line through the eye which is parallel to the plane needs to be added. Thus we can have homogeneous coordinates. And we have the decomposition of projective space into a sum of affine spaces of each dimension. Projective geometry is the space of one-dimensional subspaces, and they all include zero, thus they are the lines which go through zero. Or the hyperplanes which go through zero.
* Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
* In projective geometry, vectors are points and [[https://en.wikipedia.org/wiki/Bivector | bivectors]] are lines.
* Projective geometry transforms conics into conics.
* projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
* "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
* [1:2:0] is a point that is a "direction" (two directions)
* A vector subspace needs to contain zero. How is this related to projective geometry? Vector spaces: Two different coordinate systems agree on the origin 0.
* Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
* [[https://en.m.wikipedia.org/wiki/Homography Homography]] two approaches to projective geometry with fields or without
* A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
* Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
* Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
Ištrinta 128 eilutė:
Ištrintos 280-281 eilutės:
Ištrintos 327-354 eilutės:

Affine geometry
* Allowing only positive "coefficients" is related to positive definiteness, convexity.
* Does not assume Euclid's third and fourth axioms.
* Different coordinate systems don't agree on any origin.
* Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
** In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
** Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
* In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
* https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume
* https://en.m.wikipedia.org/wiki/Motive_(algebraic_geometry) related to the connection between affine and projective space
* Tiesė perkelta į kitą tiesę išsaugoja trijų taškų paprastą santykį (ratio).

Projective geometry
* Projective geometry relates one plane (upon which the projection is made) with another plane (where the "eye" is, the zero where all the lines come from). And thus the line through the eye which is parallel to the plane needs to be added. Thus we can have homogeneous coordinates. And we have the decomposition of projective space into a sum of affine spaces of each dimension. Projective geometry is the space of one-dimensional subspaces, and they all include zero, thus they are the lines which go through zero. Or the hyperplanes which go through zero.
* Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
* In projective geometry, vectors are points and [[https://en.wikipedia.org/wiki/Bivector | bivectors]] are lines.
* Projective geometry transforms conics into conics.
* projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
* "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
* [1:2:0] is a point that is a "direction" (two directions)
* A vector subspace needs to contain zero. How is this related to projective geometry? Vector spaces: Two different coordinate systems agree on the origin 0.
* Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
* [[https://en.m.wikipedia.org/wiki/Homography Homography]] two approaches to projective geometry with fields or without
* A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
* Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
* Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
2017 rugsėjo 23 d., 12:12 atliko AndriusKulikauskas -
Pakeistos 45-46 eilutės iš
į:
[+Linear Algebra+]

* Gelfand, [[https://www.amazon.com/Lectures-Linear-Algebra-Dover-Mathematics/dp/0486660826 | Lectures on Linear Algebra]]
Pakeistos 63-66 eilutės iš
į:
* [[http://www.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
* Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
Pakeistos 78-79 eilutės iš
į:
* Learn: affine complex varieties
Pakeistos 93-94 eilutės iš
* Sheaves
**
https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
į:
* [[https://en.wikipedia.org/wiki/Alexander_Grothendieck | Grothendieck]]
* Robin Hartshorne, Algebraic Geometry
Sheaves
* https:
//ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
* [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
Schemes
Pridėta 102 eilutė:
Pakeistos 110-112 eilutės iš
[+Homology+]
į:
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
* [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
* The [[https://en.wikipedia.org/wiki/Geometrization_conjecture | Geometrization conjecture]] and the eight Thurston geometries. Also, the [[https://en.wikipedia.org/wiki/Bianchi_classification | Bianchi classification]] of low dimensional Lie algebras.

[+Homology and Cohomology
+]
Pakeistos 117-118 eilutės iš
* Gelfand
į:
* [[https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology | Coherent sheaf cohomology]]
* [[https://en.wikipedia.org/wiki/Motive_(algebraic_geometry) | Motives]] and Universal cohomology. [[https://en.wikipedia.org/wiki/Weil_cohomology_theory | Weil cohomology theory]] and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
* spectrum - topology, cohomology
Pridėtos 125-128 eilutės:
[+Differential Geometry+]

* [[https://en.wikipedia.org/wiki/Vector_bundle | Vector bundle]]
Pakeistos 149-150 eilutės iš
* [[http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf | Pierre Cartier: Mad Day's Work: From Grothendieck to Connes and Kontsevich, The Evolution of Concepts of Space and Symmetry]]
į:
* [[http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf | Pierre Cartier: Mad Day's Work: From Grothendieck to Connes and Kontsevich, The Evolution of Concepts of Space and Symmetry]]
* [[https://www.amazon.com/Geometry-Revealed-Lester-J-Senechal-ebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me= | Berger: Geometry Revealed
]]
Pakeistos 153-155 eilutės iš
į:
* [[https://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/0387986502 | Robin Hartshorne Geometry: Euclid and Beyond]]
* [[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Foundations of geometry]]
Pridėtos 171-182 eilutės:
Symmetry
* [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
* [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
* [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
* [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli

Different geometries
* [[https://en.m.wikipedia.org/wiki/List_of_geometry_topics | List of geometry topics]]
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* [[https://en.m.wikipedia.org/wiki/Ordered_geometry | Ordered geometry]]
Pakeistos 194-196 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Conformal_geometric_algebra | Conformal geometric algebra]] includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
* [[https://en.wikipedia.org/wiki/Versor | Versor]] and sandwiching.
Pakeistos 230-270 eilutės iš
* [[https://www.amazon.com/Geometry-Revealed-Lester-J-Senechal-ebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me= | Berger: Geometry Revealed]]

* Geometry - Others
** [[http://www.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
** Robin Hartshorne Geometry: Euclid and Beyond

* Symmetry
** [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
** [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
** [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
** [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
* [[Algebraic Geometry]]
** Learn: affine complex varieties
** Robin Hartshorne, Algebraic Geometry
** [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
** [[https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology | Coherent sheaf cohomology]]

* Algebraic Topology
** [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
** [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
* [[https://en.wikipedia.org/wiki/Motive_(algebraic_geometry) | Motives]] and Universal cohomology. [[https://en.wikipedia.org/wiki/Weil_cohomology_theory | Weil cohomology theory]] and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
* [[https://en.wikipedia.org/wiki/Alexander_Grothendieck | Grothendieck]]
** Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
* [[https://en.m.wikipedia.org/wiki/List_of_geometry_topics | List of geometry topics]]
* [[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Foundations of geometry]]
* [[https://en.wikipedia.org/wiki/Conformal_geometric_algebra | Conformal geometric algebra]] includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* [[https://en.m.wikipedia.org/wiki/Ordered_geometry | Ordered geometry]]
* [[https://en.wikipedia.org/wiki/Vector_bundle | Vector bundle]]
* The [[https://en.wikipedia.org/wiki/Geometrization_conjecture | Geometrization conjecture]] and the eight Thurston geometries. Also, the [[https://en.wikipedia.org/wiki/Bianchi_classification | Bianchi classification]] of low dimensional Lie algebras.
* [[https://en.wikipedia.org/wiki/Versor | Versor]] and sandwiching.
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
* spectrum - topology, cohomology

'''Geometry'''
į:
[+Defining Geometry+]
2017 rugsėjo 23 d., 11:51 atliko AndriusKulikauskas -
Pakeista 7 eilutė iš:
* To see if there are four geometries: affine, projective, conformal and symplectic.
į:
* To distinguish four geometries: affine, projective, conformal and symplectic.
Pakeistos 89-91 eilutės iš
[+Other Geometry+]
į:
* [[http://blogs.ams.org/mathgradblog/2017/07/16/idea-scheme/ | The Idea of a Scheme]]
* [[http://www.maths.ed.ac.uk/~aar/papers/eisenbudharris.pdf | The Geometry of Schemes]], Isenbott and Harris, nicely illustrated concrete examples

[+Algebraic Topology
+]
Pakeistos 96-105 eilutės iš
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]

Noncommutative geometry
į:
Pakeistos 98-123 eilutės iš
į:
* Allen Hatcher, Algebraic Topology - free on his website

[+Homology+]

* Weibel, Homological Algebra
* Gelfand

[+Geometry and Logic+]

* Sheaves in Geometry and Logic, Medak and Macleigh

[+Other Geometry+]

Videos
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]

Noncommutative geometry

Books
Pakeistos 131-132 eilutės iš
į:
* [[http://www.springer.com/us/book/9783642192241 | A Royal Road to Algebraic Geometry]]
Pakeistos 228-234 eilutės iš
* [[http://www.springer.com/us/book/9783642192241 | A Royal Road to Algebraic Geometry]]
* [[http://www.maths.ed.ac.uk/~aar/papers/eisenbudharris.pdf | The Geometry of Schemes]], Isenbott and Harris, nicely illustrated concrete examples
* Weibel, Homological Algebra
* Gelfand
* Allen Hatcher, Algebraic Topology - free on his website
* Sheaves in Geometry and Logic, Medak and Macleigh
* [[http://blogs.ams.org/mathgradblog/2017/07/16/idea-scheme/ | The Idea of a Scheme]]
į:
2017 rugsėjo 22 d., 12:36 atliko AndriusKulikauskas -
Pridėtos 63-64 eilutės:

Attach:AlgebrajineGeometrija.png
2017 rugsėjo 22 d., 12:13 atliko AndriusKulikauskas -
Pridėtos 129-135 eilutės:
Conformal geometry
* In conformal geometry (Euclidean geometry), we have inversions. The (infinite) horizon line is a circle that we are within. Reflection takes us in and out of this circle.
* An example of conformal geometry is (universal conformal) stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere).
* Algebraic geometry presumes orthogonal basis elements, thus, perpendicularity and angles. Thus affine geometry and projective geometry should be restricted to not using algebraic geometry.
* Universal hyperbolic geometry (projective geometry with a distinguished circle) is perhaps conformal geometry. It relates two different spaces, the inside and the outside of the circle.
* [[https://www.youtube.com/watch?v=JX3VmDgiFnY | Moebius transformations revealed]].
Pridėtos 141-166 eilutės:

Symplectic geometry
* Symplectic geometry is an even dimensional geometry. It lives on even dimensional
spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional
lengths and angles that are familiar from Euclidean and Riemannian geometry. It is
naturally associated with the field of complex rather than real numbers. However, it
is not as rigid as complex geometry: one of its most intriguing aspects is its curious
mixture of rigidity (structure) and flabbiness (lack of structure). [[http://www.math.stonybrook.edu/~dusa/ewmcambrevjn23.pdf | What is Symplectic Geometry? by Dusa McDuff]]
* McDuff: First of all, what is a symplectic structure? The concept arose in the study of classical
mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a
falling apple. The trajectory of such a system is determined if one knows its position
and velocity (speed and direction of motion) at any one time. Thus for an object
of unit mass moving in a given straight line one needs two pieces of information, the
position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.
* momentum x position is angular momentum
* McDuff: This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a
single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld
quantization principle says that the area of this region is quantized, i.e. it has to be
an integral multiple of a number called Planck’s constant. Thus one can think of the
symplectic area as a measure of the entanglement of position and velocity.
* Symplectic area is orientable.
* Area (volume) is a [[https://en.wikipedia.org/wiki/Pseudoscalar | pseudoscalar]] such as the [[https://en.wikipedia.org/wiki/Triple_product#Scalar_triple_product | scalar triple product]].
* Symplectic geometry is naturally related to time because it is swept out (in one dimension) in time. And so the time (one-)dimension thereby "defines" the geometry of the area.
* Symplectic "sweep" is related to equivalence (for example, natural transformation) relevant for arguments of equality by continuity (for example, the Fundamental Theorem of Calculus, integration).
* Symplectic geometry relates a point and its line, that is, it treats the moving point as a line with an origin, and relates the relative distance between the origins and the relative momentum between the origins. Thus it is a relation between two dimensions. And the boundary of the curve can be fuzzy, as in quantum mechanics and the Heisenberg principle.
Ištrintos 335-365 eilutės:
Conformal geometry
* In conformal geometry (Euclidean geometry), we have inversions. The (infinite) horizon line is a circle that we are within. Reflection takes us in and out of this circle.
* An example of conformal geometry is (universal conformal) stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere).
* Algebraic geometry presumes orthogonal basis elements, thus, perpendicularity and angles. Thus affine geometry and projective geometry should be restricted to not using algebraic geometry.
* Universal hyperbolic geometry (projective geometry with a distinguished circle) is perhaps conformal geometry. It relates two different spaces, the inside and the outside of the circle.
* [[https://www.youtube.com/watch?v=JX3VmDgiFnY | Moebius transformations revealed]].

Symplectic geometry
* Symplectic geometry is an even dimensional geometry. It lives on even dimensional
spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional
lengths and angles that are familiar from Euclidean and Riemannian geometry. It is
naturally associated with the field of complex rather than real numbers. However, it
is not as rigid as complex geometry: one of its most intriguing aspects is its curious
mixture of rigidity (structure) and flabbiness (lack of structure). [[http://www.math.stonybrook.edu/~dusa/ewmcambrevjn23.pdf | What is Symplectic Geometry? by Dusa McDuff]]
* McDuff: First of all, what is a symplectic structure? The concept arose in the study of classical
mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a
falling apple. The trajectory of such a system is determined if one knows its position
and velocity (speed and direction of motion) at any one time. Thus for an object
of unit mass moving in a given straight line one needs two pieces of information, the
position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.
* momentum x position is angular momentum
* McDuff: This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a
single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld
quantization principle says that the area of this region is quantized, i.e. it has to be
an integral multiple of a number called Planck’s constant. Thus one can think of the
symplectic area as a measure of the entanglement of position and velocity.
* Symplectic area is orientable.
* Area (volume) is a [[https://en.wikipedia.org/wiki/Pseudoscalar | pseudoscalar]] such as the [[https://en.wikipedia.org/wiki/Triple_product#Scalar_triple_product | scalar triple product]].
* Symplectic geometry is naturally related to time because it is swept out (in one dimension) in time. And so the time (one-)dimension thereby "defines" the geometry of the area.
* Symplectic "sweep" is related to equivalence (for example, natural transformation) relevant for arguments of equality by continuity (for example, the Fundamental Theorem of Calculus, integration).
* Symplectic geometry relates a point and its line, that is, it treats the moving point as a line with an origin, and relates the relative distance between the origins and the relative momentum between the origins. Thus it is a relation between two dimensions. And the boundary of the curve can be fuzzy, as in quantum mechanics and the Heisenberg principle.
2017 rugsėjo 22 d., 12:11 atliko AndriusKulikauskas -
Pridėtos 54-61 eilutės:
Affine and Projective Geometry
* [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
* [[https://www.amazon.com/Introduction-Geometry-Wiley-Classics-Library/dp/0471504580/ref=sr_1_1?ie=UTF8&qid=1387737461&sr=8#reader_0471504580 | Introduction to Geometry]] by Coxeter.
* Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
** [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]
Pridėtos 74-80 eilutės:
'''Intuition'''

Ravi Vakil: The intuition for schemes can be built on the intuition for affine complex varieties. Allen Knutson and Terry Tao have pointed out that this involves three different simultaneous generalizations, which can be interpreted as three large themes in mathematics.
* (i) We allow nilpotents in the ring of functions, which is basically analysis (looking at near-solutions of equations instead of exact solutions).
* (ii) We glue these affine schemes together, which is what we do in differential geometry (looking at manifolds instead of coordinate patches).
* (iii) Instead of working over C (or another algebraically closed field), we work more generally over a ring that isn’t an algebraically closed field, or even a field at all, which is basically number theory (solving equations over number fields, rings of integers, etc.).
Pakeistos 85-87 eilutės iš
į:
* Sheaves
** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
Pridėtos 114-116 eilutės:
Intuition
* I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.
Pakeistos 121-126 eilutės iš
į:
* Sylvain Poirer
** [[http://settheory.net/geometry | Geometry]]
** [[http://settheory.net/geometry-axioms | Geometry axioms]]
** [[http://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
Pakeistos 132-140 eilutės iš

* Sylvain Poirer
**
[[http://settheory.net/geometry | Geometry]]
** [[http://settheory.net/geometry-axioms | Geometry axioms]]
** [[http
://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
į:
[+Symplectic Geometry+]

Books
*
[[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
* [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]

Pakeistos 141-147 eilutės iš
* Affine and Projective Geometry
** [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
* [[https://www.amazon.com/Introduction-Geometry-Wiley-Classics-Library/dp/0471504580/ref=sr_1_1?ie=UTF8&qid=1387737461&sr=8#reader_0471504580 | Introduction to Geometry]] by Coxeter.
* Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
** [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]
į:
Pakeistos 145-147 eilutės iš
* Symplectic Geometry
** [[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
** [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]
į:
Pakeistos 156-157 eilutės iš
* Sheaves
** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
į:
Pakeistos 210-211 eilutės iš
I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.
į:
Pridėtos 465-470 eilutės:

[+Geometry Intuition+]

2017 rugsėjo 22 d., 11:54 atliko AndriusKulikauskas -
Pakeistos 46-55 eilutės iš
* [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]
į:
Pridėtos 54-55 eilutės:
Books
Pakeistos 60-63 eilutės iš
į:
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]

Books
Ištrinta 68 eilutė:
Pakeistos 71-73 eilutės iš
http://mokslasplius.lt/files/GeometrineAlgebra/GA/GA.html

* Geometry
į:
[+Other Geometry+]

Videos
* [[https
://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]

Noncommutative geometry

Books

[+Relating Geometries+]

'''History of Geometry'''

Books
Pakeistos 94-97 eilutės iš
** [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
į:
'''Organizing Geometry'''

* [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
Pakeistos 101-110 eilutės iš
į:
[+Conformal Geometry+]

Books
*
[[http://mokslasplius.lt/files/GeometrineAlgebra/GA/GA.html | Geometrinė algebra]]

2017 rugsėjo 22 d., 11:45 atliko AndriusKulikauskas -
Pakeistos 43-44 eilutės iš
'''Geometry to study'''
į:
[++Geometry to study++]

* [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]

[+Plane Geometry+]

Videos
* [[https://www.youtube.com/user/UNSWelearning/playlists?view=50&shelf_id=5&sort=dd | Norman Wildberger]]
** [[https://www.youtube.com/playlist?list=PLC37ED4C488778E7E | Universal Hyperbolic Geometry]]
*** UnivHypGeom4: First steps in hyperbolic geometry: fundamental results

[+Classical Algebraic Geometry+]

Videos
* [[http://nptel.ac.in/courses/111106097/ | Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity]]

[+Modern Algebraic Geometry+]

Videos
* [[https://www.youtube.com/watch?v=93cyKWOG5Ag | Nickolas Rollick: Algebraic Geometry]]

Books
Pakeistos 142-152 eilutės iš
Geometry Videos
* [[https://www.youtube.com/playlist?list=PLC37ED4C488778E7E | Universal Hyperbolic Geometry]]
** UnivHypGeom4: First steps in hyperbolic geometry: fundamental results
* [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]
į:
2017 rugsėjo 22 d., 11:36 atliko AndriusKulikauskas -
Pridėtos 4-9 eilutės:

Overall goals:
* To understand what geometry contributes to the overall map of mathematics.
* To see if there are four geometries: affine, projective, conformal and symplectic.
* To understand the relationship between geometries and logic, the classical Lie groups/algebras, category theory, etc.
* To have a better understanding of mathematical concepts, tools, theorems and examples that would serve me in understanding all branches of mathematics.
2017 rugpjūčio 09 d., 09:38 atliko AndriusKulikauskas -
Pridėta 74 eilutė:
** Learn: affine complex varieties
2017 rugpjūčio 09 d., 09:38 atliko AndriusKulikauskas -
Pakeista 73 eilutė iš:
* Algebraic Geometry
į:
* [[Algebraic Geometry]]
2017 liepos 28 d., 20:07 atliko AndriusKulikauskas -
Pridėta 102 eilutė:
* [[http://blogs.ams.org/mathgradblog/2017/07/16/idea-scheme/ | The Idea of a Scheme]]
2017 vasario 07 d., 16:10 atliko AndriusKulikauskas -
Pridėta 43 eilutė:
* [[https://www.springer.com/la/book/9783034808972 | 5000 Years of Geometry: Mathematics in History and Culture]]] Offers in-depth insights on geometry as a chain of developments in cultural history.
2017 sausio 24 d., 00:33 atliko AndriusKulikauskas -
Pridėtos 38-39 eilutės:

http://mokslasplius.lt/files/GeometrineAlgebra/GA/GA.html
2017 sausio 22 d., 15:22 atliko AndriusKulikauskas -
Pridėtos 387-392 eilutės:

Squeeze transformacija trijuose matuose: a b c = 1. Tai simetrinė funkcija.

Transformacijos sieja nepriklausomus matus.

Affine geometry - free monoid - without negative sign (subtraction) - lattice of steps - such as Young tableaux as paths on Pascal's triangle.
2016 gruodžio 19 d., 17:23 atliko AndriusKulikauskas -
Pakeista 94 eilutė iš:
* The Geometry of Schemes, Isenbott and Harris, nicely illustrated concrete examples
į:
* [[http://www.maths.ed.ac.uk/~aar/papers/eisenbudharris.pdf | The Geometry of Schemes]], Isenbott and Harris, nicely illustrated concrete examples
2016 gruodžio 19 d., 17:20 atliko AndriusKulikauskas -
Pakeista 93 eilutė iš:
* The Royal Road to Algebraic Geometry
į:
* [[http://www.springer.com/us/book/9783642192241 | A Royal Road to Algebraic Geometry]]
2016 gruodžio 15 d., 20:12 atliko AndriusKulikauskas -
Ištrinta 98 eilutė:
* Lawryre - A Conceptual Introduction to Mathematics
2016 gruodžio 15 d., 20:10 atliko AndriusKulikauskas -
Pridėtos 93-99 eilutės:
* The Royal Road to Algebraic Geometry
* The Geometry of Schemes, Isenbott and Harris, nicely illustrated concrete examples
* Weibel, Homological Algebra
* Gelfand
* Allen Hatcher, Algebraic Topology - free on his website
* Sheaves in Geometry and Logic, Medak and Macleigh
* Lawryre - A Conceptual Introduction to Mathematics
2016 gruodžio 15 d., 16:30 atliko AndriusKulikauskas -
Pridėta 118 eilutė:
* Grothendieck categories
2016 gruodžio 13 d., 23:22 atliko AndriusKulikauskas -
Pridėtos 93-104 eilutės:

Geometry Videos
* [[https://www.youtube.com/playlist?list=PLC37ED4C488778E7E | Universal Hyperbolic Geometry]]
** UnivHypGeom4: First steps in hyperbolic geometry: fundamental results
* [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]
2016 gruodžio 13 d., 22:47 atliko AndriusKulikauskas -
Pridėtos 79-92 eilutės:
* [[https://en.wikipedia.org/wiki/Motive_(algebraic_geometry) | Motives]] and Universal cohomology. [[https://en.wikipedia.org/wiki/Weil_cohomology_theory | Weil cohomology theory]] and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
* [[https://en.wikipedia.org/wiki/Alexander_Grothendieck | Grothendieck]]
** Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
* [[https://en.m.wikipedia.org/wiki/List_of_geometry_topics | List of geometry topics]]
* [[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Foundations of geometry]]
* [[https://en.wikipedia.org/wiki/Conformal_geometric_algebra | Conformal geometric algebra]] includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* [[https://en.m.wikipedia.org/wiki/Ordered_geometry | Ordered geometry]]
* [[https://en.wikipedia.org/wiki/Vector_bundle | Vector bundle]]
* The [[https://en.wikipedia.org/wiki/Geometrization_conjecture | Geometrization conjecture]] and the eight Thurston geometries. Also, the [[https://en.wikipedia.org/wiki/Bianchi_classification | Bianchi classification]] of low dimensional Lie algebras.
* [[https://en.wikipedia.org/wiki/Versor | Versor]] and sandwiching.
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
* spectrum - topology, cohomology
2016 gruodžio 13 d., 22:39 atliko AndriusKulikauskas -
Pridėtos 49-51 eilutės:
** [[http://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
* [[https://www.amazon.com/Geometry-Revealed-Lester-J-Senechal-ebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me= | Berger: Geometry Revealed]]
2016 gruodžio 13 d., 22:38 atliko AndriusKulikauskas -
Pridėtos 67-75 eilutės:
* Algebraic Geometry
** Robin Hartshorne, Algebraic Geometry
** [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
** [[https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology | Coherent sheaf cohomology]]
* Sheaves
** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
* Algebraic Topology
** [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
** [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
2016 gruodžio 13 d., 22:34 atliko AndriusKulikauskas -
Pakeistos 37-66 eilutės iš
'''Works to study'''
į:
'''Geometry to study'''

* Geometry
* [[http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf | Pierre Cartier: Mad Day's Work: From Grothendieck to Connes and Kontsevich, The Evolution of Concepts of Space and Symmetry]]
** [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
* [[http://www-history.mcs.st-and.ac.uk/~john/ | John O'Connor]]
** [[http://www-history.mcs.st-and.ac.uk/~john/geometry/index.html | Topics in Geometry]]
** [[http://www-history.mcs.st-and.ac.uk/~john/MT4521/index.html | Geometry and Topology]]
* Sylvain Poirer
** [[http://settheory.net/geometry | Geometry]]
** [[http://settheory.net/geometry-axioms | Geometry axioms]]
* Affine and Projective Geometry
** [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
* [[https://www.amazon.com/Introduction-Geometry-Wiley-Classics-Library/dp/0471504580/ref=sr_1_1?ie=UTF8&qid=1387737461&sr=8#reader_0471504580 | Introduction to Geometry]] by Coxeter.
* Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
** [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]
* Geometry - Others
** [[http://www.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
** Robin Hartshorne Geometry: Euclid and Beyond
* Symplectic Geometry
** [[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
** [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]
* Symmetry
** [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
** [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
** [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
** [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
2016 gruodžio 13 d., 22:31 atliko AndriusKulikauskas -
Ištrintos 36-37 eilutės:
shear map takes parallelogram to square, preserves area
Pakeistos 39-168 eilutės iš
Overviews and History
* [[https://basepub.dauphine.fr/bitstream/handle/123456789/6842/polymathematics.PDF | Polymathematics: is mathematics a single science or a set of arts?]], V.I.Arnold
* [[http://matematicas.unex.es/~navarro/res/lisker1.pdf | Récoltes et Semailles, Part 1]], Alexander Grothendieck. Also, [[http://matematicas.unex.es/~navarro/res/ | translation into Spanish and other works]].
** [[http://www.landsburg.com/grothendieck/pragasz.pdf | Notes on the Life and Work of Alexander Grothendieck]] by Piotr Pragacz
* [[http://www.alainconnes.org/docs/maths.pdf | A View of Mathematics]], Alain Connes
* Geometry
* http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf
** [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
* [[http://www-history.mcs.st-and.ac.uk/~john/ | John O'Connor]]
** [[http://www-history.mcs.st-and.ac.uk/~john/geometry/index.html | Topics in Geometry]]
** [[http://www-history.mcs.st-and.ac.uk/~john/MT4521/index.html | Geometry and Topology]]
* Sylvain Poirer
** [[http://settheory.net/geometry | Geometry]]
** [[http://settheory.net/geometry-axioms | Geometry axioms]]
* Affine and Projective Geometry
** [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
* [[https://www.amazon.com/Introduction-Geometry-Wiley-Classics-Library/dp/0471504580/ref=sr_1_1?ie=UTF8&qid=1387737461&sr=8#reader_0471504580 | Introduction to Geometry]] by Coxeter.
* Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
** [[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
** [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]
* Geometry - Others
** [[http://www.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
** Robin Hartshorne Geometry: Euclid and Beyond
* Symplectic Geometry
** [[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
** [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]
* Symmetry
** [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
** [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
** [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
** [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
* Symmetric functions
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
* Information Geometry
** [[http://math.ucr.edu/home/baez/information/index.html | Information Geometry]] by John Baez
* Algebraic Geometry
** Robin Hartshorne, Algebraic Geometry
** [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
** [[https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology | Coherent sheaf cohomology]]
* Six operations
** [[https://homotopical.files.wordpress.com/2014/06/ctsaghandout.pdf | Cohomology theories in motivic stable homotopy theory]]
** [[http://mathoverflow.net/questions/170319/what-if-anything-unifies-stable-homotopy-theory-and-grothendiecks-six-functor | What unifies stable homotopy theory and six functors]]
** [[http://www.math.univ-toulouse.fr/~dcisinsk/DM.pdf | Triangulated Categories of Mixed Motives]]
** [[http://arxiv.org/abs/1509.02145 | The six operations in equivariant motivic homotopy theory]] Marc Hoyois
** [[https://arxiv.org/abs/1402.7041 | Quantization via Linear homotopy types]], Urs Schreiber
** [[http://www.math.uchicago.edu/~may/PAPERS/FormalFinalMarch.pdf | Isomorphisms between left and right adjoints]]
* Sheaves
** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
* Algebraic Topology
** [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
** [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
* Linear algebra
** [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/ | Linear Algebra Notes by Terrence Tao]]
** [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/week10.pdf | Week 10: Linear Functionals, Adjoints]] by Terrence Tao
** [[http://www.cambridge.org/gb/academic/subjects/physics/mathematical-methods/students-guide-vectors-and-tensors?format=PB&isbn=9780521171908 | A Student's Guide to Vectors and Tensors]]
* Network theory
** [[http://www.azimuthproject.org/azimuth/show/Network+theory | Network theory (wiki)]] and [[http://math.ucr.edu/home/baez/networks/ | Network theory (blog)]] by John Baez
* Langlands program
** [[http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/S0273-0979-1984-15237-6.pdf | An Elementary Introduction to the Langlands Program]] by Stephen Gelbart
** [[https://arxiv.org/pdf/hep-th/0512172v1 | Langland Frenkel]]
** [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory | 6D (2,0) superconformal field theory]] ?
* Sylvain Poirer
** [[http://settheory.net/foundations/variables-sets | Variables - sets]]
** [[http://settheory.net/foundations/theories | Theories]]
** [[http://settheory.net/foundations/classes2 | Classes]]
** [[http://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
* [[https://www.amazon.com/Geometry-Revealed-Lester-J-Senechal-ebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me= | Berger: Geometry Revealed]]
* Bott http://mathoverflow.net/questions/8800/proofs-of-bott-periodicity

Videos
* [[https://www.youtube.com/playlist?list=PLC37ED4C488778E7E | Universal Hyperbolic Geometry]]
** UnivHypGeom4: First steps in hyperbolic geometry: fundamental results
* [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[https://video.ias.edu/univalent/voevodsky | Univalent Foundations of Mathematics]]
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
* [[http://www.simonwillerton.staff.shef.ac.uk/TheCatsters/ | Catster videos]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
* [[https://www.youtube.com/watch?v=A8fsU97g3tg | ML Baker on Elliptic curves and modular forms]]
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]
* symplectic geometry videos
* [[http://www.claymath.org/library/video-catalogue | Clay Mathematics Videos]]

Concepts
* [[https://en.wikipedia.org/wiki/Six_operations | Six operations]]
** [[https://ncatlab.org/nlab/show/six+operations | Six operations at nLab]]
** [[http://math.stackexchange.com/questions/1351735/grothendiecks-yoga-of-six-operations-in-relatively-basic-terms | Six operations at Math Stack Exchange]]
* [[https://en.wikipedia.org/wiki/Motive_(algebraic_geometry) | Motives]] and Universal cohomology. [[https://en.wikipedia.org/wiki/Weil_cohomology_theory | Weil cohomology theory]] and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
* [[https://en.wikipedia.org/wiki/Alexander_Grothendieck | Grothendieck]]
** "Continuous" and "discrete" duality (derived categories and "six operations")
** Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
* [[https://en.m.wikipedia.org/wiki/List_of_geometry_topics | List of geometry topics]]
* [[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Foundations of geometry]]
* [[https://en.wikipedia.org/wiki/Conformal_geometric_algebra | Conformal geometric algebra]] includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* [[https://en.m.wikipedia.org/wiki/Ordered_geometry | Ordered geometry]]
* [[https://en.wikipedia.org/wiki/Vector_bundle | Vector bundle]]
* The [[https://en.wikipedia.org/wiki/Geometrization_conjecture | Geometrization conjecture]] and the eight Thurston geometries. Also, the [[https://en.wikipedia.org/wiki/Bianchi_classification | Bianchi classification]] of low dimensional Lie algebras.
* [[https://en.wikipedia.org/wiki/Versor | Versor]] and sandwiching.
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
* spectrum - topology, cohomology
* Constructiveness - closed sets any intersections and finite unions are open sets constructive
* Derived functors manifest the threesome, ever perfecting one's position, increasing the kernel, the zero. {$\displaystyle 0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to \cdots$}
* threesome Jacobi identity
* The kernel is the zero.
* Bott periodicity Divisions of everything are perhaps chopping up a sphere where the sphere is everything also circle folding
* natural contradiction inherent in the approach of defining categories by starting with a category of categories and then taking a category to be its object
* is there a category of universal properties?
* http://math.ucr.edu/home/baez/octonions/octonions.html
* Exact sequences http://math.stackexchange.com/questions/419329/intuitive-meaning-of-exact-sequence
* Length six https://arxiv.org/pdf/0906.1286v2
* http://cheng.staff.shef.ac.uk/morality/morality.pdf cheng math, morally
* http://cheng.staff.shef.ac.uk/misc/4000.pdf cheng architecture of math

Challenges
* a unifying perspective on cohomology
* higher order homotopy groups for sphere
* field with one element
* explanation of four classical Lie groups
į:
Pridėtos 207-208 eilutės:

shear map takes parallelogram to square, preserves area
2016 gruodžio 12 d., 16:03 atliko AndriusKulikauskas -
Pridėta 48 eilutė:
* http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf
2016 gruodžio 08 d., 22:22 atliko AndriusKulikauskas -
Pridėtos 437-440 eilutės:

Apibendrinimas yra "objekto" kūrimas.

Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia.
2016 gruodžio 08 d., 22:19 atliko AndriusKulikauskas -
Pridėtos 435-436 eilutės:

Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
2016 gruodžio 07 d., 00:16 atliko AndriusKulikauskas -
Pridėta 162 eilutė:
* http://cheng.staff.shef.ac.uk/misc/4000.pdf cheng architecture of math
2016 gruodžio 07 d., 00:08 atliko AndriusKulikauskas -
Pridėta 161 eilutė:
* http://cheng.staff.shef.ac.uk/morality/morality.pdf cheng math, morally
2016 gruodžio 06 d., 11:52 atliko AndriusKulikauskas -
Pridėta 160 eilutė:
* Length six https://arxiv.org/pdf/0906.1286v2
2016 gruodžio 06 d., 11:38 atliko AndriusKulikauskas -
Pridėta 159 eilutė:
* Exact sequences http://math.stackexchange.com/questions/419329/intuitive-meaning-of-exact-sequence
2016 gruodžio 05 d., 06:36 atliko AndriusKulikauskas -
Pakeista 430 eilutė iš:
į:
Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) turi tam tikras bendras savybes, tada tas apibendrintas savybes gali naujai priskirti naujoms jų apibudintoms sąvokoms.
2016 gruodžio 05 d., 06:33 atliko AndriusKulikauskas -
Pridėtos 427-430 eilutės:

Fiber is a Zero.
2016 gruodžio 04 d., 22:56 atliko AndriusKulikauskas -
Pridėta 130 eilutė:
2016 lapkričio 25 d., 18:57 atliko AndriusKulikauskas -
Pridėta 157 eilutė:
* http://math.ucr.edu/home/baez/octonions/octonions.html
2016 lapkričio 23 d., 14:23 atliko AndriusKulikauskas -
Pakeistos 418-425 eilutės iš
Our Father relates a left exact sequence and a right exact sequence.
į:
Exact sequence
* Our Father relates a left exact sequence and a right exact
sequence.
* Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.

Dievas žmogui yra skylė gyvenime, prasmė - neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje - homologijoje.

Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, tai sutapatinama, kaip kad "cross-cap".
2016 lapkričio 22 d., 21:57 atliko AndriusKulikauskas -
Pridėta 13 eilutė:
* In category theory, where do symmetric functions come up? What are eigenvalues understood as? What would be symmetric functions of eigenvalues?
2016 lapkričio 22 d., 20:09 atliko AndriusKulikauskas -
Pakeistos 154-155 eilutės iš
* nagural contradiction inherent in the approach of defining categories by starting with a category of categories and then taking a category to be its object
į:
* natural contradiction inherent in the approach of defining categories by starting with a category of categories and then taking a category to be its object
* is there a category of universal properties?
2016 lapkričio 22 d., 19:46 atliko AndriusKulikauskas -
Pridėta 154 eilutė:
* nagural contradiction inherent in the approach of defining categories by starting with a category of categories and then taking a category to be its object
2016 lapkričio 22 d., 12:29 atliko AndriusKulikauskas -
Pridėta 159 eilutė:
* explanation of four classical Lie groups
2016 lapkričio 21 d., 23:50 atliko AndriusKulikauskas -
Pridėta 151 eilutė:
* threesome Jacobi identity
2016 lapkričio 21 d., 23:18 atliko AndriusKulikauskas -
Pridėta 110 eilutė:
2016 lapkričio 21 d., 23:06 atliko AndriusKulikauskas -
Pridėtos 152-156 eilutės:

Challenges
* a unifying perspective on cohomology
* higher order homotopy groups for sphere
* field with one element
2016 lapkričio 21 d., 23:04 atliko AndriusKulikauskas -
Pridėta 151 eilutė:
* Bott periodicity Divisions of everything are perhaps chopping up a sphere where the sphere is everything also circle folding
2016 lapkričio 21 d., 11:30 atliko AndriusKulikauskas -
Pridėtos 81-83 eilutės:
** [[https://homotopical.files.wordpress.com/2014/06/ctsaghandout.pdf | Cohomology theories in motivic stable homotopy theory]]
** [[http://mathoverflow.net/questions/170319/what-if-anything-unifies-stable-homotopy-theory-and-grothendiecks-six-functor | What unifies stable homotopy theory and six functors]]
** [[http://www.math.univ-toulouse.fr/~dcisinsk/DM.pdf | Triangulated Categories of Mixed Motives]]
Pridėtos 85-86 eilutės:
** [[https://arxiv.org/abs/1402.7041 | Quantization via Linear homotopy types]], Urs Schreiber
** [[http://www.math.uchicago.edu/~may/PAPERS/FormalFinalMarch.pdf | Isomorphisms between left and right adjoints]]
2016 lapkričio 21 d., 11:23 atliko AndriusKulikauskas -
Pridėtos 400-401 eilutės:

Our Father relates a left exact sequence and a right exact sequence.
2016 lapkričio 19 d., 22:18 atliko AndriusKulikauskas -
Pridėtos 144-145 eilutės:
* Derived functors manifest the threesome, ever perfecting one's position, increasing the kernel, the zero. {$\displaystyle 0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to \cdots$}
* The kernel is the zero.
2016 lapkričio 18 d., 13:43 atliko AndriusKulikauskas -
Pridėtos 31-32 eilutės:

Relate Cayley's theorem to the field with one element
2016 lapkričio 18 d., 06:49 atliko AndriusKulikauskas -
Pridėta 102 eilutė:
* Bott http://mathoverflow.net/questions/8800/proofs-of-bott-periodicity
2016 lapkričio 17 d., 23:05 atliko AndriusKulikauskas -
Pridėtos 360-361 eilutės:

Cross cap introduces contradiction, which breaks the segregation between orientations, whether inside and outside, self and world, or true and false.
2016 lapkričio 17 d., 20:47 atliko AndriusKulikauskas -
Pridėtos 80-81 eilutės:
* Sheaves
** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
2016 lapkričio 12 d., 15:55 atliko AndriusKulikauskas -
Pridėta 138 eilutė:
* Constructiveness - closed sets any intersections and finite unions are open sets constructive
2016 lapkričio 09 d., 17:02 atliko AndriusKulikauskas -
2016 lapkričio 05 d., 16:35 atliko AndriusKulikauskas -
Pakeistos 177-180 eilutės iš
* Path geometry is given by A + B + C = 0 gets you back where you started from.
* Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc. Barycentric coordinates for vectors v1, v2, v3 (with scalars lambda l1, l2, l3) where the scalars are between 0 and 1 and the sum l1v1 + l2v2 + l3v3 = 1 on the triangle and <1 within it and all are 1/3 to get the center, the average.
* Angle geometry gives this a total value of
1, the total angle. And so we can accord to A, B, C a ratio that measures the opposite angle.
* Area geometry assigns an oriented area AREA to the total value
.
į:
* Path geometry is given by A + B + C = [0] gets you back where you started from. It is geometry without space, as when God thinks why, so that everything is connected by relationships, and God of himself only thinks forwards, unfolding.
* Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc. Barycentric coordinates for vectors v1, v2, v3 (with scalars lambda l1, l2, l3) where the scalars are between 0 and 1 and the sum l1v1 + l2v2 + l3v3 =
1 on the triangle and <1 within it and all are 1/3 to get the center, the average. For example, a line in a plane splits that plane into two sides, just as a plane splits a three-dimensional space. Thus this is where "holes" come from, disconnections, emptiness, homology.
* Angle geometry gives this a total value of 1, the total angle. And so we can accord to A, B, C a ratio that measures the opposite angle. This creates the inside and the outside of the triangle. Indeed, the three lines carves the plane into spaces. It's not clear how they meet at infinity.
* Area geometry assigns an oriented area AREA to the total value. Time arises as we have one side and the other swept by it
.
2016 lapkričio 03 d., 01:14 atliko AndriusKulikauskas -
Pakeista 178 eilutė iš:
* Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc.
į:
* Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc. Barycentric coordinates for vectors v1, v2, v3 (with scalars lambda l1, l2, l3) where the scalars are between 0 and 1 and the sum l1v1 + l2v2 + l3v3 = 1 on the triangle and <1 within it and all are 1/3 to get the center, the average.
2016 lapkričio 02 d., 19:43 atliko AndriusKulikauskas -
Pridėtos 175-180 eilutės:

Consider a trigon with 3 directed sides A, B, C:
* Path geometry is given by A + B + C = 0 gets you back where you started from.
* Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc.
* Angle geometry gives this a total value of 1, the total angle. And so we can accord to A, B, C a ratio that measures the opposite angle.
* Area geometry assigns an oriented area AREA to the total value.
2016 lapkričio 01 d., 20:04 atliko AndriusKulikauskas -
Pakeista 117 eilutė iš:
* http://www.claymath.org/library/video-catalogue
į:
* [[http://www.claymath.org/library/video-catalogue | Clay Mathematics Videos]]
2016 spalio 30 d., 22:58 atliko AndriusKulikauskas -
Pridėta 117 eilutė:
* http://www.claymath.org/library/video-catalogue
2016 spalio 28 d., 21:11 atliko AndriusKulikauskas -
Pridėtos 297-298 eilutės:

Relate to the six transformations in the anharmonic group of the [[https://en.wikipedia.org/wiki/Cross-ratio | cross-ratio]]. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?
2016 spalio 26 d., 16:05 atliko AndriusKulikauskas -
Pridėtos 384-393 eilutės:

>>bgcolor=#FFECC0<<

1999. I asked God which questions I should think over so as to understand why good will makes way for good heart. He responded:
* What captures attention and guides it? mažėjantis laisvumas
* What drops down upon reality and bounces away in random paths? didėjantis laisvumas
* What is wound in one direction, and lives through spinning in the opposite direction? prasmingas - kodėl
* What falls as rain day and night until there sprout and grow plants that will bear fruit? pastovus - kaip
* What like a ray reflects off of society and does not return? betarpiškas - koks
* What by its turning (in the direction of winding) commands our attention and then slips away to the side? tiesus - ar
2016 spalio 26 d., 11:46 atliko AndriusKulikauskas -
Pridėta 279 eilutė:
* Symplectic geometry relates a point and its line, that is, it treats the moving point as a line with an origin, and relates the relative distance between the origins and the relative momentum between the origins. Thus it is a relation between two dimensions. And the boundary of the curve can be fuzzy, as in quantum mechanics and the Heisenberg principle.
2016 spalio 25 d., 23:34 atliko AndriusKulikauskas -
Pakeistos 116-117 eilutės iš
į:
* symplectic geometry videos
Pridėta 136 eilutė:
* spectrum - topology, cohomology
2016 spalio 25 d., 22:36 atliko AndriusKulikauskas -
Pridėtos 305-306 eilutės:

Reflection introduces the action of Z2. It is the reflection across the boundary of self and world. (We can later also think of reflection across the horizon around us, as inversion.) This is the parity of multisets (element or not an element). And that circle S02 is then referenced by rotations and shear mapping and all work with angles. And then the relationship between two dimensions is given perhaps by Z2 x S02, the relationship between two axes: x vs. x (dilation), x vs. 1/x (squeeze) and x vs. y (translation).
2016 spalio 25 d., 22:32 atliko AndriusKulikauskas -
Pridėtos 305-306 eilutės:

Squeeze specification draws a hyperbola (x vs. 1/x). Dilation draws a line (x vs. x). Are there specifications that draw circles (rotation?), ellipses? parabolas?
2016 spalio 25 d., 22:28 atliko AndriusKulikauskas -
Pakeistos 214-215 eilutės iš
į:
* Tiesė perkelta į kitą tiesę išsaugoja trijų taškų paprastą santykį (ratio).
Pridėta 230 eilutė:
* Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
2016 spalio 25 d., 22:16 atliko AndriusKulikauskas -
Pakeista 134 eilutė iš:
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective.
į:
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
2016 spalio 25 d., 21:49 atliko AndriusKulikauskas -
Pakeista 134 eilutė iš:
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point).
į:
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective.
2016 spalio 25 d., 21:48 atliko AndriusKulikauskas -
Pakeistos 134-135 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Eccentricity_(mathematics) | Eccentricity]] defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point).
Pridėtos 301-302 eilutės:

Note that the [[https://en.wikipedia.org/wiki/SL2(R)#Classification_of_elements | classification of elements of SL2(R)]] includes elliptic (conjugate to a rotation), parabolic (shear) and hyperbolic (squeeze). Similarly, see the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Classification | classification of Moebius transformations]].
2016 spalio 25 d., 21:30 atliko AndriusKulikauskas -
Pridėta 99 eilutė:
* [[https://www.amazon.com/Geometry-Revealed-Lester-J-Senechal-ebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me= | Berger: Geometry Revealed]]
2016 spalio 25 d., 21:13 atliko AndriusKulikauskas -
Pakeista 245 eilutė iš:
* An example of conformal geometry is stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere).
į:
* An example of conformal geometry is (universal conformal) stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere).
Pridėta 248 eilutė:
* [[https://www.youtube.com/watch?v=JX3VmDgiFnY | Moebius transformations revealed]].
2016 spalio 25 d., 21:03 atliko AndriusKulikauskas -
Pridėta 269 eilutė:
* Area (volume) is a [[https://en.wikipedia.org/wiki/Pseudoscalar | pseudoscalar]] such as the [[https://en.wikipedia.org/wiki/Triple_product#Scalar_triple_product | scalar triple product]].
2016 spalio 25 d., 20:59 atliko AndriusKulikauskas -
Pridėta 244 eilutė:
* In conformal geometry (Euclidean geometry), we have inversions. The (infinite) horizon line is a circle that we are within. Reflection takes us in and out of this circle.
2016 spalio 25 d., 20:55 atliko AndriusKulikauskas -
Pakeistos 132-133 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Versor | Versor]] and sandwiching.
Pridėta 217 eilutė:
* In projective geometry, vectors are points and [[https://en.wikipedia.org/wiki/Bivector | bivectors]] are lines.
2016 spalio 25 d., 20:42 atliko AndriusKulikauskas -
Pridėta 242 eilutė:
* An example of conformal geometry is stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere).
2016 spalio 21 d., 19:30 atliko AndriusKulikauskas -
Pridėta 77 eilutė:
** [[https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology | Coherent sheaf cohomology]]
2016 spalio 21 d., 05:00 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš
[[Geometry illustrations]], [[Universal hyperbolic geometry]]
į:
[[Geometry theorems]], [[Geometries]], [[Geometry illustrations]], [[Universal hyperbolic geometry]]
Pakeista 7 eilutė iš:
* Make a list of geometries and show how they are related.
į:
* Make a list of [[geometries]] and show how they are related.
2016 spalio 18 d., 19:33 atliko AndriusKulikauskas -
Pridėta 213 eilutė:
* Projective geometry relates one plane (upon which the projection is made) with another plane (where the "eye" is, the zero where all the lines come from). And thus the line through the eye which is parallel to the plane needs to be added. Thus we can have homogeneous coordinates. And we have the decomposition of projective space into a sum of affine spaces of each dimension. Projective geometry is the space of one-dimensional subspaces, and they all include zero, thus they are the lines which go through zero. Or the hyperplanes which go through zero.
Pridėta 215 eilutė:
* Projective geometry transforms conics into conics.
2016 spalio 18 d., 17:34 atliko AndriusKulikauskas -
Pakeistos 287-289 eilutės iš
But I don't know how to think of shear or squeeze mappings in terms of a camera.
į:
But I don't know how to think of shear or squeeze mappings in terms of a camera. However, consider what a camera would do to a tiled floor.
* Shear:
* Squeeze: the camera looks out onto the horizon
.
2016 spalio 16 d., 22:32 atliko AndriusKulikauskas -
Pridėta 299 eilutė:
* Dilation brings to mind the Cartesian product A x B. There is also the inner (direct) product A + B. How is it related to the disjoint union? And there is the tensor product which I think is like an expansion in terms of A.B and so is like multiplication.
2016 spalio 15 d., 16:44 atliko AndriusKulikauskas -
Pakeistos 163-167 eilutės iš
Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity.
į:
Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity. And a different way of relating two dimensions.

Each geometry is the action of a monoid, thus a language. But that monoid may contain an inverse, which distinguishes the projective geometry from the affine geometry.

In a free monoid the theorems are equations and they are determined by what can be done with associativity. This is first order logic. A second order logic or higher order logic would be given by what can be expressed, for example, by counting various possibilities
.
2016 spalio 15 d., 16:32 atliko AndriusKulikauskas -
Pakeistos 278-283 eilutės iš
The 6 specifications
į:
The 6 specifications can be compared with cinematographic movements of a camera.
* Reflection: a camera in a mirror, a frame within a frame...
* Rotation: a camera swivels from left to right, makes a choice, like turning one's head
* Dilation: a camera zooms for the desired composition.
* Translation: a camera moves around.
But I don't know how to think of shear or squeeze mappings in terms of a camera.
2016 spalio 15 d., 16:23 atliko AndriusKulikauskas -
Pakeistos 272-273 eilutės iš
'''Transformations'''
į:
'''6 Specifications'''

The 6 specifications between 4 geometries are transformations which make one geometry more specific than another geometry by introducing orientation, angles and areas. This also makes distance more sophisticated, allowing for negative (oriented) numbers, rational (angular) numbers, and real (continuous) numbers.
Pridėtos 277-278 eilutės:

The 6 specifications
2016 spalio 15 d., 14:33 atliko AndriusKulikauskas -
Pridėta 113 eilutė:
* [[https://www.youtube.com/watch?v=zrSiyDfQhxk&list=PLS3WLIIQXmMLvfY5NOfWRs55x_H4lRTpj&index=1 | Video lectures on homological algebra]]
2016 spalio 15 d., 14:29 atliko AndriusKulikauskas -
Pridėta 112 eilutė:
* [[https://www.youtube.com/watch?v=A8fsU97g3tg | ML Baker on Elliptic curves and modular forms]]
2016 spalio 15 d., 01:07 atliko AndriusKulikauskas -
Pridėtos 203-204 eilutės:
* https://en.m.wikipedia.org/wiki/Motive_(algebraic_geometry) related to the connection between affine and projective space
2016 spalio 14 d., 09:42 atliko AndriusKulikauskas -
Pakeistos 272-273 eilutės iš
Homotopy is translation.
į:
* Harmonic analysis, periodic functions, circle are rotation.
*
Homotopy is translation.
2016 spalio 14 d., 08:02 atliko AndriusKulikauskas -
Pridėtos 271-272 eilutės:

Homotopy is translation.
2016 spalio 12 d., 23:53 atliko AndriusKulikauskas -
Pridėtos 29-30 eilutės:

Generalize this result to n-dimensions (starting with 4-dimensions): [[http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L12.html | Full finite symmetry groups in 3 dimensions]]
2016 spalio 12 d., 22:59 atliko AndriusKulikauskas -
Pridėtos 68-69 eilutės:
* Symmetric functions
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
Ištrintos 85-86 eilutės:
* Symmetric functions
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
2016 spalio 12 d., 22:59 atliko AndriusKulikauskas -
Pakeistos 42-60 eilutės iš
* [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
Geometry
* [[http://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
*
[[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
* [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
* Robin Hartshorne Geometry: Euclid and Beyond
* Robin Hartshorne, Algebraic Geometry
* [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
* [[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
* [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]
* [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
* [[http://arxiv.org/abs/1509.02145 | The six operations in equivariant motivic homotopy theory]] Marc Hoyois
* [[http://math.ucr.edu/home/baez/information/index.html | Information Geometry]] by John Baez
* [[http://www.azimuthproject.org/azimuth/show/Network+theory | Network theory (wiki)]] and [[http://math.ucr.edu/home/baez/networks/ | Network theory (blog)]] by John Baez
* [[http://www.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
* [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
* [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
* [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
į:
* Geometry
**
[[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
* [[http://www-history.mcs.st-and.ac.uk/~john/ | John O'Connor]]
** [[http://www-history.mcs.st-and.ac.uk/~john/geometry/index.html | Topics in Geometry]]
** [[http://www-history.mcs.st-and.ac.uk/~john/MT4521/index.html | Geometry and Topology]]
Pakeista 48 eilutė iš:
** [[http://settheory.net/geometry | Sylvain Poirer]]
į:
** [[http://settheory.net/geometry | Geometry]]
Pakeistos 50-61 eilutės iš
** [[http://settheory.net/foundations/variables-sets | Variables - sets]]
** [[http://settheory.net/foundations/theories | Theories]]
** [[http://settheory.net/foundations/classes2 | Classes]]
** [[http://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
* Symmetric functions
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
* [[http://www.cambridge.org/gb/academic/subjects/physics/mathematical-methods/students-guide-vectors-and-tensors?format=PB&isbn=9780521171908 | A Student's Guide to Vectors and Tensors]]
* [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/ | Linear Algebra Notes by Terrence Tao]]
** [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/week10.pdf | Week 10: Linear Functionals, Adjoints]] by Terrence Tao
* [[http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/S0273-0979-1984-15237-6.pdf | An Elementary Introduction to the Langlands Program]] by Stephen Gelbart

* [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
į:
* Affine and Projective Geometry
** [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
Pakeistos 53-58 eilutės iš
* [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
*
[[https://arxiv.org/pdf/hep-th/0512172v1 | Langland Frenkel]]
* [[http://www-history.mcs.st-and.ac.uk/~john/ | John O'Connor]]
** [[http://www-history.mcs.st-and.ac.uk/~john/geometry/index.html | Topics in Geometry]]
** [[http://www-history.mcs.st-and.ac.uk/~john/MT4521/index.html | Geometry and Topology]]
*
[[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory | 6D (2,0) superconformal field theory]] ?
į:
* Norman Wildberger
**
[[http://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
**
[[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
** [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]
* Geometry - Others
** [[http://www
.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
** Robin Hartshorne Geometry: Euclid and Beyond
* Symplectic
Geometry
** [[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
** [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]
* Symmetry
** [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
** [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
** [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
** [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
* Information Geometry
** [[http://math.ucr.edu/home/baez/information/index.html | Information Geometry]] by John Baez
* Algebraic Geometry
** Robin Hartshorne, Algebraic Geometry
** [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
* Six operations
** [[http://arxiv.org/abs/1509.02145 | The six operations in equivariant motivic homotopy theory]] Marc Hoyois
* Algebraic Topology
** [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
** [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
* Linear algebra
** [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/ | Linear Algebra Notes by Terrence Tao]]
** [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/week10.pdf | Week 10: Linear Functionals, Adjoints]] by Terrence Tao
?format=PB&isbn=9780521171908 | A Student's Guide to Vectors and Tensors]]
* Network theory
** [[http://www.azimuthproject.org/azimuth/show/Network+theory | Network theory (wiki)]] and [[http://math.ucr.edu/home/baez/networks/ | Network theory (blog)]] by John Baez
* Symmetric functions
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
* Langlands program
** [[http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/S0273-0979-1984-15237-6.pdf | An Elementary Introduction to the Langlands Program]] by Stephen Gelbart
** [[https://arxiv.org/pdf/hep-th/0512172v1 | Langland Frenkel]]
** [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory | 6D (2,0) superconformal field theory]] ?
* Sylvain Poirer
** [[http://settheory.net/foundations/variables-sets | Variables - sets]]
** [[http://settheory.net/foundations/theories | Theories]]
** [[http://settheory.net/foundations/classes2 | Classes]]
** [[http://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
2016 spalio 12 d., 22:49 atliko AndriusKulikauskas -
Pakeistos 79-81 eilutės iš
* http://www-history.mcs.st-and.ac.uk/~john/geometry/index.html
į:
* [[http://www-history.mcs.st-and.ac.uk/~john/ | John O'Connor]]
** [[http:
//www-history.mcs.st-and.ac.uk/~john/geometry/index.html | Topics in Geometry]]
** [[http://www-history.mcs.st-and.ac.uk/~john/MT4521/index.html | Geometry and Topology]]
2016 spalio 11 d., 15:24 atliko AndriusKulikauskas -
Pridėta 238 eilutė:
* Symplectic "sweep" is related to equivalence (for example, natural transformation) relevant for arguments of equality by continuity (for example, the Fundamental Theorem of Calculus, integration).
2016 spalio 11 d., 15:22 atliko AndriusKulikauskas -
Pridėtos 175-176 eilutės:
These geometries show how to relate (ever more tightly) two distinct dimensions.
Pakeistos 236-237 eilutės iš
* Symplectic area is orientable?
į:
* Symplectic area is orientable.
* Symplectic geometry is naturally related to time because it is swept out (in one dimension) in time. And so the time (one-)dimension thereby "defines" the geometry of the area.
2016 spalio 11 d., 12:07 atliko AndriusKulikauskas -
Pakeista 6 eilutė iš:
* Make a list of geometry theorems and sort them by geometry.
į:
* Make a list of [[geometry theorems]] and sort them by geometry.
2016 spalio 11 d., 11:28 atliko AndriusKulikauskas -
Pridėta 80 eilutė:
* [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory | 6D (2,0) superconformal field theory]] ?
2016 spalio 10 d., 17:51 atliko AndriusKulikauskas -
Pridėta 175 eilutė:
* Allowing only positive "coefficients" is related to positive definiteness, convexity.
Pridėta 185 eilutė:
* Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
2016 spalio 10 d., 17:41 atliko AndriusKulikauskas -
Pridėtos 4-8 eilutės:

I should
* Make a list of geometry theorems and sort them by geometry.
* Make a list of geometries and show how they are related.
* Study Bezier curves and Bernstein polynomials.
2016 spalio 07 d., 09:24 atliko AndriusKulikauskas -
Pridėta 74 eilutė:
* http://www-history.mcs.st-and.ac.uk/~john/geometry/index.html
2016 spalio 06 d., 23:52 atliko AndriusKulikauskas -
Pridėta 73 eilutė:
* [[https://arxiv.org/pdf/hep-th/0512172v1 | Langland Frenkel]]
2016 spalio 04 d., 15:52 atliko AndriusKulikauskas -
Pridėta 72 eilutė:
* [[http://www.springer.com/us/book/9781402084478 | From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept]]
2016 spalio 04 d., 15:45 atliko AndriusKulikauskas -
Pridėtos 70-71 eilutės:
* [[http://math.stackexchange.com/questions/204533/applications-of-the-fundamental-theorems-of-affine-and-projective-geometry?rq=1 | Fundamental theorems of affine and projective geometry]]
* [[https://www.amazon.com/Introduction-Geometry-Wiley-Classics-Library/dp/0471504580/ref=sr_1_1?ie=UTF8&qid=1387737461&sr=8#reader_0471504580 | Introduction to Geometry]] by Coxeter.
2016 spalio 03 d., 14:13 atliko AndriusKulikauskas -
Pakeistos 66-67 eilutės iš
* [[Homotopy type theory]]
*
į:
* [[http://www.cambridge.org/gb/academic/subjects/physics/mathematical-methods/students-guide-vectors-and-tensors?format=PB&isbn=9780521171908 | A Student's Guide to Vectors and Tensors]]
Ištrinta 83 eilutė:
* [[https://www.youtube.com/watch?v=E3steS2Hr1Y | An Intuitive Introduction to Motivic Homotopy Theory]] Vladimir Voevodsky [2002]
2016 spalio 03 d., 00:17 atliko AndriusKulikauskas -
Pridėta 57 eilutė:
** [[http://settheory.net/geometry | Sylvain Poirer]]
2016 spalio 02 d., 21:09 atliko AndriusKulikauskas -
Pridėta 68 eilutė:
** [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/week10.pdf | Week 10: Linear Functionals, Adjoints]] by Terrence Tao
2016 spalio 02 d., 21:04 atliko AndriusKulikauskas -
Pakeista 67 eilutė iš:
* http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/
į:
* [[http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/ | Linear Algebra Notes by Terrence Tao]]
2016 spalio 02 d., 21:03 atliko AndriusKulikauskas -
Pridėta 68 eilutė:
* [[http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/S0273-0979-1984-15237-6.pdf | An Elementary Introduction to the Langlands Program]] by Stephen Gelbart
2016 spalio 02 d., 20:30 atliko AndriusKulikauskas -
Pakeista 300 eilutė iš:
A variable is an "atom" as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.
į:
A variable is an "atom" of meaning as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.
2016 spalio 02 d., 20:30 atliko AndriusKulikauskas -
Pridėtos 299-300 eilutės:

A variable is an "atom" as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.
2016 rugsėjo 30 d., 13:57 atliko AndriusKulikauskas -
Pridėtos 26-27 eilutės:

shear map takes parallelogram to square, preserves area
2016 rugsėjo 28 d., 07:59 atliko AndriusKulikauskas -
Pridėta 80 eilutė:
* [[https://www.youtube.com/watch?v=E3steS2Hr1Y | An Intuitive Introduction to Motivic Homotopy Theory]] Vladimir Voevodsky [2002]
2016 rugsėjo 28 d., 01:25 atliko AndriusKulikauskas -
Pridėta 65 eilutė:
* http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/
2016 rugsėjo 28 d., 01:17 atliko AndriusKulikauskas -
Pridėta 64 eilutė:
2016 rugsėjo 27 d., 14:26 atliko AndriusKulikauskas -
Pakeistos 62-63 eilutės iš
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and
Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
į:
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
2016 rugsėjo 27 d., 14:25 atliko AndriusKulikauskas -
Pridėta 64 eilutė:
* [[Homotopy type theory]]
2016 rugsėjo 27 d., 14:18 atliko AndriusKulikauskas -
Pridėtos 20-23 eilutės:

Bernstein polynomials
* x = 1/2 get simplex
* x = 1/3 or 2/3 get cube and cross-polytope
2016 rugsėjo 27 d., 14:13 atliko AndriusKulikauskas -
Pridėtos 57-59 eilutės:
* Symmetric functions
** [[http://www.math.drexel.edu/~rboyer/boyer_thiel.pdf | Generalized Bernstein Polynomials and
Symmetric Functions]], Boyer and Thiel, about [[https://en.wikipedia.org/wiki/Bernstein_polynomial | Bernstein polynomials]] and Pascal's triangle.
2016 rugsėjo 26 d., 22:23 atliko AndriusKulikauskas -
Pridėta 26 eilutė:
* [[https://basepub.dauphine.fr/bitstream/handle/123456789/6842/polymathematics.PDF | Polymathematics: is mathematics a single science or a set of arts?]], V.I.Arnold
2016 rugsėjo 26 d., 11:03 atliko AndriusKulikauskas -
Pakeista 47 eilutė iš:
* [[ Symmetry and the Standard Model]] Matthew Robinson
į:
* [[http://link.springer.com/book/10.1007%2F978-1-4419-8267-4 | Symmetry and the Standard Model]] Matthew Robinson
2016 rugsėjo 26 d., 11:00 atliko AndriusKulikauskas -
Pridėta 47 eilutė:
* [[ Symmetry and the Standard Model]] Matthew Robinson
2016 rugsėjo 19 d., 15:54 atliko AndriusKulikauskas -
Pridėta 90 eilutė:
* the ways that our expectations can be related, thus how we are related to each other
Pridėta 94 eilutė:
* the ways that a vector space is grounded
2016 rugsėjo 19 d., 15:22 atliko AndriusKulikauskas -
Pridėta 93 eilutė:
* the relationship between two spaces, for example, points, lines, planes
2016 rugsėjo 19 d., 15:11 atliko AndriusKulikauskas -
Pakeistos 277-278 eilutės iš
------------------
į:
'''Notes'''

What is the significance of a triangle or a trilateral? They are the fourth row of Pascal's triangle.

A triangle on a sphere together with its antipodes (defined in terms of the center) defines eight triangles, an octahedron. A triangle in three dimensional space defines a demicube (simplex) in terms of the origin. A triangle with its center defines a simplex. How is a triangle related to a cube?
2016 rugsėjo 18 d., 07:37 atliko AndriusKulikauskas -
Pridėta 181 eilutė:
* Algebraic geometry presumes orthogonal basis elements, thus, perpendicularity and angles. Thus affine geometry and projective geometry should be restricted to not using algebraic geometry.
2016 rugsėjo 13 d., 19:49 atliko AndriusKulikauskas -
Pakeista 90 eilutė iš:
* the relationship between our old and new search
į:
* the relationship between our old and new search. And search is triggered by constancy, which is the representation of the nullsome which is related to anything and thus to calm and expectations, space and time, etc.
2016 rugsėjo 13 d., 14:47 atliko AndriusKulikauskas -
Pridėtos 218-221 eilutės:

* Flip around our search, turn vector around: (reflection)
* Turn a corner into another dimension
* Sweep a new dimension in terms of an old dimension (translation)
2016 rugsėjo 13 d., 14:43 atliko AndriusKulikauskas -
Pridėta 90 eilutė:
* the relationship between our old and new search
2016 rugsėjo 13 d., 12:36 atliko AndriusKulikauskas -
2016 rugsėjo 13 d., 12:36 atliko AndriusKulikauskas -
Pridėta 152 eilutė:
* https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume
2016 rugsėjo 13 d., 12:15 atliko AndriusKulikauskas -
Pridėta 217 eilutė:
* http://settheory.net/geometry#transf
Pakeistos 228-229 eilutės iš
į:
* https://en.m.wikipedia.org/wiki/Möbius_transformation combines
translation, inversion, reflection, rotation, homothety
2016 rugsėjo 13 d., 12:04 atliko AndriusKulikauskas -
Pridėtos 162-164 eilutės:

Projective geometry adds points at infinity to affine geometry. Conformal geometry or inversive geometry adds a distinguished circle. Symplectic geometry adds an area product. Moebius strip plays with the distinguished circle changing orientation if you go around.
2016 rugsėjo 13 d., 11:18 atliko AndriusKulikauskas -
Pakeistos 47-48 eilutės iš
* [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
į:
* [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
* Sylvain Poirer
** [[http://settheory.net/geometry-axioms | Geometry axioms]]
** [[http://settheory.net/foundations/variables-sets | Variables - sets]]
** [[http://settheory.net/foundations/theories | Theories]]
** [[http://settheory.net/foundations/classes2 | Classes]]
** [[http://spoirier.lautre.net/no12.pdf | Geometry: in French]]
** [[http://spoirier.lautre.net/no3.pdf | Geometry: in French]]
Pridėtos 239-263 eilutės:

'''Sylvain Poirer'''

Some key ideas, probably you know, but just in case:
The (n+p-1)-dimensional projective space associated with a quadratic
space with signature (n,p), is divided by its (n+p-2)-dimensional
surface (images of null vectors), which is a conformal space with
signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p)
and positive curvature, the other with dimension (n,p-1) and negative
curvature.
Just by changing convention, the one with signature (n-1,p) and
positive curvature can also seen as a space with signature (p,n-1) and
negative curvature; and similarly for the other.

Affine representations of that quadric are classified by the choice of
the horizon, or equivalently the polar point of that horizon (the
point representing in the projective space the direction orthogonal to
that hyperplane). So there are 3 possibilities.
The null one sees it as a paraboloid and gives it an affine geometry.
The 2 others, with the different signs, see it as a quadric whose
center is the polar point, and give it 2 different curved geometries

We can understand the stereographic projection as the effect of the
projective transformation of the space, which changes the sphere into
a paraboloid, itself projected into an affine space.
2016 rugsėjo 12 d., 14:09 atliko AndriusKulikauskas -
Pridėta 47 eilutė:
* [[https://www.amazon.com/Symmetries-Group-Theory-Particle-Physics/dp/3642154816 | Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli
2016 rugsėjo 12 d., 14:06 atliko AndriusKulikauskas -
Pridėta 46 eilutė:
* [[http://www.wall.org/~aron/blog/the-ten-symmetries-of-spacetime/ | Ten symmetries of space time]]
2016 rugsėjo 11 d., 09:35 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
[[Geometry illustrations]]
į:
[[Geometry illustrations]], [[Universal hyperbolic geometry]]
2016 rugsėjo 10 d., 16:48 atliko AndriusKulikauskas -
Pridėta 45 eilutė:
* [[http://www.cut-the-knot.org/geometry.shtml | Geometry at Cut-the-Knot]]
2016 rugpjūčio 30 d., 22:15 atliko AndriusKulikauskas -
Pridėta 19 eilutė:
* How is the Zariski topology related to the Binomial theorem?
2016 rugpjūčio 30 d., 22:14 atliko AndriusKulikauskas -
Pridėta 186 eilutė:
* Symplectic area is orientable?
2016 rugpjūčio 30 d., 11:30 atliko AndriusKulikauskas -
Pakeista 79 eilutė iš:
* how to expand our vision (from a smaller space to a larger space)
į:
* how to expand our vision (from a smaller space to a larger space) (Tadashi Tokieda)
Pridėta 82 eilutė:
2016 rugpjūčio 27 d., 14:50 atliko AndriusKulikauskas -
Pridėtos 55-56 eilutės:
* [[http://www.alainconnes.org/en/videos.php | Alain Connes: The Music of Shapes]]
* [[http://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html | Alain Connes: Noncommutative Geometry and Physics]]
2016 rugpjūčio 27 d., 14:45 atliko AndriusKulikauskas -
Pridėta 28 eilutė:
2016 rugpjūčio 25 d., 23:45 atliko AndriusKulikauskas -
Pridėta 42 eilutė:
* [[http://www.azimuthproject.org/azimuth/show/Network+theory | Network theory (wiki)]] and [[http://math.ucr.edu/home/baez/networks/ | Network theory (blog)]] by John Baez
2016 rugpjūčio 25 d., 23:39 atliko AndriusKulikauskas -
Pakeistos 40-41 eilutės iš
* [[http://arxiv.org/abs/1509.02145 | The six operations in equivariant motivic homotopy theory]]
Marc Hoyois
į:
* [[http://arxiv.org/abs/1509.02145 | The six operations in equivariant motivic homotopy theory]] Marc Hoyois
2016 rugpjūčio 25 d., 23:26 atliko AndriusKulikauskas -
Pridėta 42 eilutė:
* [[http://math.ucr.edu/home/baez/information/index.html | Information Geometry]] by John Baez
2016 rugpjūčio 23 d., 12:17 atliko AndriusKulikauskas -
Pridėtos 207-211 eilutės:

Grothendieck's six operations:
* pushforward along a morphism and its left adjoint
* compactly supported pushforward and its right adjoint
* tensor product and its adjoint internal hom
2016 rugpjūčio 23 d., 11:07 atliko AndriusKulikauskas -
Pridėtos 40-41 eilutės:
* [[http://arxiv.org/abs/1509.02145 | The six operations in equivariant motivic homotopy theory]]
Marc Hoyois
2016 rugpjūčio 22 d., 17:05 atliko AndriusKulikauskas -
Pridėta 45 eilutė:
* [[https://video.ias.edu/univalent/voevodsky | Univalent Foundations of Mathematics]]
2016 rugpjūčio 22 d., 15:42 atliko AndriusKulikauskas -
Pakeistos 17-18 eilutės iš
į:
* Try to use the tetrahedron as a way to model the 4th dimension so as to imagine how a trefoil knot could be untangled.
* Try to imagine what a 3-sphere looks like as we pass through it from time t = -1 to 1.
2016 rugpjūčio 22 d., 14:12 atliko AndriusKulikauskas -
Pakeista 66 eilutė iš:
* [[
į:
* The [[https://en.wikipedia.org/wiki/Geometrization_conjecture | Geometrization conjecture]] and the eight Thurston geometries. Also, the [[https://en.wikipedia.org/wiki/Bianchi_classification | Bianchi classification]] of low dimensional Lie algebras.
2016 rugpjūčio 22 d., 14:10 atliko AndriusKulikauskas -
Pridėta 44 eilutė:
* [[http://athome.harvard.edu/threemanifolds/watch.html | The Geometry of 3-Manifolds]]
Pridėta 66 eilutė:
* [[
2016 rugpjūčio 21 d., 10:00 atliko AndriusKulikauskas -
Pridėta 16 eilutė:
* How can you cut in half a topological object if you have no metric? How can you be sure whether you will get two or three pieces?
2016 rugpjūčio 21 d., 09:43 atliko AndriusKulikauskas -
Pridėta 15 eilutė:
* Topological product (for a torus) is a list, has an order. In general, a Cartesian product is a list. What if such a product is unordered? How do we get there in the limit to F1?
2016 rugpjūčio 21 d., 09:41 atliko AndriusKulikauskas -
Pridėta 14 eilutė:
* What is the topological quotient for an equilateral triangle or a simplex?
2016 rugpjūčio 21 d., 09:38 atliko AndriusKulikauskas -
Pakeista 207 eilutė iš:
Quotient is gluing is equivalence on a boundary.
į:
Quotient is gluing is equivalence on a boundary. Topology is the creation of a smaller space from a larger space.
2016 rugpjūčio 21 d., 09:38 atliko AndriusKulikauskas -
Pridėtos 206-207 eilutės:

Quotient is gluing is equivalence on a boundary.
2016 rugpjūčio 21 d., 09:35 atliko AndriusKulikauskas -
Pakeista 205 eilutė iš:
square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball)
į:
square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball: pi-to-the-n/2 over (n/2)!
2016 rugpjūčio 21 d., 09:34 atliko AndriusKulikauskas -
Pridėtos 204-205 eilutės:

square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball)
2016 rugpjūčio 21 d., 05:56 atliko AndriusKulikauskas -
Pakeista 203 eilutė iš:
A 0-sphere is 2 points, much as generated by the center of a cross-polytope.
į:
A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
2016 rugpjūčio 21 d., 05:52 atliko AndriusKulikauskas -
Pridėta 13 eilutė:
* What happens to the corners of the shapes?
2016 rugpjūčio 21 d., 05:46 atliko AndriusKulikauskas -
Pridėta 12 eilutė:
* In topology product rule d(MxN) = dM x N union MxN addition is union (whereas in the Zariski topology multiplication is union). Why? The product rule is related to the deRham cohomology.
2016 rugpjūčio 21 d., 05:16 atliko AndriusKulikauskas -
Pridėtos 200-201 eilutės:

A 0-sphere is 2 points, much as generated by the center of a cross-polytope.
2016 rugpjūčio 21 d., 04:56 atliko AndriusKulikauskas -
Pridėtos 196-199 eilutės:

'''Figuring things out'''

Tadashi Tokieda: Basic strategy of topology. When a problem has degeneracies, then deform (or perturb) to a problem without degeneracies, then deform back. We can use the same approach to show some problems are unsolvable.
2016 rugpjūčio 20 d., 23:56 atliko AndriusKulikauskas -
Pridėta 36 eilutė:
* [[https://www.youtube.com/playlist?list=PLC37ED4C488778E7E | Universal Hyperbolic Geometry]]
2016 rugpjūčio 20 d., 23:54 atliko AndriusKulikauskas -
Pakeista 18 eilutė iš:
į:
Overviews and History
Pakeistos 22-23 eilutės iš
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
į:
* [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
Geometry
Pakeista 27 eilutė iš:
* UnivHypGeom4: First steps in hyperbolic geometry: fundamental results
į:
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
Ištrinta 29 eilutė:
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
Pakeistos 34-35 eilutės iš
* Catster videos
* [[http
://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
į:
Videos
** UnivHypGeom4: First steps in hyperbolic geometry: fundamental results
* [[https:
//www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ | Tadashi Tokieda, Topology and Geometry]]
* [[http://www.simonwillerton.staff.shef.ac.uk/TheCatsters/ | Catster videos]]
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic | Lectures on the Geometric Anatomy of Theoretical Physics]]
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
* [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory]]
]]
2016 rugpjūčio 18 d., 16:26 atliko AndriusKulikauskas -
Pridėta 141 eilutė:
* Universal hyperbolic geometry (projective geometry with a distinguished circle) is perhaps conformal geometry. It relates two different spaces, the inside and the outside of the circle.
2016 rugpjūčio 18 d., 10:28 atliko AndriusKulikauskas -
Pridėta 56 eilutė:
* how to expand our vision (from a smaller space to a larger space)
2016 rugpjūčio 17 d., 14:44 atliko AndriusKulikauskas -
Pakeistos 11-12 eilutės iš
į:
* Relate sheaves and vector bundles.
Pridėta 51 eilutė:
* [[https://en.wikipedia.org/wiki/Vector_bundle | Vector bundle]]
2016 rugpjūčio 17 d., 14:17 atliko AndriusKulikauskas -
Pridėta 34 eilutė:
* [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf | Felix Klein, Erlangen program]]
2016 rugpjūčio 17 d., 13:54 atliko AndriusKulikauskas -
Pakeista 32 eilutė iš:
į:
* [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf | Sheaf Theory by Tom Lovering]]
2016 rugpjūčio 17 d., 13:47 atliko AndriusKulikauskas -
Pakeistos 18-19 eilutės iš
*
į:
* [[http://matematicas.unex.es/~navarro/res/lisker1.pdf | Récoltes et Semailles, Part 1]], Alexander Grothendieck. Also, [[http://matematicas.unex.es/~navarro/res/ | translation into Spanish and other works]].
** [[http://www.landsburg.com/grothendieck/pragasz.pdf | Notes on the Life and Work of Alexander Grothendieck]] by Piotr Pragacz
2016 rugpjūčio 17 d., 13:42 atliko AndriusKulikauskas -
Pridėta 18 eilutė:
*
Pridėtos 38-41 eilutės:
* [[https://en.wikipedia.org/wiki/Motive_(algebraic_geometry) | Motives]] and Universal cohomology. [[https://en.wikipedia.org/wiki/Weil_cohomology_theory | Weil cohomology theory]] and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
* [[https://en.wikipedia.org/wiki/Alexander_Grothendieck | Grothendieck]]
** "Continuous" and "discrete" duality (derived categories and "six operations")
** Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
2016 rugpjūčio 17 d., 12:33 atliko AndriusKulikauskas -
Pridėtos 34-36 eilutės:
* [[https://en.wikipedia.org/wiki/Six_operations | Six operations]]
** [[https://ncatlab.org/nlab/show/six+operations | Six operations at nLab]]
** [[http://math.stackexchange.com/questions/1351735/grothendiecks-yoga-of-six-operations-in-relatively-basic-terms | Six operations at Math Stack Exchange]]
2016 rugpjūčio 17 d., 11:08 atliko AndriusKulikauskas -
Pridėtos 9-10 eilutės:
* Relate the first Betti number with my version of the Euler characteristic, C - V + E - F + T.
* Think of how transformations act on 0, 1, infinity, for example, translations can take 0 to 1, but infinity to infinity.
2016 rugpjūčio 17 d., 10:46 atliko AndriusKulikauskas -
Pridėtos 28-29 eilutės:
* Catster videos
2016 rugpjūčio 16 d., 16:22 atliko AndriusKulikauskas -
Pridėtos 78-82 eilutės:

* Affine geometry supposes the integers
* Projective geometry supposes the rationals
* Conformal (Euclidean) geometry supposes the reals
* Symplectic geometry supposes the complexes
2016 rugpjūčio 16 d., 16:14 atliko AndriusKulikauskas -
Pridėta 8 eilutė:
* How does the geometric product in a Clifford Algebra model angular momentum, the basis for symplectic geometry, which is otherwise typically described by the cross product?
2016 rugpjūčio 16 d., 16:10 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš
* [[
į:
Pridėtos 126-137 eilutės:
* McDuff: First of all, what is a symplectic structure? The concept arose in the study of classical
mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a
falling apple. The trajectory of such a system is determined if one knows its position
and velocity (speed and direction of motion) at any one time. Thus for an object
of unit mass moving in a given straight line one needs two pieces of information, the
position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.
* momentum x position is angular momentum
* McDuff: This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a
single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld
quantization principle says that the area of this region is quantized, i.e. it has to be
an integral multiple of a number called Planck’s constant. Thus one can think of the
symplectic area as a measure of the entanglement of position and velocity.
2016 rugpjūčio 16 d., 15:59 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš
į:
* [[
Pridėtos 121-126 eilutės:
* Symplectic geometry is an even dimensional geometry. It lives on even dimensional
spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional
lengths and angles that are familiar from Euclidean and Riemannian geometry. It is
naturally associated with the field of complex rather than real numbers. However, it
is not as rigid as complex geometry: one of its most intriguing aspects is its curious
mixture of rigidity (structure) and flabbiness (lack of structure). [[http://www.math.stonybrook.edu/~dusa/ewmcambrevjn23.pdf | What is Symplectic Geometry? by Dusa McDuff]]
2016 rugpjūčio 16 d., 14:18 atliko AndriusKulikauskas -
Pridėtos 25-26 eilutės:
* [[http://arxiv.org/abs/1112.2378 | Clifford Algebras in Symplectic Geometry and Quantum Mechanics]]
* [[https://www.researchgate.net/publication/225390483_Generalized_Clifford_algebras_Orthogonal_and_symplectic_cases | Generalized Clifford algebras: Orthogonal and symplectic cases]]
2016 rugpjūčio 16 d., 13:19 atliko AndriusKulikauskas -
Pridėta 29 eilutė:
* [[https://en.wikipedia.org/wiki/Conformal_geometric_algebra | Conformal geometric algebra]] includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
2016 rugpjūčio 16 d., 11:34 atliko AndriusKulikauskas -
Pridėta 49 eilutė:
* Algebraic geometry is the study of spaces of solutions to algebraic equations.
2016 rugpjūčio 16 d., 06:46 atliko AndriusKulikauskas -
Pakeistos 9-13 eilutės iš
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* https://en.m.wikipedia.org/wiki/Ordered_geometry
į:
Pakeistos 29-32 eilutės iš
į:
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* [[https://en.m.wikipedia.org/wiki/Ordered_geometry | Ordered geometry]]
Pakeistos 79-80 eilutės iš
[[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Absolute geometry]], also known as neutral geometry,
į:
Ordered geometry features the concept of intermediacy. It is a common foundation for affine, Euclidean, absolute geometry and hyperbolic geometry, but not projective geometry. Like projective geometry, it omits the notion of measurement.

[[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Absolute geometry]], also known as neutral geometry, is based on the axioms of Euclidean geometry (including the first four of Euclid's axioms) but with the parallel postulate removed.
Pridėta 84 eilutė:
* Does not assume Euclid's third and fourth axioms.
2016 rugpjūčio 16 d., 06:39 atliko AndriusKulikauskas -
Pakeistos 16-18 eilutės iš
Works to study:
į:
'''Works to study'''

Pridėtos 30-33 eilutės:
Concepts
* [[https://en.m.wikipedia.org/wiki/List_of_geometry_topics | List of geometry topics]]
* [[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Foundations of geometry]]
Ištrinta 74 eilutė:
Pridėtos 79-80 eilutės:

[[https://en.m.wikipedia.org/wiki/Foundations_of_geometry | Absolute geometry]], also known as neutral geometry,
2016 rugpjūčio 16 d., 06:34 atliko AndriusKulikauskas -
Pridėtos 9-13 eilutės:
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* https://en.m.wikipedia.org/wiki/Ordered_geometry
Ištrintos 137-141 eilutės:

* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* https://en.m.wikipedia.org/wiki/Ordered_geometry
2016 rugpjūčio 16 d., 06:34 atliko AndriusKulikauskas -
Pakeistos 127-128 eilutės iš
* https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}},
in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
į:
* https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
2016 rugpjūčio 16 d., 06:33 atliko AndriusKulikauskas -
Pridėtos 47-48 eilutės:
Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity.
Ištrinta 52 eilutė:
Pakeista 54 eilutė iš:
* An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality.
į:
* An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality. This is geometry without any field, and without any zeros - what does this mean for the correspondence with the polynomial ring?
Pakeistos 56-58 eilutės iš
* Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center.
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, perhaps as oriented bounded spaces, in which case they would be preserved by symplectic geometry.
į:
* Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center. They ground infinite limits, thus the reals.
* Dn - Demicubes have neither
a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, perhaps as oriented bounded spaces, in which case they would be preserved by symplectic geometry. The duality mirror grounds the duality between points (vertices) and lines (origins).
Pakeistos 64-65 eilutės iš
Each of these four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.
į:

Lines, Angles, Areas require a Field whereas directions do not. Lines are translations and Angles are rotations. Together they define the Complexes. Are they key to Dn? Study visual [[complex analysis]].
A Field allows, for example, proportionality and other transformations - multiplications - consider!
Pridėta 83 eilutė:
* [[https://en.m.wikipedia.org/wiki/Homography Homography]] two approaches to projective geometry with fields or without
Pakeistos 85-86 eilutės iš
į:
* Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
Pridėtos 99-102 eilutės:
Conformal geometry

Symplectic geometry
Pridėtos 109-110 eilutės:
Each of the four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.
Ištrinta 113 eilutė:
Ištrintos 137-161 eilutės:

------------------

----------------------

Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.

Geometry is defined with regard to the Center:
* An: Directions (vectors) are defined by the relationships between the Center and the vertex it generates.
* Cn: Lines are defined by the relationships between the Center and pairs of vertices it generates.
* Bn: Angles are defined by the Center/Volume? and the perpendicular angles created with each new mirror.
* Dn: Areas are defined by the lines and angles of the hemicube?

Lines, Angles, Areas require a Field whereas directions do not. Lines are translations and Angles are rotations. Together they define the Complexes. Are they key to Dn? Study visual [[complex analysis]].

A Field allows, for example, proportionality and other transformations - multiplications - consider!

https://en.m.wikipedia.org/wiki/Homography two approaches to projective geometry with fields or without

Geometry
2016 rugpjūčio 16 d., 06:23 atliko AndriusKulikauskas -
Pakeistos 23-24 eilutės iš
'''Geometries'''
į:
'''Geometry'''

Geometry is:
* how to embed a lower dimensional space into a higher dimensional space
* the construction of sets of roots of polynomials

Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.
* [[Tensor | Tensors]] give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
* A [[https://ncatlab.org/nlab/show/geometric+embedding | geometric embedding]] is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.

Definitions of geometry
* Wikipedia defines geometry as "concerned with questions of shape, size, relative position of figures, and the properties of space".
* MathWorld defines geometry as "the study of figures in a space of a given number of dimensions and of a given type", and formally, as "a complete locally homogeneous Riemannian manifold".
* nLab seems to define it as part of an Isbell duality between geometry (presheaves) and algebra (copresheaves) where presheaves (contravariant functors C->Set) and copresheaves (functors on C) are identified with each other and thus glued together (for some category C).
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''

I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.

Construction of the continuum
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.

'''A System of Geometries'''

* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.

Four infinite families of polytopes can be distinguished by how they are extended in each new dimension. They seem to relate to four different geometries and four different classical Lie algebras:
* An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality.
* Cn - Cross-polytopes (such as the octahedron) are extended when the Center creates two new vertices ("opposites") and thereby defines a line in two directions, which is preserved by projective geometry. Cross-polytopes have a Center but no Totality.
* Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center.
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, perhaps as oriented bounded spaces, in which case they would be preserved by symplectic geometry.

* Simplex (1+1)^N
* Cross-polytopes (1+2)^N
* Cubes (2+1)^N
* Half-cubes (2+2)^N

Each of these four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.

'''Distinct
Geometries'''
Pakeistos 69-73 eilutės iš
į:
* Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
** In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
** Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
* In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
Pakeista 78 eilutė iš:
* A vector subspace needs to contain zero. How is this related to projective geometry?
į:
* A vector subspace needs to contain zero. How is this related to projective geometry? Vector spaces: Two different coordinate systems agree on the origin 0.
Ištrintos 94-98 eilutės:
Vector spaces
* Two different coordinate systems agree on the origin 0.

Pakeistos 99-105 eilutės iš

* Simplex (1+1)^N
* Cross-polytopes (1+2)^N
* Cubes (2+1)^N
* Half-cubes (2+2)^N
į:
'''Pairs of Geometries'''

Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.

Pridėtos 107-108 eilutės:
I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry. And I imagine that they are as intuitive as the various ways that we interpret multiplication in arithmetic. I suppose that they may include translation, rotation, scaling, homothety, similarity, reflection and shear.
Pakeistos 111-120 eilutės iš
į:
* In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
* [[https://en.wikipedia.org/wiki/Squeeze_mapping | Squeeze mapping]]
* Isometry
* Special conformal is reflection and inversion
* [[https://en.wikipedia.org/wiki/Homography | Homography]] is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation.
* [[https://en.m.wikipedia.org/wiki/Affine_transformation | Affine transformation]]
* [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]] is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
* https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}},
in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
Ištrintos 123-152 eilutės:
I asked the following question at Math Overflow but it was deleted.

'''What is geometry?'''

I wish to know what geometers and other mathematicians consider geometry.
* Wikipedia defines geometry as "concerned with questions of shape, size, relative position of figures, and the properties of space".
* MathWorld defines geometry as "the study of figures in a space of a given number of dimensions and of a given type", and formally, as "a complete locally homogeneous Riemannian manifold".
* nLab seems to define it as part of an Isbell duality between geometry (presheaves) and algebra (copresheaves) where presheaves (contravariant functors C->Set) and copresheaves (functors on C) are identified with each other and thus glued together (for some category C).
I don't understand the latter but I would try if somebody might explain. My own Ph.D. is in algebraic combinatorics. I will explain how I am coming to think of geometry. I appreciate thoughts on how I might develop my understanding further.

I study conceptual frameworks by which we think and live. I am applying such frameworks to think about the "implicit math" in our minds by which we figure things out in mathematics or interpret mathematics as, for example, algebraic combinatorialists analyze an equation to figure out what it is counting. I am also interested in how math can express and model such conceptual frameworks.

As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]]. I tried to identify which areas depended on which areas. I noticed that differential geometry and algebraic geometry depend on geometry. But what is geometry and what basic concepts does it contribute?

I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.

In my essay, [[http://www.ms.lt/sodas/Book/DiscoveryInMathematics | Discovery in Mathematics: A System of Deep Structure]], I notice that four infinite families of polytopes can be distinguished by how they are extended in each new dimension. They seem to relate to four different geometries and four different classical Lie algebras:
* An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality.
* Cn - Cross-polytopes (such as the octahedron) are extended when the Center creates two new vertices ("opposites") and thereby defines a line in two directions, which is preserved by projective geometry. Cross-polytopes have a Center but no Totality.
* Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center.
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, perhaps as oriented bounded spaces, in which case they would be preserved by symplectic geometry.

Each of these four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.

As I mention in my essay, I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry. And I imagine that they are as intuitive as the various ways that we interpret multiplication in arithmetic. I suppose that they may include translation, rotation, scaling, homothety, similarity, reflection and shear.

I thus ask how geometers think of geometry and what it contributes to the big picture in math. I wish for my own philosophical speculations to be more fruitful. I wonder how to pursue them further mathematically. What should I study?

Ištrintos 125-131 eilutės:

[[https://en.m.wikipedia.org/wiki/Affine_transformation | Affine transformation]]
*
Pakeistos 129-152 eilutės iš
[[https://en.wikipedia.org/wiki/Squeeze_mapping | Squeeze mapping]]

Isometry

Special conformal is reflection and inversion

[[https://en.wikipedia.org/wiki/Homography | Homography]] is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation.

[[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]] is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.

[[http://www.math.ucr.edu/home/baez/groupoidification/ | Erlangen program]]

https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends

{\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}},
in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.

In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
į:
Pakeistos 132-137 eilutės iš
'''Pairs of geometries'''

Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.

In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
į:
Pakeistos 137-140 eilutės iš
Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
* In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
* Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
į:
Ištrintos 154-177 eilutės:

* [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]]

'''What is geometry?'''

* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.

Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.
* [[Tensor | Tensors]] give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
* A [[https://ncatlab.org/nlab/show/geometric+embedding | geometric embedding]] is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.

Congruence
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''

Construction of the continuum
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.

---------
2016 rugpjūčio 16 d., 06:02 atliko AndriusKulikauskas -
Pakeistos 34-36 eilutės iš
į:
* A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
Pakeistos 43-44 eilutės iš
compare to: Beltrami-Klein model of hyperbolic geometry
į:
Compare to: Beltrami-Klein model of hyperbolic geometry
Ištrinta 122 eilutė:
A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
2016 rugpjūčio 16 d., 05:58 atliko AndriusKulikauskas -
Pridėtos 3-10 eilutės:
>>bgcolor=#FFFFC0<<

* Look at Wildberger's three binormal forms.
* Do the six natural bases of the symmetric functions correspond to the six transformations?
* Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?

>><<
Pakeistos 56-57 eilutės iš
Look at Wildberger's three binormal forms.
į:

* Simplex (1+1)^N
* Cross-polytopes (1+2)^N
* Cubes (2+1)^N
* Half-cubes (2+2)^N

'''Transformations'''

* Translation - does not affect vectors
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively. Complex number dilation (rotating). Rotations are multiplicative but not additive. This brings to mind the field with one element
.
Pakeistos 99-104 eilutės iš
>>bgcolor=#FFFFC0<<

Do the six natural bases of the symmetric functions correspond to the six transformations?

Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?
į:
Pakeistos 105-108 eilutės iš
Transformations
* Translation - does not affect vectors
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively. Complex number dilation (rotating). Rotations are multiplicative but not additive. This brings to mind the field with one element.
į:
Pakeistos 193-196 eilutės iš
* Simplex (1+1)^N
* Cross-polytopes (1+2)^N
* Cubes (2+1)^N
* Half-cubes (2+2)^N
į:
2016 rugpjūčio 16 d., 05:44 atliko AndriusKulikauskas -
Pakeistos 15-16 eilutės iš
į:
'''Geometries'''

Affine geometry
* Different coordinate systems don't agree on any origin.

Projective geometry
* projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
* "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
* [1:2:0] is a point that is a "direction" (two directions)
* A vector subspace needs to contain zero. How is this related to projective geometry?
* Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
Pakeistos 42-49 eilutės iš
Affine geometry
* Different coordinate systems don't agree on any origin.

Projective geometry
* projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
* "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
* [1:2:0] is a point that is a "direction" (two directions)
į:
Pakeistos 89-91 eilutės iš
A vector subspace needs to contain zero. How is this related to projective geometry?

Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
į:
2016 rugpjūčio 14 d., 23:34 atliko AndriusKulikauskas -
Pridėta 13 eilutė:
* [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/euler-characteristics.pdf | Understanding Euler Characteristic]], Ong Yen Chin
2016 rugpjūčio 14 d., 23:23 atliko AndriusKulikauskas -
Pridėta 5 eilutė:
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf | Combinatorial Algebraic Topology]]
2016 rugpjūčio 14 d., 22:54 atliko AndriusKulikauskas -
Pridėta 4 eilutė:
* [[http://www.alainconnes.org/docs/maths.pdf | A View of Mathematics]], Alain Connes
2016 rugpjūčio 13 d., 16:03 atliko AndriusKulikauskas -
Pridėta 10 eilutė:
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by | NPTEL videos on algebraic geometry]]
2016 rugpjūčio 13 d., 12:20 atliko AndriusKulikauskas -
Pakeistos 8-9 eilutės iš
į:
* Robin Hartshorne Geometry: Euclid and Beyond
* Robin Hartshorne, Algebraic Geometry
2016 rugpjūčio 13 d., 09:33 atliko AndriusKulikauskas -
Pakeista 39 eilutė iš:
į:
Look at Wildberger's three binormal forms.
2016 rugpjūčio 11 d., 19:02 atliko AndriusKulikauskas -
Pridėta 33 eilutė:
* [1:2:0] is a point that is a "direction" (two directions)
2016 rugpjūčio 11 d., 18:29 atliko AndriusKulikauskas -
Pakeistos 31-34 eilutės iš
*

projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
į:
* projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
* "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
2016 rugpjūčio 11 d., 18:05 atliko AndriusKulikauskas -
Pakeistos 12-17 eilutės iš
* hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
** perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
** orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
** most important theorem: Pythagoras q=q1+q2 - q1q2
** second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3
į:
Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
* perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
* orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
* most important theorem: Pythagoras q=q1+q2 - q1q2
* second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3
Pakeistos 20-24 eilutės iš
* Euclidean geometry: empty space + tools: straightedge, compass, area measurer
** most important theorem: Pythagoras q=q1+q2
** (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
į:
Euclidean geometry: empty space + tools: straightedge, compass, area measurer
* most important theorem: Pythagoras q=q1+q2
* (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)

Vector spaces
* Two different coordinate systems agree on the origin 0.

Affine geometry
* Different coordinate systems don't agree on any origin.

Projective geometry
*
Pridėtos 39-40 eilutės:
2016 rugpjūčio 11 d., 13:22 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the parametrization of the circle.
* What about the sphere? (1 + ti)(1 + ti)(1 + ti) ?
į:
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the rational parametrization of the circle.
* What about the sphere? The stereographic projection of the circle onto the plane in Cartesian coordinates is given by (1 + xi + yj)(1 + xi + yj) where ij + ji = 1, that is, i and j anticommute.
2016 rugpjūčio 11 d., 11:13 atliko AndriusKulikauskas -
Pakeista 29 eilutė iš:
* Note also that infinity is the flip side of zero - they make a pair
į:
* Note also that infinity is the flip side of zero - they make a pair. They are alternate ways of linking together the positive and negative values.
2016 rugpjūčio 11 d., 11:12 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the parametrization of the circle. What about the sphere?
(1 + ti)(1 + ti)(1 + ti) ?
į:
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the parametrization of the circle.
*
What about the sphere? (1 + ti)(1 + ti)(1 + ti) ?
* Note also that infinity is the flip side of zero - they make a pair
2016 rugpjūčio 11 d., 11:11 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the parametrization of the circle.
į:
(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the parametrization of the circle. What about the sphere?
(1 + ti)(1 + ti)(1 + ti) ?
2016 rugpjūčio 11 d., 11:10 atliko AndriusKulikauskas -
Pridėtos 26-27 eilutės:

(1 + ti)(1 + ti) = (1 - t2) + (2t) i is the parametrization of the circle.
2016 rugpjūčio 11 d., 11:02 atliko AndriusKulikauskas -
Pridėtos 23-25 eilutės:

projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
2016 rugpjūčio 10 d., 11:06 atliko AndriusKulikauskas -
Pakeistos 7-11 eilutės iš
į:
* UnivHypGeom4: First steps in hyperbolic geometry: fundamental results

* hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
** perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
** orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
** most important theorem: Pythagoras q=q1+q2 - q1q2
** second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3

compare to: Beltrami-Klein model of hyperbolic geometry

* Euclidean geometry: empty space + tools: straightedge, compass, area measurer
** most important theorem: Pythagoras q=q1+q2
** (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)

------------------
Ištrintos 135-144 eilutės:
Review:
* UnivHypGeom4: First steps in hyperbolic geometry: fundamental results

* hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
** perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
** orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.

compare to: Beltrami-Klein model of hyperbolic geometry

* Euclidean geometry: empty space + tools: straightedge, compass, area measurer
2016 rugpjūčio 10 d., 11:02 atliko AndriusKulikauskas -
Pakeista 128 eilutė iš:
į:
compare to: Beltrami-Klein model of hyperbolic geometry
2016 rugpjūčio 10 d., 10:58 atliko AndriusKulikauskas -
Pakeista 126 eilutė iš:
į:
** orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
2016 rugpjūčio 10 d., 10:48 atliko AndriusKulikauskas -
Pridėtos 125-129 eilutės:
** perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa

2016 rugpjūčio 10 d., 10:46 atliko AndriusKulikauskas -
Pakeistos 121-122 eilutės iš
į:
Review:
* UnivHypGeom4: First steps in hyperbolic geometry: fundamental results

* hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
* Euclidean geometry: empty space + tools: straightedge, compass, area measurer
2016 rugpjūčio 09 d., 10:30 atliko AndriusKulikauskas -
Pridėtos 47-48 eilutės:

A vector subspace needs to contain zero. How is this related to projective geometry?
2016 rugpjūčio 08 d., 10:47 atliko AndriusKulikauskas -
Pakeistos 3-12 eilutės iš
I want to ask the following question at Math Overflow:
į:
Works to study:
* [[http
://web.maths.unsw.edu.au/~norman/papers/AffineProjArXiV.pdf | Affine and projective universal geometry]] by Norman Wildberger
* [[http://web.maths.unsw.edu.au/~norman/papers/OneDimensionalArXiV.pdf | One dimensional metrical geometry]]
* [[http://web.maths.unsw.edu.au/~norman/papers/Chromogeometry.pdf | Chromogeometry]]

I asked the following question at Math Overflow but it was deleted.
2016 rugpjūčio 04 d., 10:22 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
[[Geometry illustrations]]
2016 liepos 28 d., 14:34 atliko AndriusKulikauskas -
Pridėtos 36-37 eilutės:

Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
2016 liepos 18 d., 14:41 atliko AndriusKulikauskas -
Pakeista 39 eilutė iš:
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively. Complex number dilation (rotating).
į:
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively. Complex number dilation (rotating). Rotations are multiplicative but not additive. This brings to mind the field with one element.
2016 liepos 18 d., 14:41 atliko AndriusKulikauskas -
Pakeista 39 eilutė iš:
* Dilation (scaling) including negative (flipping)
į:
* Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively. Complex number dilation (rotating).
2016 liepos 18 d., 14:39 atliko AndriusKulikauskas -
Pridėtos 36-39 eilutės:

Transformations
* Translation - does not affect vectors
* Dilation (scaling) including negative (flipping)
2016 liepos 16 d., 16:26 atliko AndriusKulikauskas -
Pridėtos 123-128 eilutės:
---------

* Simplex (1+1)^N
* Cross-polytopes (1+2)^N
* Cubes (2+1)^N
* Half-cubes (2+2)^N
2016 liepos 08 d., 11:23 atliko AndriusKulikauskas -
Pakeistos 29-31 eilutės iš
į:
>>bgcolor=#FFFFC0<<

Do the six natural bases of the symmetric functions correspond to the six transformations?

Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?

>>bgcolor=#EEEEEE<<
Pakeista 68 eilutė iš:
>>bgcolor=#EEEEEE<<
į:
------------------
2016 birželio 28 d., 10:43 atliko AndriusKulikauskas -
Pakeistos 35-36 eilutės iš
[[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
į:
* [[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
* [[https://en.m.wikipedia.org/wiki/Incidence_structure | Incidence structure]]
* https://en.m.wikipedia.org/wiki/Ordered_geometry
2016 birželio 28 d., 10:09 atliko AndriusKulikauskas -
Pridėta 35 eilutė:
[[https://en.m.wikipedia.org/wiki/Incidence_geometry | Incidence geometry]]
2016 birželio 26 d., 23:06 atliko AndriusKulikauskas -
Pakeistos 21-22 eilutės iš
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, in which case they would be preserved by symplectic geometry.
į:
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, perhaps as oriented bounded spaces, in which case they would be preserved by symplectic geometry.
Pakeistos 25-42 eilutės iš
I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry.

I wish for my philosophical speculations to be more fruitful.

I learned...
And why, intuitively,
are there four classical Lie groups/algebras?

...in terms of Center and Totality. In studying simplexes An, I realized that the -1 simplex could be interpreted as the Center of a simplex which ever generates the next simplex by ever adding one new vertex along with edges to existing vertices and then becoming the Center of the newly created simplex. We can think of the Center as the simplex with no vertices which, for each n, is the dual of the Totality, the simplex with n vertices.

* recopy whole
* rescale whole
* rescale multiple
* redistribute set
* redistribute multiple
* redistribute whole
į:
As I mention in my essay, I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry. And I imagine that they are as intuitive as the various ways that we interpret multiplication in arithmetic. I suppose that they may include translation, rotation, scaling, homothety, similarity, reflection and shear.

I thus ask how geometers think of geometry and what it contributes to the big picture in math. I wish for my own philosophical speculations to be more fruitful. I wonder how to pursue them further mathematically. What should I study?

Pakeistos 33-39 eilutės iš
* translation
* scaling
* homothety
* similarity
* reflection
* rotation
* shear
į:
*
2016 birželio 26 d., 22:40 atliko AndriusKulikauskas -
Pakeista 21 eilutė iš:
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I suppose these demicubes could be thought to define areas, in which case they would be preserved by symplectic geometry.
į:
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, in which case they would be preserved by symplectic geometry.
2016 birželio 26 d., 22:38 atliko AndriusKulikauskas -
Pridėta 28 eilutė:
I wish for my philosophical speculations to be more fruitful.
2016 birželio 26 d., 22:37 atliko AndriusKulikauskas -
Pakeistos 23-24 eilutės iš
Each of these four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space.
į:
Each of these four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.
Pakeistos 27-29 eilutės iš
ways of embedding of one space into another, which may lead to different ways of embedding the same space, due to symmetry

tensors and triviality
į:
2016 birželio 26 d., 22:35 atliko AndriusKulikauskas -
Pakeistos 15-16 eilutės iš
I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.
į:
I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.
Pakeistos 21-25 eilutės iš
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the
į:
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I suppose these demicubes could be thought to define areas, in which case they would be preserved by symplectic geometry.

Each of these four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space.

I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry.
2016 birželio 26 d., 22:25 atliko AndriusKulikauskas -
Pridėtos 14-21 eilutės:

I am somewhat aware of Felix Klein's [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]] whereby we consider transformation groups which leave geometric properties invariant, and also [[http://www.math.ucr.edu/home/baez/groupoidification/ | groupoidification and geometric representation]], [[https://en.wikipedia.org/wiki/Moving_frame | moving frames]], [[[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]], principal connection and Ehresmann connection. But I'm wondering if there is a more fundamental way to think about geometry. I like the idea that we can get a geometry for each of the Dynkin diagrams.

In my essay, [[http://www.ms.lt/sodas/Book/DiscoveryInMathematics | Discovery in Mathematics: A System of Deep Structure]], I notice that four infinite families of polytopes can be distinguished by how they are extended in each new dimension. They seem to relate to four different geometries and four different classical Lie algebras:
* An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality.
* Cn - Cross-polytopes (such as the octahedron) are extended when the Center creates two new vertices ("opposites") and thereby defines a line in two directions, which is preserved by projective geometry. Cross-polytopes have a Center but no Totality.
* Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center.
* Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the
2016 birželio 26 d., 19:28 atliko AndriusKulikauskas -
Pakeistos 52-54 eilutės iš
...[[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]]
į:
[[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]] is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
2016 birželio 26 d., 19:27 atliko AndriusKulikauskas -
Pakeista 47 eilutė iš:
Homography
į:
[[https://en.wikipedia.org/wiki/Homography | Homography]] is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation.
2016 birželio 26 d., 19:24 atliko AndriusKulikauskas -
Pakeista 32 eilutė iš:
Affine transformation https://en.m.wikipedia.org/wiki/Affine_transformation
į:
[[https://en.m.wikipedia.org/wiki/Affine_transformation | Affine transformation]]
Pakeistos 41-42 eilutės iš
į:
[[https://en.wikipedia.org/wiki/Squeeze_mapping | Squeeze mapping]]
Pakeistos 54-57 eilutės iš
http://www.math.ucr.edu/home/baez/groupoidification/ Erlangen program

https://en.m.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends
į:
[[http://www.math.ucr.edu/home/baez/groupoidification/ | Erlangen program]]

https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends
2016 birželio 26 d., 19:21 atliko AndriusKulikauskas -
Pakeistos 33-43 eilutės iš
* translation scaling homothety similarity reflection rotation shear

Translation is an affine transformation

Reflection

Rotation

Scaling

Shear
į:
* translation
*
scaling
*
homothety
*
similarity
*
reflection
*
rotation
*
shear
2016 birželio 26 d., 16:16 atliko AndriusKulikauskas -
2016 birželio 26 d., 15:49 atliko AndriusKulikauskas -
Pakeistos 24-29 eilutės iš
*
*
*
*
*
*
į:
* recopy whole
* rescale whole
* rescale multiple
* redistribute set
* redistribute multiple
* redistribute whole
2016 birželio 26 d., 15:43 atliko AndriusKulikauskas -
2016 birželio 26 d., 15:43 atliko AndriusKulikauskas -
Pridėtos 23-33 eilutės:

*
*
*
*
*
*

Affine transformation https://en.m.wikipedia.org/wiki/Affine_transformation
* translation scaling homothety similarity reflection rotation shear
2016 birželio 26 d., 15:20 atliko AndriusKulikauskas -
2016 birželio 26 d., 15:20 atliko AndriusKulikauskas -
Pridėtos 14-17 eilutės:

ways of embedding of one space into another, which may lead to different ways of embedding the same space, due to symmetry

tensors and triviality
2016 birželio 26 d., 15:16 atliko AndriusKulikauskas -
Pakeista 13 eilutė iš:
As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]]. I tried to identify which areas depended on which areas. I noticed that differential geometry and algebraic geometry depend on geometry. But what is geometry and what key concepts does it contribute?
į:
As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]]. I tried to identify which areas depended on which areas. I noticed that differential geometry and algebraic geometry depend on geometry. But what is geometry and what basic concepts does it contribute?
2016 birželio 25 d., 11:26 atliko AndriusKulikauskas -
Pridėtos 40-41 eilutės:

http://www.math.ucr.edu/home/baez/groupoidification/ Erlangen program
2016 birželio 25 d., 01:08 atliko AndriusKulikauskas -
Pridėtos 33-37 eilutės:

Homography

A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
2016 birželio 25 d., 01:04 atliko AndriusKulikauskas -
Pridėtos 29-32 eilutės:

Isometry

Special conformal is reflection and inversion
2016 birželio 25 d., 01:01 atliko AndriusKulikauskas -
Pridėtos 19-28 eilutės:

Translation is an affine transformation

Reflection

Rotation

Scaling

Shear
2016 birželio 25 d., 00:50 atliko AndriusKulikauskas -
Pridėtos 22-28 eilutės:

https://en.m.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends

{\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}},
in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.

In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
2016 birželio 25 d., 00:03 atliko AndriusKulikauskas -
Pakeista 13 eilutė iš:
As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]]. I tried to identify which areas depended on which areas. I noticed that differential geometry and algebraic geometry depend on geometry. But what is geometry?
į:
As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]]. I tried to identify which areas depended on which areas. I noticed that differential geometry and algebraic geometry depend on geometry. But what is geometry and what key concepts does it contribute?
2016 birželio 24 d., 22:58 atliko AndriusKulikauskas -
Pridėtos 19-20 eilutės:

...[[https://en.wikipedia.org/wiki/Cartan_connection | Cartan connection]]
2016 birželio 24 d., 19:13 atliko AndriusKulikauskas -
Pakeista 18 eilutė iš:
in terms of Center and Totality. In studying simplexes An, I realized that the -1 simplex could be interpreted as the Center of a simplex which ever generates the next simplex by ever adding one new vertex along with edges to existing vertices and then becoming the Center of the newly created simplex. We can think of the Center as the simplex with no vertices which, for each n, is the dual of the Totality, the simplex with n vertices.
į:
...in terms of Center and Totality. In studying simplexes An, I realized that the -1 simplex could be interpreted as the Center of a simplex which ever generates the next simplex by ever adding one new vertex along with edges to existing vertices and then becoming the Center of the newly created simplex. We can think of the Center as the simplex with no vertices which, for each n, is the dual of the Totality, the simplex with n vertices.
2016 birželio 24 d., 19:05 atliko AndriusKulikauskas -
Pridėtos 14-18 eilutės:

I learned...
And why, intuitively, are there four classical Lie groups/algebras?

in terms of Center and Totality. In studying simplexes An, I realized that the -1 simplex could be interpreted as the Center of a simplex which ever generates the next simplex by ever adding one new vertex along with edges to existing vertices and then becoming the Center of the newly created simplex. We can think of the Center as the simplex with no vertices which, for each n, is the dual of the Totality, the simplex with n vertices.
2016 birželio 24 d., 18:47 atliko AndriusKulikauskas -
Pakeista 13 eilutė iš:
As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]].
į:
As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]]. I tried to identify which areas depended on which areas. I noticed that differential geometry and algebraic geometry depend on geometry. But what is geometry?
2016 birželio 24 d., 18:46 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
I don't understand the latter but I would try if somebody might explain. My own Ph.D. is in algebraic combinatorics. I will explain how I am coming to think of geometry.
į:
I don't understand the latter but I would try if somebody might explain. My own Ph.D. is in algebraic combinatorics. I will explain how I am coming to think of geometry. I appreciate thoughts on how I might develop my understanding further.
2016 birželio 24 d., 18:45 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
I don't understand the latter but I would try if somebody might explain.
į:
I don't understand the latter but I would try if somebody might explain. My own Ph.D. is in algebraic combinatorics. I will explain how I am coming to think of geometry.
2016 birželio 24 d., 18:43 atliko AndriusKulikauskas -
Pridėtos 1-15 eilutės:
I want to ask the following question at Math Overflow:

'''What is geometry?'''

I wish to know what geometers and other mathematicians consider geometry.
* Wikipedia defines geometry as "concerned with questions of shape, size, relative position of figures, and the properties of space".
* MathWorld defines geometry as "the study of figures in a space of a given number of dimensions and of a given type", and formally, as "a complete locally homogeneous Riemannian manifold".
* nLab seems to define it as part of an Isbell duality between geometry (presheaves) and algebra (copresheaves) where presheaves (contravariant functors C->Set) and copresheaves (functors on C) are identified with each other and thus glued together (for some category C).
I don't understand the latter but I would try if somebody might explain.

I study conceptual frameworks by which we think and live. I am applying such frameworks to think about the "implicit math" in our minds by which we figure things out in mathematics or interpret mathematics as, for example, algebraic combinatorialists analyze an equation to figure out what it is counting. I am also interested in how math can express and model such conceptual frameworks.

As part of that, I made a [[http://www.ms.lt/derlius/MatematikosSakosDidelis.png | diagram of the areas in math]] listed in the [[https://en.wikipedia.org/wiki/Mathematics_Subject_Classification | Mathematics Subject Classification]].
Pakeistos 45-48 eilutės iš
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
* [[https://en.wikipedia.org/wiki/Klein_geometry | Klein geometry]]
* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
į:
Pridėtos 49-57 eilutės:

'''What is geometry?'''

* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
Pakeistos 59-60 eilutės iš
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
į:
Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.
* [[Tensor | Tensors]] give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring
. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space
.
Pakeistos 64-65 eilutės iš
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
į:
Congruence
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''

Construction of the continuum
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
2016 birželio 23 d., 16:41 atliko AndriusKulikauskas -
Pridėtos 3-4 eilutės:
'''Pairs of geometries'''
Pridėtos 6-9 eilutės:

In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.

----------------------
2016 birželio 23 d., 16:33 atliko AndriusKulikauskas -
Pridėtos 4-7 eilutės:

Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
* In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
* Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
2016 birželio 23 d., 14:06 atliko AndriusKulikauskas -
Pakeistos 5-31 eilutės iš
Given any field F
,
2
one can construct the
n
-dimensional projective space Pn
(
F
)
as the space of lines through the origin in
F
n
+
1
. Equivalently, points in
P
n
(
F
)
are equivalence classes of nonzero points in
F
n
+
1
modulo multiplication by nonzero scalars.
į:
Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
Ištrinta 16 eilutė:
Pakeistos 19-29 eilutės iš
į:
Geometry
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
* [[https://en.wikipedia.org/wiki/Klein_geometry | Klein geometry]]
* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
* [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]]
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
* A [[https://ncatlab.org/nlab/show/geometric+embedding | geometric embedding]] is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
2016 birželio 23 d., 09:45 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
Think of pairs of geometries as defining equivalence classes variously.
į:
Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.
2016 birželio 23 d., 09:43 atliko AndriusKulikauskas -
Pakeistos 5-6 eilutės iš
Given any field
F
į:
Given any field F
2016 birželio 23 d., 08:47 atliko AndriusKulikauskas -
Pridėta 3 eilutė:
Think of pairs of geometries as defining equivalence classes variously.
2016 birželio 23 d., 08:46 atliko AndriusKulikauskas -
Pridėtos 2-30 eilutės:

Given any field
F
,
2
one can construct the
n
-dimensional projective space Pn
(
F
)
as the space of lines through the origin in
F
n
+
1
. Equivalently, points in
P
n
(
F
)
are equivalence classes of nonzero points in
F
n
+
1
modulo multiplication by nonzero scalars.
2016 birželio 23 d., 03:08 atliko AndriusKulikauskas -
Pridėtos 13-14 eilutės:

https://en.m.wikipedia.org/wiki/Homography two approaches to projective geometry with fields or without
2016 birželio 22 d., 17:18 atliko AndriusKulikauskas -
Pridėtos 1-17 eilutės:
>>bgcolor=#EEEEEE<<

Geometry is defined with regard to the Center:
* An: Directions (vectors) are defined by the relationships between the Center and the vertex it generates.
* Cn: Lines are defined by the relationships between the Center and pairs of vertices it generates.
* Bn: Angles are defined by the Center/Volume? and the perpendicular angles created with each new mirror.
* Dn: Areas are defined by the lines and angles of the hemicube?

Lines, Angles, Areas require a Field whereas directions do not. Lines are translations and Angles are rotations. Together they define the Complexes. Are they key to Dn? Study visual [[complex analysis]].

A Field allows, for example, proportionality and other transformations - multiplications - consider!

>><<

#### Geometry

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 Puslapis paskutinį kartą pakeistas 2019 balandžio 29 d., 00:34