Geometry theorems, Geometries, Geometry illustrations, Universal hyperbolic geometry
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.
- 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?
- In category theory, where do symmetric functions come up? What are eigenvalues understood as? What would be symmetric functions of eigenvalues?
- 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?
- 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.
- Relate sheaves and vector bundles.
- 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 use the tetrahedron as a way to model the 4th dimension so as to imagine how a trefoil knot could be untangled.
- Try to imagine what a 3-sphere looks like as we pass through it from time t = -1 to 1.
- How is the Zariski topology related to the Binomial theorem?
Bernstein polynomials
- x = 1/2 get simplex
- x = 1/3 or 2/3 get cube and cross-polytope
Generalize this result to n-dimensions (starting with 4-dimensions): Full finite symmetry groups in 3 dimensions
Relate Cayley's theorem to the field with one element
Geometry to study
http://mokslasplius.lt/files/GeometrineAlgebra/GA/GA.html
- Geometry
- Pierre Cartier: Mad Day's Work: From Grothendieck to Connes and Kontsevich, The Evolution of Concepts of Space and Symmetry
- 5000 Years of Geometry: Mathematics in History and Culture] Offers in-depth insights on geometry as a chain of developments in cultural history.
- John O'Connor
- Sylvain Poirer
- Berger: Geometry Revealed
- Affine and Projective Geometry
- Introduction to Geometry by Coxeter.
- Norman Wildberger
- Geometry - Others
- Symplectic Geometry
- Symmetry
- Algebraic Geometry
- Sheaves
- Algebraic Topology
- Motives and Universal cohomology. Weil cohomology theory and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
- Grothendieck
- Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
- List of geometry topics
- Foundations of geometry
- Conformal geometric algebra includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
- Incidence geometry
- Incidence structure
- Ordered geometry
- Vector bundle
- The Geometrization conjecture and the eight Thurston geometries. Also, the Bianchi classification of low dimensional Lie algebras.
- Versor and sandwiching.
- 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
- A 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
Geometry Videos
Geometry
Geometry is:
- the ways that our expectations can be related, thus how we are related to each other
- 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.
- how to expand our vision (from a smaller space to a larger space) (Tadashi Tokieda)
- how to embed a lower dimensional space into a higher dimensional space
- the ways that a vector space is grounded
- the relationship between two spaces, for example, points, lines, planes
- the construction of sets of roots of polynomials
- a quadratic subject, with quadratic concepts: quadrance and spread. (Norman Wildberger)
- Grothendieck categories
Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.
- 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 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 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.
- Algebraic geometry is the study of spaces of solutions to algebraic equations.
I am somewhat aware of Felix Klein's Erlangen program whereby we consider transformation groups which leave geometric properties invariant, and also groupoidification and geometric representation, moving frames, 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
Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity. And a different way of relating two dimensions.
Each geometry is the action of a monoid, thus a language. But that monoid may contain an inverse, which distinguishes the projective geometry from the affine geometry.
In a free monoid the theorems are equations and they are determined by what can be done with associativity. This is first order logic. A second order logic or higher order logic would be given by what can be expressed, for example, by counting various possibilities.
Consider a trigon with 3 directed sides A, B, C:
- Path geometry is given by A + B + C = [0] gets you back where you started from. It is geometry without space, as when God thinks why, so that everything is connected by relationships, and God of himself only thinks forwards, unfolding.
- Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc. Barycentric coordinates for vectors v1, v2, v3 (with scalars lambda l1, l2, l3) where the scalars are between 0 and 1 and the sum l1v1 + l2v2 + l3v3 = 1 on the triangle and <1 within it and all are 1/3 to get the center, the average. For example, a line in a plane splits that plane into two sides, just as a plane splits a three-dimensional space. Thus this is where "holes" come from, disconnections, emptiness, homology.
- Angle geometry gives this a total value of 1, the total angle. And so we can accord to A, B, C a ratio that measures the opposite angle. This creates the inside and the outside of the triangle. Indeed, the three lines carves the plane into spaces. It's not clear how they meet at infinity.
- Area geometry assigns an oriented area AREA to the total value. Time arises as we have one side and the other swept by it.
- 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.”
- John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
- Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.
Four infinite families of polytopes can be distinguished by how they are extended in each new dimension. They seem to relate to four different geometries and four different classical Lie algebras:
- An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality. This is geometry without any field, and without any zeros - what does this mean for the correspondence with the polynomial ring?
- Cn - Cross-polytopes (such as the octahedron) are extended when the Center creates two new vertices ("opposites") and thereby defines a line in two directions, which is preserved by projective geometry. Cross-polytopes have a Center but no Totality.
- Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center. 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).
- Simplex (1+1)^N
- Cross-polytopes (1+2)^N
- Cubes (2+1)^N
- Half-cubes (2+2)^N
- Affine geometry supposes the integers
- Projective geometry supposes the rationals
- Conformal (Euclidean) geometry supposes the reals
- Symplectic geometry supposes the complexes
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!
Distinct Geometries
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.
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.
These geometries show how to relate (ever more tightly) two distinct dimensions.
Affine geometry
- Allowing only positive "coefficients" is related to positive definiteness, convexity.
- Does not assume Euclid's third and fourth axioms.
- Different coordinate systems don't agree on any origin.
- Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
- In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
- Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
- In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
- https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume
- https://en.m.wikipedia.org/wiki/Motive_(algebraic_geometry) related to the connection between affine and projective space
- Tiesė perkelta į kitą tiesę išsaugoja trijų taškų paprastą santykį (ratio).
Projective geometry
- Projective geometry relates one plane (upon which the projection is made) with another plane (where the "eye" is, the zero where all the lines come from). And thus the line through the eye which is parallel to the plane needs to be added. Thus we can have homogeneous coordinates. And we have the decomposition of projective space into a sum of affine spaces of each dimension. Projective geometry is the space of one-dimensional subspaces, and they all include zero, thus they are the lines which go through zero. Or the hyperplanes which go through zero.
- Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
- In projective geometry, vectors are points and bivectors are lines.
- Projective geometry transforms conics into conics.
- projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
- "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
- [1:2:0] is a point that is a "direction" (two directions)
- A vector subspace needs to contain zero. How is this related to projective geometry? Vector spaces: Two different coordinate systems agree on the origin 0.
- Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
- https://en.m.wikipedia.org/wiki/Homography Homography two approaches to projective geometry with fields or without
- A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
- Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
- Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
Projective geometry adds points at infinity to affine geometry. Conformal geometry or inversive geometry adds a distinguished circle. Symplectic geometry adds an area product. Moebius strip plays with the distinguished circle changing orientation if you go around.
Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
- perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
- orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
- most important theorem: Pythagoras q=q1+q2 - q1q2
- second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3
Compare to: Beltrami-Klein model of hyperbolic geometry
Euclidean geometry: empty space + tools: straightedge, compass, area measurer
- most important theorem: Pythagoras q=q1+q2
- (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
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.
- Moebius transformations revealed.
Symplectic geometry
- Symplectic geometry is an even dimensional geometry. It lives on even dimensional
spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional
lengths and angles that are familiar from Euclidean and Riemannian geometry. It is
naturally associated with the field of complex rather than real numbers. However, it
is not as rigid as complex geometry: one of its most intriguing aspects is its curious
mixture of rigidity (structure) and flabbiness (lack of structure). What is Symplectic Geometry? by Dusa McDuff
- McDuff: First of all, what is a symplectic structure? The concept arose in the study of classical
mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a
falling apple. The trajectory of such a system is determined if one knows its position
and velocity (speed and direction of motion) at any one time. Thus for an object
of unit mass moving in a given straight line one needs two pieces of information, the
position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.
- momentum x position is angular momentum
- McDuff: This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a
single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld
quantization principle says that the area of this region is quantized, i.e. it has to be
an integral multiple of a number called Planck’s constant. Thus one can think of the
symplectic area as a measure of the entanglement of position and velocity.
- Symplectic area is orientable.
- Area (volume) is a pseudoscalar such as the 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.
(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.
Pairs of Geometries
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.
Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.
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.
I expect that there are six transformations by which one geometry reinterprets a perspective from another geometry. And I imagine that they are as intuitive as the various ways that we interpret multiplication in arithmetic. I suppose that they may include translation, rotation, scaling, homothety, similarity, reflection and shear.
Relate to the six transformations in the anharmonic group of the cross-ratio. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?
shear map takes parallelogram to square, preserves area
The 6 specifications can be compared with cinematographic movements of a camera.
- Reflection: a camera in a mirror, a frame within a frame...
- Rotation: a camera swivels from left to right, makes a choice, like turning one's head
- Dilation: a camera zooms for the desired composition.
- Translation: a camera moves around.
But I don't know how to think of shear or squeeze mappings in terms of a camera. However, consider what a camera would do to a tiled floor.
- Shear:
- Squeeze: the camera looks out onto the horizon.
Note that the classification of elements of SL2(R) includes elliptic (conjugate to a rotation), parabolic (shear) and hyperbolic (squeeze). Similarly, see the classification of Moebius 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?
- Harmonic analysis, periodic functions, circle are rotation.
- Homotopy is translation.
- 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)
- http://settheory.net/geometry#transf
- 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.
- 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
- Squeeze mapping
- Isometry
- Special conformal is reflection and inversion
- 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.
- Affine transformation
- Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
- https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
- 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
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.
A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball: pi-to-the-n/2 over (n/2)!
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.
Sylvain Poirer
Some key ideas, probably you know, but just in case:
The (n+p-1)-dimensional projective space associated with a quadratic
space with signature (n,p), is divided by its (n+p-2)-dimensional
surface (images of null vectors), which is a conformal space with
signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p)
and positive curvature, the other with dimension (n,p-1) and negative
curvature.
Just by changing convention, the one with signature (n-1,p) and
positive curvature can also seen as a space with signature (p,n-1) and
negative curvature; and similarly for the other.
Affine representations of that quadric are classified by the choice of
the horizon, or equivalently the polar point of that horizon (the
point representing in the projective space the direction orthogonal to
that hyperplane). So there are 3 possibilities.
The null one sees it as a paraboloid and gives it an affine geometry.
The 2 others, with the different signs, see it as a quadric whose
center is the polar point, and give it 2 different curved geometries
We can understand the stereographic projection as the effect of the
projective transformation of the space, which changes the sphere into
a paraboloid, itself projected into an affine space.
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?
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.
Exact sequence
- Our Father relates a left exact sequence and a right exact sequence.
- Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.
Dievas žmogui yra skylė gyvenime, prasmė - neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje - homologijoje.
Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, tai sutapatinama, kaip kad "cross-cap".
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.
Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
Apibendrinimas yra "objekto" kūrimas.
Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia.
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.
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