Geometry theorems, Geometries, Geometry illustrations, Universal hyperbolic geometry

Overall goals:

- To understand what geometry contributes to the overall map of mathematics.http://www.ms.lt/sodas/Book/Geometry?action=diff
- To distinguish 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.

I should

- Make a list of geometry theorems and sort them by geometry.
- Make a list of geometries and show how they are related.
- Why are rings important for geometry rather than just groups? Because want to work with ideals and not subrings, because we are dealing with what is not as well as what is, because we are constructing both top-down and bottom-up.

Four kinds of geometry

**General notions**

- 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.

**Path geometry**

- 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.
- Unions of spaces.

**Line geometry**

- 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?
- Linear equations are intersections of hyperplanes.
- 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 bivectors are lines.
- Projective geometry transforms conics into conics.
- Fundamental theorems of affine and projective geometry
- Fundamental Theorem of Affine geometry. let {$X,X'$} be two finite dimensional affine spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:X\to X'$} be a bijection that sends collinear points to collinear points, i.e. such that for all {$a,b,c\in X$} that are collinear, {$f(a),f(b),f(c)$} are collinear too. Then {$f$} is a semi-affine isomorphism.
- This means that there is a field isomorphism {$\sigma:K\to K'$} such that for any point {$a\in X$} the map induced by {$f_a: X_a\to X'_{f(a)}$} is a {$\sigma$}-semi-linear isomorphism.
- Fundamental Theorem of Projective geometry. let {$P(X),P(X')$} be two finite dimensional projective spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:P(X)\to P(X')$} be a bijection that sends collinear points to collinear points. Then {$f$} is a semi-linear isomorphism.

- Projective geometry: Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).

**Angle geometry**

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

**Oriented area geometry**

Geometry to study

Linear Algebra

- 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.

- Gelfand, Lectures on Linear Algebra

Plane Geometry

- Look at Wildberger's three binormal forms.

Videos

- Norman Wildberger
- Triangle geometry
- Universal Hyperbolic Geometry
- UnivHypGeom4: First steps in hyperbolic geometry: fundamental results

- Algebraic calculus one

Affine and Projective Geometry

- Introduction to Geometry by Coxeter.
- Norman Wildberger
- Affine and projective universal geometry by Norman Wildberger
- One dimensional metrical geometry
- Chromogeometry

- Geometry at Cut-the-Knot
- 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.

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).
- 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.
- Affine geometry - free monoid - without negative sign (subtraction) - lattice of steps - such as Young tableaux as paths on Pascal's triangle.

- What does it mean that the point at infinity is a zero of a polynomial? Is that establishing and modeling the limiting process?
- The link between projective geometry and fractions (as equivalence classes).

- Projective Geometry. A Short Introduction
- Projective Geometry: From Foundations to Applications Beutelspacher and Rosenbaum

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 - 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.
- Sylvain Poirier: Some key ideas, probably you know, but just in case:

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

- Sylvain Poirier: 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 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.

- (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.
- 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)!

Classical Algebraic Geometry

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

Videos

- Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
- NPTEL videos on algebraic geometry

Books

- Algebraic Curves: An Introduction to Algebraic Geometry, William Fulton - section 2.10 has exercises about exact sequences
- Learn: affine complex varieties

Modern Algebraic Geometry

**Intuition**

Ravi Vakil: The intuition for schemes can be built on the intuition for affine complex varieties. Allen Knutson and Terry Tao have pointed out that this involves three different simultaneous generalizations, which can be interpreted as three large themes in mathematics.

- (i) We allow nilpotents in the ring of functions, which is basically analysis (looking at near-solutions of equations instead of exact solutions).
- (ii) We glue these affine schemes together, which is what we do in differential geometry (looking at manifolds instead of coordinate patches).
- (iii) Instead of working over C (or another algebraically closed field), we work more generally over a ring that isn’t an algebraically closed field, or even a field at all, which is basically number theory (solving equations over number fields, rings of integers, etc.).

http://drmichaelrobinson.net/sheaftutorial/index.html

Ideas

- Fiber is a Zero.

Videos

Books

- Grothendieck
- Robin Hartshorne, Algebraic Geometry

Sheaves

- https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks
- Sheaf Theory by Tom Lovering

Schemes

- The Idea of a Scheme
- The Geometry of Schemes, Isenbott and Harris, nicely illustrated concrete examples

Algebraic Topology

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

Videos

Books

- Allen Hatcher, Algebraic Topology - free on his website
- Combinatorial Algebraic Topology
- Understanding Euler Characteristic, Ong Yen Chin
- The Geometrization conjecture and the eight Thurston geometries. Also, the Bianchi classification of low dimensional Lie algebras.

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.
- Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, (kaip kad ramybe - lūkesčių nebuvimu), tai sutapatinama, kaip kad "cross-cap".
- 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.
- 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.

Homology and Cohomology

- Weibel, Homological Algebra
- Coherent sheaf cohomology
- Motives and Universal cohomology. Weil cohomology theory and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
- spectrum - topology, cohomology
- Our Father relates a left exact sequence and a right exact sequence.
- Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
- Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.
- Long exact sequence from short exact sequence: derived functors.
- Dievas žmogui yra skylė gyvenime, prasmė - neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje - homologijoje.
- Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?

Geometry and Logic

- Sheaves in Geometry and Logic, Medak and Macleigh

Differential Geometry

- Relate sheaves and vector bundles.

Other Geometry

Videos

- The Geometry of 3-Manifolds
- Lectures on the Geometric Anatomy of Theoretical Physics
- Geometric Representation Theory
- Alain Connes: The Music of Shapes
- Alain Connes: Noncommutative Geometry and Physics
- Video lectures on homological algebra

Noncommutative geometry

Books

Relating Geometries

**History of Geometry**

Books

- Pierre Cartier: Mad Day's Work: From Grothendieck to Connes and Kontsevich, The Evolution of Concepts of Space and Symmetry
- Berger: Geometry Revealed
- 5000 Years of Geometry: Mathematics in History and Culture] Offers in-depth insights on geometry as a chain of developments in cultural history.
- A Royal Road to Algebraic Geometry
- Robin Hartshorne Geometry: Euclid and Beyond
- Foundations of geometry

**Organizing Geometry**

Intuition

- 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.
- Felix Klein, Erlangen program
- John O'Connor
- Sylvain Poirier

Symmetry

- From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept
- Ten symmetries of space time
- Symmetry and the Standard Model Matthew Robinson
- Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries by Giovanni Costa, Gianluigi Fogli

Different geometries

Conformal Geometry

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.

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

Books

- Geometrinė algebra
- Conformal geometric algebra includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations
- Versor and sandwiching.

Symplectic Geometry

- 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?

Videos

- Symmetric geometry and classical mechanics, Tobias Osborne

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.

Books

- Clifford Algebras in Symplectic Geometry and Quantum Mechanics
- Generalized Clifford algebras: Orthogonal and symplectic cases

Defining 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

- 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.
- 2 dimensions - Scalar curvature R
- 3 dimensions - Ricci curvature Rij
- 4 dimensions - Riemann curvature Rijk

- 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!
- 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.

Geometry challenges

- Dimension 3: relate Jones quantum invariants (knots, any manifold) with Perlman-Thurston.
- Dimension 4: understand the structure of simply-connected 4-manifolds and the relation to physics.
- Atiyah speculation: Space + Circle = 4 dimensions (Riemannian). Donaldson theory -> geometric models of matter? Signature of 4-manifold = electric charge. Second Betti number = number of protons + neutrons.

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**

Ideas

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

**Four Basic Geometries**

Center and Totality

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

Generalize this result to n-dimensions (starting with 4-dimensions): Full finite symmetry groups in 3 dimensions

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?

**Distinct Geometries**

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.
- 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.

**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**

- 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?

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.

Sources to think about

- 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.
- Möbius transformation combines translation, inversion, reflection, rotation, homothety. See the classification of Moebius transformations. Note also that the 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 cross-ratio. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?
- The 6 specifications can be compared with cinematographic movements of a camera. But I don't know how to think of shear or squeeze mappings in terms of a camera. However, consider what a camera would do to a tiled floor. Shear? Squeeze: the camera looks out onto the horizon?
- Reflection: a camera in a mirror, a frame within a frame...
- Rotation: a camera swivels from left to right, makes a choice, like turning one's head
- Dilation: a camera zooms for the desired composition.
- Translation: a camera moves around.

- 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.

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.

Reflection

- Flip around our search, turn vector around: (reflection)

Shear

- Shear map takes parallelogram to square, preserves area
- Turn a corner into another dimension

Rotation

- Harmonic analysis, periodic functions, circle are rotation.
- Rotations are multiplicative but not additive. This brings to mind the field with one element.

Dilation

- Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively.

Complex number dilation (rotating).

- Homothety is related to dilation. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
- https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
- Dilation brings to mind the Cartesian product A x B. There is also the inner (direct) product A + B. How is it related to the disjoint union? And there is the tensor product which I think is like an expansion in terms of A.B and so is like multiplication.

Squeeze

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

Translation

- Homotopy is translation.
- Sweep a new dimension in terms of an old dimension (translation)
- Translation - does not affect vectors

Other transformations

- Special conformal is reflection and inversion
- Isometry
- 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.

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

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

Parsiųstas iš http://www.ms.lt/sodas/Book/Geometry

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