Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
Book.Geometry istorijaPaslėpti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2019 balandžio 29 d., 00:34
atliko 
Pridėtos 184185 eilutės:
http://drmichaelrobinson.net/sheaftutorial/index.html 2019 sausio 19 d., 11:14
atliko 
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 
Pakeista 3 eilutė iš:
[[https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6  [++++几何++++]]] _ _ _ _ [+++ 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 
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 03 d., 14:27
atliko 
Pakeistos 2526 eilutės iš
* 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 
Pakeistos 3031 eilutės iš
į:
* Unions of spaces. Pridėta 35 eilutė:
* Linear equations are intersections of hyperplanes. 2019 sausio 03 d., 11:37
atliko 
Pridėtos 2225 eilutės:
'''General notions''' * Defining equivalence. For example, what makes shapes equivalent? 2019 sausio 03 d., 11:23
atliko 
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 
Pridėtos 3033 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 113115 eilutės:
* 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 
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 
Pridėta 29 eilutė:
* Map lines to lines. Projective geometry additionally maps zero to zero. And infinity to infinity? Pakeistos 3436 eilutės iš
į:
* Projective geometry: Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio). Ištrintos 8384 eilutės:
Ištrinta 120 eilutė:
2019 sausio 03 d., 10:47
atliko 
Pakeistos 2935 eilutės iš
[[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?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 semiaffine 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 {$\sigma$}semilinear 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 semilinear isomorphism. į:
* [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?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 semiaffine 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 {$\sigma$}semilinear 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 semilinear isomorphism. 2019 sausio 03 d., 10:47
atliko 
Pakeistos 2930 eilutės iš
į:
[[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] Ištrinta 75 eilutė:
2019 sausio 03 d., 10:46
atliko 
Pridėtos 2834 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 semiaffine 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 {$\sigma$}semilinear 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 semilinear isomorphism. 2019 sausio 03 d., 10:40
atliko 
Pridėtos 2038 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 
Pakeistos 8283 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 
Pridėtos 8182 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 
Pridėta 78 eilutė:
* The link between projective geometry and fractions (as equivalence classes). 2018 rugsėjo 11 d., 08:50
atliko 
Pakeista 96 eilutė iš:
* Sylvain į:
* Sylvain Poirier: Some key ideas, probably you know, but just in case: Pakeista 105 eilutė iš:
* Sylvain į:
* Sylvain Poirier: Affine representations of that quadric are classified by the choice of Pakeista 112 eilutė iš:
* Sylvain į:
* Sylvain Poirier: We can understand the stereographic projection as the effect of the Pakeista 257 eilutė iš:
* Sylvain į:
* Sylvain Poirier 2018 liepos 14 d., 12:16
atliko 
Pakeistos 543544 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, vectorspace notions are what are needed to tell us what this local geometry is, the vector space in question being the ndimensional 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, vectorspace notions are what are needed to tell us what this local geometry is, the vector space in question being the ndimensional 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 
Pridėtos 542544 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, vectorspace notions are what are needed to tell us what this local geometry is, the vector space in question being the ndimensional 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 
Pridėtos 303305 eilutės:
Videos * [[https://www.youtube.com/watch?v=pXGTevGJ01o  Symmetric geometry and classical mechanics]], Tobias Osborne 2017 lapkričio 11 d., 19:43
atliko 
Pakeista 437 eilutė iš:
į:
Understanding the demicubes Pridėtos 439445 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 
Ištrintos 1822 eilutės:
* 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 443452 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 
Pakeistos 2122 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 
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 
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 
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 
Pridėtos 1819 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 
Pakeista 137 eilutė iš:
* [[http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf  Algebraic Curves: An Introduction to Algebraic Geometry]], William 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 
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 
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 
Pridėta 363 eilutė:
* Atiyah speculation: Space + Circle = 4 dimensions (Riemannian). Donaldson theory > geometric models of matter? Signature of 4manifold = electric charge. Second Betti number = number of protons + neutrons. 2017 spalio 25 d., 11:18
atliko 
Pridėtos 359362 eilutės:
Geometry challenges * Dimension 3: relate Jones quantum invariants (knots, any manifold) with PerlmanThurston. * Dimension 4: understand the structure of simplyconnected 4manifolds and the relation to physics. 2017 spalio 25 d., 11:07
atliko 
Pakeista 353 eilutė iš:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n1. Icosahedron is the fake sphere in 3dimensions 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 n1. Icosahedron is the fake sphere in 3dimensions 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 
Pakeista 353 eilutė iš:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n1. į:
* Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n1. Icosahedron is the fake sphere in 3dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture. 2017 spalio 25 d., 11:03
atliko 
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 n1. 2017 spalio 25 d., 10:55
atliko 
Pridėtos 349352 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 
Pakeistos 188190 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 202203 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 415418 eilutės:
* Unmarked opposites: crosspolytope. Each dimension independently + or  (all or nothing). * Cube: all vertices have a genealogy, a combination of +s and s. * Halfcube defines + for all, thus defines marked opposites. Pridėtos 421429 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 crosspolytope. (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). * Crosspolytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory). 2017 spalio 24 d., 07:43
atliko 
Pakeistos 187188 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 379380 eilutės iš
į:
* Trikampis  išauga požiūrių skaičius apibudinant: affinetaškai0, projectivetiesės1, conformalkampai2, symplecticplotai3. 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 
Pakeistos 117118 eilutės iš
į:
* squarerootofpi is gammaofnegativeonehalf (relate this to the volume of an odddimensional ball: pitothen/2 over (n/2)! Ištrinta 324 eilutė:
Pridėtos 375378 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 409411 eilutės:
Crosspolytope * A 0sphere is 2 points, much as generated by the center of a crosspolytope. We get a product of circles. And circles have no boundary. So there is no totality for the crosspolytope. Ištrintos 422423 eilutės:
Ištrinta 428 eilutė:
Ištrintos 488513 eilutės:
'''Figuring things out''' Crosspolytope * A 0sphere is 2 points, much as generated by the center of a crosspolytope. We get a product of circles. And circles have no boundary. So there is no totality for the crosspolytope. squarerootofpi is gammaofnegativeonehalf (relate this to the volume of an odddimensional ball: pitothen/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 
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 507510 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 
Pakeistos 451454 eilutės iš
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:
Ideas about transformations * 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:
į:
Other transformations Ištrintos 509511 eilutės:
Transformacijos sieja nepriklausomus matus. 2017 spalio 24 d., 07:22
atliko 
Pakeistos 7172 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 145147 eilutės:
Ideas * Fiber is a Zero. Pakeistos 185186 eilutės iš
į:
* Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, (kaip kad ramybe  lūkesčių nebuvimu), tai sutapatinama, kaip kad "crosscap". Pakeistos 195196 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 496510 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 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 508509 eilutės:
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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 
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 
Pridėtos 7075 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 
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 
Pridėtos 2427 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 
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 topdown and bottomup. 2017 spalio 04 d., 11:16
atliko 
Pridėta 116 eilutė:
** [[https://www.youtube.com/watch?v=MASKnQriQo&list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by  Videos on YouTube]] 2017 spalio 04 d., 11:12
atliko 
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 
Pridėtos 1617 eilutės:
Why are rings important for geometry rather than just groups? 2017 spalio 04 d., 10:27
atliko 
Pakeistos 1429 eilutės iš
* 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 crosspolytope Generalize this result to ndimensions (starting with 4dimensions): [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/Lectures/L12.html  Full finite symmetry groups in 3 dimensions]] į:
Pridėtos 337348 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 crosspolytope. * 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 ndimensions (starting with 4dimensions): [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/Lectures/L12.html  Full finite symmetry groups in 3 dimensions]] >><< 2017 spalio 04 d., 10:24
atliko 
Ištrintos 1722 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 1921 eilutės iš
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Pridėtos 155163 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 3sphere looks like as we pass through it from time t = 1 to 1. >><< 2017 spalio 04 d., 10:21
atliko 
Ištrintos 1518 eilutės:
* 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 1718 eilutės iš
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Pakeistos 3435 eilutės iš
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Pridėtos 125130 eilutės:
>>bgcolor=#FFFFC0<< * How is the Zariski topology related to the Binomial theorem? >><< Pridėtos 401408 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 
Pridėtos 101124 eilutės:
* Sylvain Poirer: Some key ideas, probably you know, but just in case: The (n+p1)dimensional projective space associated with a quadratic space with signature (n,p), is divided by its (n+p2)dimensional surface (images of null vectors), which is a conformal space with signature (n1,p1), into 2 curved spaces: one with signature (n1,p) and positive curvature, the other with dimension (n,p1) and negative curvature. Just by changing convention, the one with signature (n1,p) and positive curvature can also seen as a space with signature (p,n1) 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 180182 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 390393 eilutės iš
* 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 452454 eilutės iš
A 0sphere is 2 points, much as generated by the center of a crosspolytope. We get a product of circles. And circles have no boundary. So there is no totality for the crosspolytope. į:
Crosspolytope * A 0sphere is 2 points, much as generated by the center of a crosspolytope. We get a product of circles. And circles have no boundary. So there is no totality for the crosspolytope. Ištrintos 458482 eilutės:
Some key ideas, probably you know, but just in case: The (n+p1)dimensional projective space associated with a quadratic space with signature (n,p), is divided by its (n+p2)dimensional surface (images of null vectors), which is a conformal space with signature (n1,p1), into 2 curved spaces: one with signature (n1,p) and positive curvature, the other with dimension (n,p1) and negative curvature. Just by changing convention, the one with signature (n1,p) and positive curvature can also seen as a space with signature (p,n1) 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 467470 eilutės iš
* 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 486491 eilutės:
[+Geometry Intuition+] 2017 spalio 04 d., 10:12
atliko 
Pridėtos 145149 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 379395 eilutės iš
Relate to the six transformations in the anharmonic group of the shear map takes parallelogram to square, preserves area The * 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 But I don't know how to think of shear or squeeze mappings in terms of * Shear: * Squeeze: Note that the [[https://en 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/Crossratio  crossratio]]. If ratio is affine invariant, and crossratio 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 398400 eilutės iš
* Homotopy is translation. į:
Reflection Pridėtos 400401 eilutės:
Shear * Shear map takes parallelogram to square, preserves area Pridėtos 403415 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 homothetytranslations. 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 416417 eilutės:
* http://settheory.net/geometry#transf Pakeistos 418422 eilutės iš
* 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 425432 eilutės iš
* 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:
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Pakeistos 434436 eilutės iš
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 
Pridėtos 229235 eilutės:
Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry * perpendicularity via Appolonius polepolar 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: BeltramiKlein model of hyperbolic geometry Pakeistos 309315 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. 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 distancepreserving 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 Gstructure, 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 325326 eilutės iš
į:
Pakeista 337 eilutė iš:
* Affine geometry supposes the į:
* Affine geometry supposes the natural numbers Pridėtos 345346 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 349352 eilutės iš
[[https://en 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 356370 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 polepolar 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: BeltramiKlein 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 
Ištrinta 15 eilutė:
Pakeista 19 eilutė iš:
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Pridėtos 5055 eilutės:
>>bgcolor=#FFFFC0<< * Look at Wildberger's three binormal forms. >><< Pridėtos 158163 eilutės:
>>bgcolor=#FFFFC0<< * Relate sheaves and vector bundles. >><< Pridėtos 236240 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 
Pridėtos 6995 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 onedimensional 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 1dimensional subspace in a 2dimensional space or a 3dimensional 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 ndimensional 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:
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Ištrintos 327354 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 onedimensional 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 1dimensional subspace in a 2dimensional space or a 3dimensional 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 ndimensional 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 
Pakeistos 4546 eilutės iš
į:
[+Linear Algebra+] * Gelfand, [[https://www.amazon.com/LecturesLinearAlgebraDoverMathematics/dp/0486660826  Lectures on Linear Algebra]] Pakeistos 6366 eilutės iš
į:
* [[http://www.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] * [[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 7879 eilutės iš
į:
* Learn: affine complex varieties Pakeistos 9394 eilutės iš
* ** 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 110112 eilutės iš
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/eulercharacteristics.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 117118 eilutės iš
* 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, ladic, crystalline) * spectrum  topology, cohomology Pridėtos 125128 eilutės:
[+Differential Geometry+] * [[https://en.wikipedia.org/wiki/Vector_bundle  Vector bundle]] Pakeistos 149150 eilutės iš
* [[http://www.ams.org/bull/20013804/S0273097901009132/S0273097901009132.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/20013804/S0273097901009132/S0273097901009132.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/GeometryRevealedLesterJSenechalebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me=  Berger: Geometry Revealed]] Pakeistos 153155 eilutės iš
į:
* [[https://www.amazon.com/GeometryEuclidBeyondUndergraduateMathematics/dp/0387986502  Robin Hartshorne Geometry: Euclid and Beyond]] * [[https://en.m.wikipedia.org/wiki/Foundations_of_geometry  Foundations of geometry]] Pridėtos 171182 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/thetensymmetriesofspacetime/  Ten symmetries of space time]] * [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson * [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime 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 194196 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 230270 eilutės iš
* Geometry  Others ** [[http://www.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] ** 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/thetensymmetriesofspacetime/  Ten symmetries of space time]] ** [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson ** [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime 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/eulercharacteristics.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, ladic, 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 
Pakeista 7 eilutė iš:
* To į:
* To distinguish four geometries: affine, projective, conformal and symplectic. Pakeistos 8991 eilutės iš
į:
* [[http://blogs.ams.org/mathgradblog/2017/07/16/ideascheme/  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 96105 eilutės iš
* [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic  Lectures on the Geometric Anatomy of Theoretical Physics]] * [[http://math.ucr.edu/home/baez/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 * [[http://www.alainconnes.org/en/downloads.php  Alain Connes downloads]] į:
Pakeistos 98123 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 3Manifolds]] * [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic  Lectures on the Geometric Anatomy of Theoretical Physics]] * [[http://math.ucr.edu/home/baez/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 * [[http://www.alainconnes.org/en/downloads.php  Alain Connes downloads]] Books Pakeistos 131132 eilutės iš
į:
* [[http://www.springer.com/us/book/9783642192241  A Royal Road to Algebraic Geometry]] Pakeistos 228234 eilutės iš
* [[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/ideascheme/  The Idea of a Scheme]] į:
2017 rugsėjo 22 d., 12:13
atliko 
Pridėtos 129135 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 141166 eilutės:
Symplectic geometry * Symplectic geometry is an even dimensional geometry. It lives on even dimensional spaces, and measures the sizes of 2dimensional objects rather than the 1dimensional 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 BohrSommerfeld 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 335365 eilutės:
* 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 2dimensional objects rather than the 1dimensional 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 BohrSommerfeld 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 
Pridėtos 5461 eilutės:
Affine and Projective Geometry * [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] * [[https://www.amazon.com/IntroductionGeometryWileyClassicsLibrary/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 7480 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 nearsolutions 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 8587 eilutės iš
į:
* Sheaves ** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks Pridėtos 114116 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 121126 eilutės iš
į:
* Sylvain Poirer ** [[http://settheory.net/geometry  Geometry]] ** [[http://settheory.net/geometryaxioms  Geometry axioms]] ** [[http://spoirier.lautre.net/no12.pdf  Geometry: in French]] ** [[http://spoirier.lautre.net/no3.pdf  Geometry: in French]] Pakeistos 132140 eilutės iš
* Sylvain Poirer ** * ** [[http 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 141147 eilutės iš
** [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] * [[https://www.amazon.com/IntroductionGeometryWileyClassicsLibrary/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 145147 eilutės iš
** [[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 156157 eilutės iš
** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks į:
Pakeistos 210211 eilutės iš
į:
Pridėtos 465470 eilutės:
[+Geometry Intuition+] 2017 rugsėjo 22 d., 11:54
atliko 
Pakeistos 4655 eilutės iš
* [[http://athome.harvard.edu/threemanifolds/watch.html  The Geometry of 3Manifolds]] * [[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/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 5455 eilutės:
Books Pakeistos 6063 eilutės iš
į:
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by  NPTEL videos on algebraic geometry]] Books Ištrinta 68 eilutė:
Pakeistos 7173 eilutės iš
* Geometry į:
[+Other Geometry+] Videos * [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0ilf3fYXF5woAbrzJ  Tadashi Tokieda, Topology and Geometry]] * [[http://athome.harvard.edu/threemanifolds/watch.html  The Geometry of 3Manifolds]] * [[https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic  Lectures on the Geometric Anatomy of Theoretical Physics]] * [[http://math.ucr.edu/home/baez/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 * [[http://www.alainconnes.org/en/downloads.php  Alain Connes downloads]] Books [+Relating Geometries+] '''History of Geometry''' Books Pakeistos 9497 eilutės iš
į:
'''Organizing Geometry''' * [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf  Felix Klein, Erlangen program]] Pakeistos 101110 eilutės iš
į:
[+Conformal Geometry+] Books * [[http://mokslasplius.lt/files/GeometrineAlgebra/GA/GA.html  Geometrinė algebra]] 2017 rugsėjo 22 d., 11:45
atliko 
Pakeistos 4344 eilutės iš
į:
[++Geometry to study++] * [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0ilf3fYXF5woAbrzJ  Tadashi Tokieda, Topology and Geometry]] * [[http://athome.harvard.edu/threemanifolds/watch.html  The Geometry of 3Manifolds]] * [[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/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 142152 eilutės iš
* [[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_09L0ilf3fYXF5woAbrzJ  Tadashi Tokieda, Topology and Geometry]] * [[http://athome.harvard.edu/threemanifolds/watch.html  The Geometry of 3Manifolds]] * [[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/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 
Pridėtos 49 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 
Pakeista 73 eilutė iš:
* Algebraic į:
* [[Algebraic Geometry]] 2017 liepos 28 d., 20:07
atliko 
Pridėta 102 eilutė:
* [[http://blogs.ams.org/mathgradblog/2017/07/16/ideascheme/  The Idea of a Scheme]] 2017 vasario 07 d., 16:10
atliko 
Pridėta 43 eilutė:
* [[https://www.springer.com/la/book/9783034808972  5000 Years of Geometry: Mathematics in History and Culture]]] Offers indepth insights on geometry as a chain of developments in cultural history. 2017 sausio 24 d., 00:33
atliko 
Pridėtos 3839 eilutės:
http://mokslasplius.lt/files/GeometrineAlgebra/GA/GA.html 2017 sausio 22 d., 15:22
atliko 
Pridėtos 387392 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 
Pakeista 94 eilutė iš:
* 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 
Pakeista 93 eilutė iš:
* į:
* [[http://www.springer.com/us/book/9783642192241  A Royal Road to Algebraic Geometry]] 2016 gruodžio 15 d., 20:12
atliko 
Ištrinta 98 eilutė:
2016 gruodžio 15 d., 20:10
atliko 
Pridėtos 9399 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 13 d., 23:22
atliko 
Pridėtos 93104 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_09L0ilf3fYXF5woAbrzJ  Tadashi Tokieda, Topology and Geometry]] * [[http://athome.harvard.edu/threemanifolds/watch.html  The Geometry of 3Manifolds]] * [[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/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.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 
Pridėtos 7992 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, ladic, 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 
Pridėtos 4951 eilutės:
** [[http://spoirier.lautre.net/no12.pdf  Geometry: in French]] ** [[http://spoirier.lautre.net/no3.pdf  Geometry: in French]] * [[https://www.amazon.com/GeometryRevealedLesterJSenechalebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me=  Berger: Geometry Revealed]] 2016 gruodžio 13 d., 22:38
atliko 
Pridėtos 6775 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/eulercharacteristics.pdf  Understanding Euler Characteristic]], Ong Yen Chin 2016 gruodžio 13 d., 22:34
atliko 
Pakeistos 3766 eilutės iš
''' į:
'''Geometry to study''' * Geometry * [[http://www.ams.org/bull/20013804/S0273097901009132/S0273097901009132.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://wwwhistory.mcs.stand.ac.uk/~john/  John O'Connor]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/index.html  Topics in Geometry]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/MT4521/index.html  Geometry and Topology]] ** [[http://www.alainconnes.org/en/downloads.php  Alain Connes downloads]] * Sylvain Poirer ** [[http://settheory.net/geometry  Geometry]] ** [[http://settheory.net/geometryaxioms  Geometry axioms]] * Affine and Projective Geometry ** [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] * [[https://www.amazon.com/IntroductionGeometryWileyClassicsLibrary/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.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] ** 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/thetensymmetriesofspacetime/  Ten symmetries of space time]] ** [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson ** [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli 2016 gruodžio 13 d., 22:31
atliko 
Ištrintos 3637 eilutės:
Pakeistos 39168 eilutės iš
* [[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 * [[http://www.alainconnes.org/en/downloads.php  Alain Connes downloads]] * Geometry * http://www.ams.org/bull/20013804/S0273097901009132/S0273097901009132.pdf ** [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf  Felix Klein, Erlangen program]] * [[http://wwwhistory.mcs.stand.ac.uk/~john/  John O'Connor]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/index.html  Topics in Geometry]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/MT4521/index.html  Geometry and Topology]] * Sylvain Poirer ** [[http://settheory.net/geometry  Geometry]] ** [[http://settheory.net/geometryaxioms  Geometry axioms]] * Affine and Projective Geometry ** [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] * [[https://www.amazon.com/IntroductionGeometryWileyClassicsLibrary/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.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] ** 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/thetensymmetriesofspacetime/  Ten symmetries of space time]] ** [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson ** [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime 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/whatifanythingunifiesstablehomotopytheoryandgrothendieckssixfunctor  What unifies stable homotopy theory and six functors]] ** [[http://www.math.univtoulouse.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/eulercharacteristics.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/mathematicalmethods/studentsguidevectorsandtensors?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/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart ** [[https://arxiv.org/pdf/hepth/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/variablessets  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/GeometryRevealedLesterJSenechalebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me=  Berger: Geometry Revealed]] * Bott http://mathoverflow.net/questions/8800/proofsofbottperiodicity * http://mathematics.stanford.edu/wpcontent/uploads/2013/08/BosmanHonorsThesis2012.pdf 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_09L0ilf3fYXF5woAbrzJ  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 3Manifolds]] * [[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/qgfall2007/  Geometric Representation Theory]] * [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html  Alain Connes: Noncommutative Geometry and Physics]] * [[https://m.facebook.com/notes/scienceandmathematics/mathematicslecturevideosforundergraduatesandgraduates/321667781262483/  Math Videos]] * [[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/videocatalogue  Clay Mathematics Videos]] * https://m.youtube.com/watch?v=lJGUMlgCxz8 cheng 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/grothendiecksyogaofsixoperationsinrelativelybasicterms  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, ladic, 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/intuitivemeaningofexactsequence * 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 207208 eilutės:
shear map takes parallelogram to square, preserves area 2016 gruodžio 12 d., 16:03
atliko 
Pridėta 48 eilutė:
* http://www.ams.org/bull/20013804/S0273097901009132/S0273097901009132.pdf 2016 gruodžio 08 d., 22:22
atliko 
Pridėtos 437440 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 
Pridėtos 435436 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 
Pridėta 162 eilutė:
* http://cheng.staff.shef.ac.uk/misc/4000.pdf cheng architecture of math 2016 gruodžio 07 d., 00:08
atliko 
Pridėta 161 eilutė:
* http://cheng.staff.shef.ac.uk/morality/morality.pdf cheng math, morally 2016 gruodžio 06 d., 11:52
atliko 
Pridėta 160 eilutė:
* Length six https://arxiv.org/pdf/0906.1286v2 2016 gruodžio 06 d., 11:38
atliko 
Pridėta 159 eilutė:
* Exact sequences http://math.stackexchange.com/questions/419329/intuitivemeaningofexactsequence 2016 gruodžio 05 d., 06:36
atliko 
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 04 d., 22:56
atliko 
Pridėta 130 eilutė:
* https://m.youtube.com/watch?v=lJGUMlgCxz8 cheng 2016 lapkričio 25 d., 18:57
atliko 
Pridėta 157 eilutė:
* http://math.ucr.edu/home/baez/octonions/octonions.html 2016 lapkričio 23 d., 14:23
atliko 
Pakeistos 418425 eilutės iš
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 "crosscap". 2016 lapkričio 22 d., 21:57
atliko 
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 
Pakeistos 154155 eilutės iš
* 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 
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 21 d., 23:18
atliko 
Pridėta 110 eilutė:
* http://mathematics.stanford.edu/wpcontent/uploads/2013/08/BosmanHonorsThesis2012.pdf 2016 lapkričio 21 d., 23:06
atliko 
Pridėtos 152156 eilutės:
Challenges * a unifying perspective on cohomology * higher order homotopy groups for sphere * field with one element 2016 lapkričio 21 d., 23:04
atliko 
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 
Pridėtos 8183 eilutės:
** [[https://homotopical.files.wordpress.com/2014/06/ctsaghandout.pdf  Cohomology theories in motivic stable homotopy theory]] ** [[http://mathoverflow.net/questions/170319/whatifanythingunifiesstablehomotopytheoryandgrothendieckssixfunctor  What unifies stable homotopy theory and six functors]] ** [[http://www.math.univtoulouse.fr/~dcisinsk/DM.pdf  Triangulated Categories of Mixed Motives]] Pridėtos 8586 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 
Pridėtos 400401 eilutės:
Our Father relates a left exact sequence and a right exact sequence. 2016 lapkričio 19 d., 22:18
atliko 
Pridėtos 144145 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 
Pridėtos 3132 eilutės:
Relate Cayley's theorem to the field with one element 2016 lapkričio 18 d., 06:49
atliko 
Pridėta 102 eilutė:
* Bott http://mathoverflow.net/questions/8800/proofsofbottperiodicity 2016 lapkričio 17 d., 23:05
atliko 
Pridėtos 360361 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 
Pridėtos 8081 eilutės:
* Sheaves ** https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks 2016 lapkričio 12 d., 15:55
atliko 
Pridėta 138 eilutė:
* Constructiveness  closed sets any intersections and finite unions are open sets constructive 2016 lapkričio 09 d., 17:02
atliko  2016 lapkričio 05 d., 16:35
atliko 
Pakeistos 177180 eilutės iš
* Path geometry is given by A + B + C = 0 gets you back where you started from. * Angle geometry gives this a total value of * 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 threedimensional 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 
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 
Pridėtos 175180 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 
Pakeista 117 eilutė iš:
* http://www.claymath.org/library/video į:
* [[http://www.claymath.org/library/videocatalogue  Clay Mathematics Videos]] 2016 spalio 30 d., 22:58
atliko 
Pridėta 117 eilutė:
* http://www.claymath.org/library/videocatalogue 2016 spalio 28 d., 21:11
atliko 
Pridėtos 297298 eilutės:
Relate to the six transformations in the anharmonic group of the [[https://en.wikipedia.org/wiki/Crossratio  crossratio]]. If ratio is affine invariant, and crossratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant? 2016 spalio 26 d., 16:05
atliko 
Pridėtos 384393 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 
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 
Pakeistos 116117 eilutės iš
į:
* symplectic geometry videos Pridėta 136 eilutė:
* spectrum  topology, cohomology 2016 spalio 25 d., 22:36
atliko 
Pridėtos 305306 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 
Pridėtos 305306 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 
Pakeistos 214215 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 
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 
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 
Pakeistos 134135 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 301302 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 
Pridėta 99 eilutė:
* [[https://www.amazon.com/GeometryRevealedLesterJSenechalebook/dp/B00DGEFHL4/ref=mt_kindle?_encoding=UTF8&me=  Berger: Geometry Revealed]] 2016 spalio 25 d., 21:13
atliko 
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 
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 
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 
Pakeistos 132133 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 
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 
Pridėta 77 eilutė:
** [[https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology  Coherent sheaf cohomology]] 2016 spalio 21 d., 05:00
atliko 
Pakeistos 12 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 
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 onedimensional 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 
Pakeistos 287289 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 
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 
Pakeistos 163167 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 
Pakeistos 278283 eilutės iš
The 6 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 
Pakeistos 272273 eilutės iš
''' 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 277278 eilutės:
The 6 specifications 2016 spalio 15 d., 14:33
atliko 
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 
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 
Pridėtos 203204 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 
Pakeistos 272273 eilutės iš
Homotopy is translation. į:
* Harmonic analysis, periodic functions, circle are rotation. * Homotopy is translation. 2016 spalio 12 d., 23:53
atliko 
Pridėtos 2930 eilutės:
Generalize this result to ndimensions (starting with 4dimensions): [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/Lectures/L12.html  Full finite symmetry groups in 3 dimensions]] 2016 spalio 12 d., 22:59
atliko 
Pridėtos 6869 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 8586 eilutės:
** [[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 
Pakeistos 4260 eilutės iš
* * * [[http:// * [[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/eulercharacteristics.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.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] * [[http://www.wall.org/~aron/blog/thetensymmetriesofspacetime/  Ten symmetries of space time]] * [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson * [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli į:
* Geometry ** [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf  Felix Klein, Erlangen program]] * [[http://wwwhistory.mcs.stand.ac.uk/~john/  John O'Connor]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/index.html  Topics in Geometry]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/MT4521/index.html  Geometry and Topology]] Pakeista 48 eilutė iš:
** [[http://settheory.net/geometry  į:
** [[http://settheory.net/geometry  Geometry]] Pakeistos 5061 eilutės iš
* ** [[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/mathematicalmethods/studentsguidevectorsandtensors?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/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart * [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] į:
* Affine and Projective Geometry ** [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] Pakeistos 5358 eilutės iš
* * * [[http:// * ** * 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.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] ** 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/thetensymmetriesofspacetime/  Ten symmetries of space time]] ** [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson ** [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime 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/eulercharacteristics.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/mathematicalmethods/studentsguidevectorsandtensors?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/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart ** [[https://arxiv.org/pdf/hepth/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/variablessets  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 
Pakeistos 7981 eilutės iš
* http://wwwhistory.mcs.stand.ac.uk/~john/ į:
* [[http://wwwhistory.mcs.stand.ac.uk/~john/  John O'Connor]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/geometry/index.html  Topics in Geometry]] ** [[http://wwwhistory.mcs.stand.ac.uk/~john/MT4521/index.html  Geometry and Topology]] 2016 spalio 11 d., 15:24
atliko 
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 
Pridėtos 175176 eilutės:
These geometries show how to relate (ever more tightly) two distinct dimensions. Pakeistos 236237 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 
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 
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 
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 
Pridėtos 48 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 
Pridėta 74 eilutė:
* http://wwwhistory.mcs.stand.ac.uk/~john/geometry/index.html 2016 spalio 06 d., 23:52
atliko 
Pridėta 73 eilutė:
* [[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]] 2016 spalio 04 d., 15:52
atliko 
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 
Pridėtos 7071 eilutės:
* [[http://math.stackexchange.com/questions/204533/applicationsofthefundamentaltheoremsofaffineandprojectivegeometry?rq=1  Fundamental theorems of affine and projective geometry]] * [[https://www.amazon.com/IntroductionGeometryWileyClassicsLibrary/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 
Pakeistos 6667 eilutės iš
* [[ * į:
* [[http://www.cambridge.org/gb/academic/subjects/physics/mathematicalmethods/studentsguidevectorsandtensors?format=PB&isbn=9780521171908  A Student's Guide to Vectors and Tensors]] Ištrinta 83 eilutė:
2016 spalio 03 d., 00:17
atliko 
Pridėta 57 eilutė:
** [[http://settheory.net/geometry  Sylvain Poirer]] 2016 spalio 02 d., 21:09
atliko 
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 
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 
Pridėta 68 eilutė:
* [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart 2016 spalio 02 d., 20:30
atliko 
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 
Pridėtos 299300 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 
Pridėtos 2627 eilutės:
shear map takes parallelogram to square, preserves area 2016 rugsėjo 28 d., 07:59
atliko 
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 
Pridėta 65 eilutė:
* http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/ 2016 rugsėjo 28 d., 01:17
atliko 
Pridėta 64 eilutė:
* http://www.cambridge.org/gb/academic/subjects/physics/mathematicalmethods/studentsguidevectorsandtensors?format=PB&isbn=9780521171908 2016 rugsėjo 27 d., 14:26
atliko 
Pakeistos 6263 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., 14:18
atliko 
Pridėtos 2023 eilutės:
Bernstein polynomials * x = 1/2 get simplex * x = 1/3 or 2/3 get cube and crosspolytope 2016 rugsėjo 27 d., 14:13
atliko 
Pridėtos 5759 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 
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 
Pakeista 47 eilutė iš:
* [[ Symmetry and the Standard Model]] Matthew Robinson į:
* [[http://link.springer.com/book/10.1007%2F9781441982674  Symmetry and the Standard Model]] Matthew Robinson 2016 rugsėjo 26 d., 11:00
atliko 
Pridėta 47 eilutė:
* [[ Symmetry and the Standard Model]] Matthew Robinson 2016 rugsėjo 19 d., 15:54
atliko 
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 
Pridėta 93 eilutė:
* the relationship between two spaces, for example, points, lines, planes 2016 rugsėjo 19 d., 15:11
atliko 
Pakeistos 277278 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 
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 
Pakeista 90 eilutė iš:
* the relationship between our old and new 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 
Pridėtos 218221 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 
Pridėta 90 eilutė:
* the relationship between our old and new search 2016 rugsėjo 13 d., 12:36
atliko  2016 rugsėjo 13 d., 12:36
atliko 
Pridėta 152 eilutė:
* https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume 2016 rugsėjo 13 d., 12:15
atliko 
Pridėta 217 eilutė:
* http://settheory.net/geometry#transf Pakeistos 228229 eilutės iš
į:
* https://en.m.wikipedia.org/wiki/Möbius_transformation combines translation, inversion, reflection, rotation, homothety 2016 rugsėjo 13 d., 12:04
atliko 
Pridėtos 162164 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 
Pakeistos 4748 eilutės iš
* [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli į:
* [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli * Sylvain Poirer ** [[http://settheory.net/geometryaxioms  Geometry axioms]] ** [[http://settheory.net/foundations/variablessets  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 239263 eilutės:
'''Sylvain Poirer''' Some key ideas, probably you know, but just in case: The (n+p1)dimensional projective space associated with a quadratic space with signature (n,p), is divided by its (n+p2)dimensional surface (images of null vectors), which is a conformal space with signature (n1,p1), into 2 curved spaces: one with signature (n1,p) and positive curvature, the other with dimension (n,p1) and negative curvature. Just by changing convention, the one with signature (n1,p) and positive curvature can also seen as a space with signature (p,n1) 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 
Pridėta 47 eilutė:
* [[https://www.amazon.com/SymmetriesGroupTheoryParticlePhysics/dp/3642154816  Symmetries and Group Theory in Particle Physics: An Introduction to SpaceTime and Internal Symmetries]] by Giovanni Costa, Gianluigi Fogli 2016 rugsėjo 12 d., 14:06
atliko 
Pridėta 46 eilutė:
* [[http://www.wall.org/~aron/blog/thetensymmetriesofspacetime/  Ten symmetries of space time]] 2016 rugsėjo 11 d., 09:35
atliko 
Pakeista 1 eilutė iš:
[[Geometry illustrations]] į:
[[Geometry illustrations]], [[Universal hyperbolic geometry]] 2016 rugsėjo 10 d., 16:48
atliko 
Pridėta 45 eilutė:
* [[http://www.cuttheknot.org/geometry.shtml  Geometry at CuttheKnot]] 2016 rugpjūčio 30 d., 22:15
atliko 
Pridėta 19 eilutė:
* How is the Zariski topology related to the Binomial theorem? 2016 rugpjūčio 30 d., 11:30
atliko 
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:
* a quadratic subject, with quadratic concepts: quadrance and spread. (Norman Wildberger) 2016 rugpjūčio 27 d., 14:50
atliko 
Pridėtos 5556 eilutės:
* [[http://www.alainconnes.org/en/videos.php  Alain Connes: The Music of Shapes]] * [[http://www.theorie.physik.unimuenchen.de/activities/special_lecture_s/lectureseries_connes/videos_connes/index.html  Alain Connes: Noncommutative Geometry and Physics]] 2016 rugpjūčio 27 d., 14:45
atliko 
Pridėta 28 eilutė:
* [[http://www.alainconnes.org/en/downloads.php  Alain Connes downloads]] 2016 rugpjūčio 25 d., 23:45
atliko 
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 
Pakeistos 4041 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 
Pridėta 42 eilutė:
* [[http://math.ucr.edu/home/baez/information/index.html  Information Geometry]] by John Baez 2016 rugpjūčio 23 d., 12:17
atliko 
Pridėtos 207211 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 
Pridėtos 4041 eilutės:
* [[http://arxiv.org/abs/1509.02145  The six operations in equivariant motivic homotopy theory]] Marc Hoyois 2016 rugpjūčio 22 d., 17:05
atliko 
Pridėta 45 eilutė:
* [[https://video.ias.edu/univalent/voevodsky  Univalent Foundations of Mathematics]] 2016 rugpjūčio 22 d., 15:42
atliko 
Pakeistos 1718 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 3sphere looks like as we pass through it from time t = 1 to 1. 2016 rugpjūčio 22 d., 14:12
atliko 
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 
Pridėta 44 eilutė:
* [[http://athome.harvard.edu/threemanifolds/watch.html  The Geometry of 3Manifolds]] Pridėta 66 eilutė:
* [[ 2016 rugpjūčio 21 d., 10:00
atliko 
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 
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 
Pridėta 14 eilutė:
* What is the topological quotient for an equilateral triangle or a simplex? 2016 rugpjūčio 21 d., 09:38
atliko 
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 
Pridėtos 206207 eilutės:
Quotient is gluing is equivalence on a boundary. 2016 rugpjūčio 21 d., 09:35
atliko 
Pakeista 205 eilutė iš:
squarerootofpi is gammaofnegativeonehalf (relate this to the volume of an odddimensional ball) į:
squarerootofpi is gammaofnegativeonehalf (relate this to the volume of an odddimensional ball: pitothen/2 over (n/2)! 2016 rugpjūčio 21 d., 09:34
atliko 
Pridėtos 204205 eilutės:
squarerootofpi is gammaofnegativeonehalf (relate this to the volume of an odddimensional ball) 2016 rugpjūčio 21 d., 05:56
atliko 
Pakeista 203 eilutė iš:
A 0sphere is 2 points, much as generated by the center of a crosspolytope. į:
A 0sphere is 2 points, much as generated by the center of a crosspolytope. We get a product of circles. And circles have no boundary. So there is no totality for the crosspolytope. 2016 rugpjūčio 21 d., 05:52
atliko 
Pridėta 13 eilutė:
* What happens to the corners of the shapes? 2016 rugpjūčio 21 d., 05:46
atliko 
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 
Pridėtos 200201 eilutės:
A 0sphere is 2 points, much as generated by the center of a crosspolytope. 2016 rugpjūčio 21 d., 04:56
atliko 
Pridėtos 196199 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 
Pridėta 36 eilutė:
* [[https://www.youtube.com/playlist?list=PLC37ED4C488778E7E  Universal Hyperbolic Geometry]] 2016 rugpjūčio 20 d., 23:54
atliko 
Pakeista 18 eilutė iš:
į:
Overviews and History Pakeistos 2223 eilutės iš
* [[http:// į:
* [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf  Felix Klein, Erlangen program]] Geometry Pakeista 27 eilutė iš:
* į:
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf  Combinatorial Algebraic Topology]] Ištrinta 29 eilutė:
Pakeistos 3435 eilutės iš
* * [[http į:
Videos ** UnivHypGeom4: First steps in hyperbolic geometry: fundamental results * [[https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0ilf3fYXF5woAbrzJ  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/qgfall2007/  Geometric Representation Theory]] * [[https://m.facebook.com/notes/scienceandmathematics/mathematicslecturevideosforundergraduatesandgraduates/321667781262483/  Math Videos]] 2016 rugpjūčio 18 d., 16:26
atliko 
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 
Pridėta 56 eilutė:
* how to expand our vision (from a smaller space to a larger space) 2016 rugpjūčio 17 d., 14:44
atliko 
Pakeistos 1112 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 
Pridėta 34 eilutė:
* [[http://matematicas.unex.es/~navarro/erlangenenglish.pdf  Felix Klein, Erlangen program]] 2016 rugpjūčio 17 d., 13:54
atliko 
Pakeista 32 eilutė iš:
* https:// į:
* [[https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf  Sheaf Theory by Tom Lovering]] 2016 rugpjūčio 17 d., 13:47
atliko 
Pakeistos 1819 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 
Pridėta 18 eilutė:
* Pridėtos 3841 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, ladic, 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 
Pridėtos 3436 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/grothendiecksyogaofsixoperationsinrelativelybasicterms  Six operations at Math Stack Exchange]] 2016 rugpjūčio 17 d., 11:08
atliko 
Pridėtos 910 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 
Pridėtos 2829 eilutės:
* https://www.google.lt/url?sa=t&source=web&rct=j&url=https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf&ved=0ahUKEwjwxIa68sfOAhULWywKHUzzAJ0QFgglMAE&usg=AFQjCNGnhOOwYoNt9SVQU3NAP59oy3fdWQ * Catster videos 2016 rugpjūčio 16 d., 16:22
atliko 
Pridėtos 7882 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 
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 
Pakeistos 2728 eilutės iš
į:
Pridėtos 126137 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 BohrSommerfeld 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 
Pakeistos 2728 eilutės iš
į:
* [[ Pridėtos 121126 eilutės:
* Symplectic geometry is an even dimensional geometry. It lives on even dimensional spaces, and measures the sizes of 2dimensional objects rather than the 1dimensional 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 
Pridėtos 2526 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 
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 
Pridėta 49 eilutė:
* Algebraic geometry is the study of spaces of solutions to algebraic equations. 2016 rugpjūčio 16 d., 06:46
atliko 
Pakeistos 913 eilutės iš
* [[https://en.m.wikipedia.org/wiki/Incidence_structure  Incidence structure]] * https://en.m.wikipedia.org/wiki/Ordered_geometry į:
Pakeistos 2932 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 7980 eilutės iš
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 
Pakeistos 1618 eilutės iš
Works to study į:
'''Works to study''' Readings and videos Pridėtos 3033 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 7980 eilutės:
[[https://en.m.wikipedia.org/wiki/Foundations_of_geometry  Absolute geometry]], also known as neutral geometry, 2016 rugpjūčio 16 d., 06:34
atliko 
Pridėtos 913 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 137141 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 
Pakeistos 127128 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 homothetytranslations. 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 homothetytranslations. 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 
Pridėtos 4748 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 5658 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. 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 6465 eilutės iš
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 8586 eilutės iš
į:
* Given any field F,2 one can construct the ndimensional 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 99102 eilutės:
Conformal geometry Symplectic geometry Pridėtos 109110 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 137161 eilutės:
  Given any field F,2 one can construct the ndimensional 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 
Pakeistos 2324 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 distancepreserving 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 Gstructure, 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  Crosspolytopes (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. Crosspolytopes 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 * Crosspolytopes (1+2)^N * Cubes (2+1)^N * Halfcubes (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 6973 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 9498 eilutės:
* Two different coordinate systems agree on the origin 0. Pakeistos 99105 eilutės iš
* Simplex (1+1)^N * Crosspolytopes (1+2)^N * Cubes (2+1)^N * Halfcubes (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 107108 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 111120 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 LeviCivita 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 homothetytranslations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L. Ištrintos 123152 eilutės:
'''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  Crosspolytopes (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. Crosspolytopes 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 125131 eilutės:
[[https://en.m.wikipedia.org/wiki/Affine_transformation  Affine transformation]] * Pakeistos 129152 eilutės iš
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 LeviCivita 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 homothetytranslations. 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 132137 eilutės iš
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 137140 eilutės iš
* 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 154177 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 distancepreserving 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 Gstructure, 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 
Pakeistos 3436 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 4344 eilutės iš
į:
Compare to: BeltramiKlein model of hyperbolic geometry Ištrinta 122 eilutė:
2016 rugpjūčio 16 d., 05:58
atliko 
Pridėtos 310 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 5657 eilutės iš
į:
* Simplex (1+1)^N * Crosspolytopes (1+2)^N * Cubes (2+1)^N * Halfcubes (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 99104 eilutės iš
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 105108 eilutės iš
* 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 193196 eilutės iš
* Crosspolytopes (1+2)^N * Cubes (2+1)^N * Halfcubes (2+2)^N į:
2016 rugpjūčio 16 d., 05:44
atliko 
Pakeistos 1516 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 1dimensional subspace in a 2dimensional space or a 3dimensional 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 4249 eilutės iš
* 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 8991 eilutės iš
Projective geometry: way of embedding a 1dimensional subspace in a 2dimensional space or a 3dimensional 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 
Pridėta 13 eilutė:
* [[http://www.math.nus.edu.sg/~matwml/courses/Graduate/MA5209%20Algebraic%20Topology/Interesting_Stuff/eulercharacteristics.pdf  Understanding Euler Characteristic]], Ong Yen Chin 2016 rugpjūčio 14 d., 23:23
atliko 
Pridėta 5 eilutė:
* [[http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf  Combinatorial Algebraic Topology]] 2016 rugpjūčio 14 d., 22:54
atliko 
Pridėta 4 eilutė:
* [[http://www.alainconnes.org/docs/maths.pdf  A View of Mathematics]], Alain Connes 2016 rugpjūčio 13 d., 16:03
atliko 
Pridėta 10 eilutė:
* [[https://www.youtube.com/playlist?list=PLbMVogVj5nJSNj24jdPGivlJtxbxua2by  NPTEL videos on algebraic geometry]] 2016 rugpjūčio 13 d., 12:20
atliko 
Pakeistos 89 eilutės iš
į:
* Robin Hartshorne Geometry: Euclid and Beyond * Robin Hartshorne, Algebraic Geometry 2016 rugpjūčio 13 d., 09:33
atliko 
Pakeista 39 eilutė iš:
į:
Look at Wildberger's three binormal forms. 2016 rugpjūčio 11 d., 19:02
atliko 
Pridėta 33 eilutė:
* [1:2:0] is a point that is a "direction" (two directions) 2016 rugpjūčio 11 d., 18:29
atliko 
Pakeistos 3134 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 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 
Pakeistos 1217 eilutės iš
* * * į:
Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry * perpendicularity via Appolonius polepolar 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 2024 eilutės iš
* * 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 3940 eilutės:
2016 rugpjūčio 11 d., 13:22
atliko 
Pakeistos 2728 eilutės iš
(1 + ti)(1 + ti) = (1  t2) + (2t) i is the parametrization of the circle. * What about the sphere? 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 
Pakeista 29 eilutė iš:
* Note also that infinity is the flip side of zero  they make a 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 
Pakeistos 2728 eilutės iš
(1 + ti)(1 + ti) = (1  t2) + (2t) i is the parametrization of the circle. What about the sphere? 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 
Pakeistos 2728 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 
Pridėtos 2627 eilutės:
(1 + ti)(1 + ti) = (1  t2) + (2t) i is the parametrization of the circle. 2016 rugpjūčio 11 d., 11:02
atliko 
Pridėtos 2325 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 
Pakeistos 711 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 polepolar 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: BeltramiKlein 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 135144 eilutės:
* UnivHypGeom4: First steps in hyperbolic geometry: fundamental results * hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry ** perpendicularity via Appolonius polepolar 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: BeltramiKlein model of hyperbolic geometry * Euclidean geometry: empty space + tools: straightedge, compass, area measurer 2016 rugpjūčio 10 d., 11:02
atliko 
Pakeista 128 eilutė iš:
į:
compare to: BeltramiKlein model of hyperbolic geometry 2016 rugpjūčio 10 d., 10:58
atliko 
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 
Pridėtos 125129 eilutės:
** perpendicularity via Appolonius polepolar duality: dual of point is line and vice versa 2016 rugpjūčio 10 d., 10:46
atliko 
Pakeistos 121122 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 
Pridėtos 4748 eilutės:
A vector subspace needs to contain zero. How is this related to projective geometry? 2016 rugpjūčio 08 d., 10:47
atliko 
Pakeistos 312 eilutės iš
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 liepos 28 d., 14:34
atliko 
Pridėtos 3637 eilutės:
Projective geometry: way of embedding a 1dimensional subspace in a 2dimensional space or a 3dimensional 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 
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 
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 
Pridėtos 3639 eilutės:
Transformations * Translation  does not affect vectors * Dilation (scaling) including negative (flipping) 2016 liepos 16 d., 16:26
atliko 
Pridėtos 123128 eilutės:
 * Simplex (1+1)^N * Crosspolytopes (1+2)^N * Cubes (2+1)^N * Halfcubes (2+2)^N 2016 liepos 08 d., 11:23
atliko 
Pakeistos 2931 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:
į:
 2016 birželio 28 d., 10:43
atliko 
Pakeistos 3536 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 
Pridėta 35 eilutė:
[[https://en.m.wikipedia.org/wiki/Incidence_geometry  Incidence geometry]] 2016 birželio 26 d., 23:06
atliko 
Pakeistos 2122 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 2542 eilutės iš
I I wish for my philosophical speculations to be more fruitful. I And why, intuitively, ...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 3339 eilutės iš
* * scaling * homothety * similarity * reflection * rotation * shear į:
* 2016 birželio 26 d., 22:40
atliko 
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 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 
Pridėta 28 eilutė:
I wish for my philosophical speculations to be more fruitful. 2016 birželio 26 d., 22:37
atliko 
Pakeistos 2324 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 2729 eilutės iš
tensors and triviality į:
2016 birželio 26 d., 22:35
atliko 
Pakeistos 1516 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]], 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 2125 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 
Pridėtos 1421 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  Crosspolytopes (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. Crosspolytopes 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 
Pakeistos 5254 eilutės iš
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 LeviCivita 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 
Pakeista 47 eilutė iš:
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 
Pakeista 32 eilutė iš:
į:
[[https://en.m.wikipedia.org/wiki/Affine_transformation  Affine transformation]] Pakeistos 4142 eilutės iš
į:
[[https://en.wikipedia.org/wiki/Squeeze_mapping  Squeeze mapping]] Pakeistos 5457 eilutės iš
http://www.math.ucr.edu/home/baez/groupoidification/ Erlangen program https://en 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 
Pakeistos 3343 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  2016 birželio 26 d., 15:49
atliko 
Pakeistos 2429 eilutės iš
* į:
* recopy whole * rescale whole * rescale multiple * redistribute set * redistribute multiple * redistribute whole 2016 birželio 26 d., 15:43
atliko  2016 birželio 26 d., 15:43
atliko 
Pridėtos 2333 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  2016 birželio 26 d., 15:20
atliko 
Pridėtos 1417 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 
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 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 
Pridėtos 4041 eilutės:
http://www.math.ucr.edu/home/baez/groupoidification/ Erlangen program 2016 birželio 25 d., 01:08
atliko 
Pridėtos 3337 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 
Pridėtos 2932 eilutės:
Isometry Special conformal is reflection and inversion 2016 birželio 25 d., 01:01
atliko 
Pridėtos 1928 eilutės:
Translation is an affine transformation Reflection Rotation Scaling Shear 2016 birželio 25 d., 00:50
atliko 
Pridėtos 2228 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 homothetytranslations. 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 
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 
Pridėtos 1920 eilutės:
...[[https://en.wikipedia.org/wiki/Cartan_connection  Cartan connection]] 2016 birželio 24 d., 19:13
atliko 
Pakeista 18 eilutė iš:
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 
Pridėtos 1418 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 
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 
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 
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 
Pridėtos 115 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 4548 eilutės iš
* [[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 4957 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 5960 eilutės iš
* 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 6465 eilutės iš
* 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 
Pridėtos 34 eilutės:
'''Pairs of geometries''' Pridėtos 69 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 
Pridėtos 47 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 
Pakeistos 531 eilutės iš
Given any field F 2 n F n 1 n F n 1 modulo multiplication by nonzero scalars. į:
Given any field F,2 one can construct the ndimensional 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 1929 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 distancepreserving 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 Gstructure, 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 
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., 08:47
atliko 
Pridėta 3 eilutė:
Think of pairs of geometries as defining equivalence classes variously. 2016 birželio 23 d., 08:46
atliko 
Pridėtos 230 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 
Pridėtos 1314 eilutės:
https://en.m.wikipedia.org/wiki/Homography two approaches to projective geometry with fields or without 2016 birželio 22 d., 17:18
atliko 
Pridėtos 117 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! >><< 
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Puslapis paskutinį kartą pakeistas 2019 balandžio 29 d., 00:34
