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Affine space

Affine space Any vector space may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations.

This suggests that the unitary space is the quintessential affine space as the special linear Lie algebra is built on displacements and has no zero.

Four Geometries

I am trying to distinguish four geometries: affine, projective, conformal, symplectic.

Ideas

• Erlangen program identifies a geometry with the group of transformations which do not change it. I want to identify a geometry with the monoid of actions that can be taken within it. It is a free monoid in the case of affine geometry but becomes a group with the introduction of inverses, transformations.

Six Specifications

Projective geometry

The set of linear transformations, without restriction. It can be identified with the vector space.

Wikipedia: Theorems in projective geometry

• Pappus's theorem
• Pascal's theorem
• Desargues theorem

Conformal geometry

• Why and how is Universal Hyperbolic Geometry related to conformal geometry?
• A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series.
• When are two vectors, lines, etc. perpendicular? When they are distinguished by i ?
• Conformal groups. Orthogonal.
• Terrence Tao: It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another.
• Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.

Symplectic geometry

Symplectic establishes the distinction between inside and outside (orientation).

Similarity geometry

A similarity transformation or similitude is an affine map which preserves angles.

A similarity transformation can be written Translation (Ta) composed with positive scalar (lambda) x Orthogonal (linear) transformation. Since it preserves angles, all vectors must be stretched by the same amount, lambda.

The well-known theorem of Pythagoras can be proved by "similar triangle" methods.

Euclidean geometry

Group I(Rn)

Transformations

Analyze Wikipedia category: Transformations and the article Wikipedia: Transformations.

Požiūrio

• reflection: atskyrimas - vienybė - požiūrio atskyrimas nuo visumos
• shear: išvertimas - dvejybė
• rotation: išplėtimas - trejybė
• dilation: atsisakymas - ketverybė
• squeeze: apvertimas - penkerybė
• translation: įsisavinimas - šešerybė
• sudūrimas - septynerybė
• suvedimas grandine, pasiklydusiu vaiku - nulybė?

I am looking for 6 transformations of perspective which link the structure of one geometry with the dynamics of another geometry. I think these 6 transformations relate to ways of interpreting multiplication, to restructurings of sequences, hierarchies and networks, and to the axioms of set theory which define sets.

• The (affine) whole is (projectively) recopied.
• The (affine) whole is (conformally) rescaled.
• The (projective) multiple is (conformally) rescaled.
• The (conformal) set is (symplectically) redistributed.
• The (projective) multiple is (symplectically) redistributed.
• The (affine) whole is (symplectically) redistributed.

Transformations in cinematography

• Shear: sideshot

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?
• Trikampis - išauga požiūrių skaičius apibudinant: affine-taškai-0, projective-tiesės-1, conformal-kampai-2, symplectic-plotai-3.

Consider a triangle 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.

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 natural numbers
• 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!

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.

• Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).
• Cube: all vertices have a genealogy, a combination of +s and -s.
• Half-cube defines + for all, thus defines marked opposites.

Cross-polytope

• A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
• Symmetric group action on an octahedron is marked, 1 and -1, the octahedron itself is unmarked.
• Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the cross-polytope. (Or consider Bernstein's polynomials.)

Understanding the 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.?

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.

Dual:

• Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory).
• Cross-polytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory).
• 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 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?

Notes

• Projective map - preserves a single line (what does that mean?)
• Conformal map - preserves pairs of lines (the angle between them)
• Symplectomorphism - preserves triplets of lines (their oriented volume), the noncollinearity of a triangle

Four coordinate systems

• In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective.
• Geometry: Options for introducing a coordinate system (none, one, two, three). No coordinate system is the case of tensors.
• 1 coordinate system = 1 side of a triangle = Length. Shrinking the side can lead to a point - the two points become equal. This is like homotopy?
• 2 coordinate systems = 2 sides of a triangle = Angle. Note that turning (rotating) one side around by 2 pi gets it back to where it was, and this is true for each 2 pi forwards and backwards. So by this equivalence we generate the integers Z as the winding numbers.
• 3 coordinate systems = 3 sides of a triangle = Oriented area (the systems are ordered). What equivalence does this support and what does it yield? Is it related to e?

All you can do

• with 0 coordinate system (affine)
• with 1 coordinate system (projective) is reflection,
• with 2 coordinates (conformal) is rotation and shear, (the origins match)
• with 3 coordinates (symplectic) is dilation (scales change), squeeze (scales change), translation (origins move).

The coordinate systems (0,1,2,3) separate the level and metalevel. Study the 6 transformations between these sets of coordinate systems.

• How do the 4 geometries (in terms of coordinate systems) relate to the 4 classical root systems?
• Substantiate: Affine geometry defines no perspective, projective geometry defines one perspective, conformal geometry defines a perspective on a perspective, symplectic geometry defines a perspective on a perspective on a perspective.
• Relate motion to bundles. Symplectic geometry, looseness, etc. All 4 geometries.
• Think of harmonic pencil types as the basis for the root systems
  • {$A_n$} {$\pm(x-y)$} dual
  • {$B_n$} {$\pm x,\pm y$} yields {$x\pm y$}
  • {$C_n$} {$x\pm y$} yields {$\pm2x,\pm2y$}
  • {$D_n$} {$\pm(x\pm y)$} dual dual
• In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math?
• Determinant expresses oriented volume, oriented area. Real numbers: distances. Complex numbers: angles. What do quaternions express?
• V: Vladimir I. Arnold. Polymathematics: complexification, symplectification and all that. 1988 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe." 39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem.
• Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list?
• Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.)
• Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world.
• A_n points and sets
• B_n inside: perpendicular (angles) and
• C_n outside: line and surface area
• D_n points and position
• Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?

Steven Lehar

• Clifford algebra: A Visual Introduction A Clifford Algebra over {$ℝ^3$} may describe the rigid motions in space (namely, conjugation acts as a reflection by a plane).
• A Clifford Algebra over {$ℝ^4$} may describe the projective geometry in space.
• A Clifford Algebra over {$ℝ^5$} may describe the conformal geometry in space.
• Symplectic?
• Double Conformal Mapping: A Finite Mathematics to Model an Infinite World
• Coexter labeled a fourth family as {$\delta_n$}, the infinite tessellations of hypercubes.
• https://en.wikipedia.org/wiki/Hypercubic_honeycomb

Geometry and logic: relation between level and metalevel is given by the number of coordinate systems

• 0 coordinate systems. Overlay of level and metalevel yields contradiction.
• 1 coordinate system. Metalevel yields a model for the level.
• 2 coordinate systems. Metalevel allows for reversal of actions in level.
• 3 coordinate systems. Metalevel allows for definition of variables in level.
• affine relates ?
• projective relates circle and line
• conformal relates (Riemannian) sphere and (complex) plane
• symplectic relates S3 ? and (quaternionic?) R3 ?

4 logics for 4 geometries

• no simplification - no distance between metalevel and level - affine - contradiction
• simplify by one perspective relative to the center - get model
• simplify by two perspectives - get directions, forward and backward
• simplify by three perspectives - get variables, defined from by the side view

Relations between geometries

• Sylvain Poirier: 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.
• Time is no perspective, space is p, p2, p3, these are the four geometries. So time is affine.

Time is affine, without origin, a clock, a loop. Time defines what steps mean. It is the lack of a coordinate system. Whereas each dimension of space adds a coordinate system, yielding projective (one coordinate system), conformal (two coordinate system), symplectic (three coordinate system) geometries. Space is finite and constructured by adding a perspective. Whereas we cut time, we don't extend beyond it. Time is infinite by nature. The speed of light makes space from time.

Indian time

• No coordinate system - time
• First coordinate system - perspective - spatial dimension
• Second coordinate system - perspective on perspective - a second spatial dimension
• Third coordinate system - perspective on perspective on perspective - a third spatial dimension
• Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1. Icosahedron is the fake sphere in 3-dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture. Icosahedron would be in A5 but reality is given by A4 and so A5 is insolvable!
• Symplectic geometry (P3) expresses slack. Affine geometry (P0) expresses everything (slack at the limits of everything). What do P1 (projective geometry includes the boundary limit of everything) and P2 express

John Baez. Week 181. Grassmanians. An series are symmetry groups of projective geometry and the Bn and Dn series are symmetry groups of conformal geometry, the Cn series are symmetry groups of "projective symplectic" geometry.