Monday, October 09, 2023

About uniform tessellations of hyperbolic, Euclidean, spherical space

It has become clear that 3-D hyperbolic tessellations are probably very important in the TGD framework. The 3-D hyperbolic space H3 is realized as a mass shell or cosmic time constant hyperboloid, and an interesting conjecture is that particles with a fixed value of mass squared are associated with the vertices of a hyperbolic tessellation. This would give rise to a quantization of momenta consistent with the number theoretic discretization. Hyperbolic tessellations could also appear in cosmological scales. So-called icosatetrahedral tessellation seems to provide a model of genetic code and DNA and suggests that genetic code is not restricted in biology but universal and realized in all scales Euclidean.

Uniform tilings/tessellations/honeycombs can be identified by their vertex configuration given a list n1.n2.n3... for the numbers n of vertices of regular n-polygons associated with the vertex. Uniform tilings can be regular, meaning that they are both vertex- and edge-transitive. Quasi-regular tessellations are only vertex-transitive and semiregular tessellations are neither vertex- nor face-transitive. Non-regular tessellations give, for instance, Archimedean solids obtained from Platonic solids by operations like truncation. In this case, the vertices are obtained by symmetries from each other but not the faces, which need not be identical anymore.

There exists an extremely general construction of tessellation of hyperbolic, Euclidean, and spherical tessellations which works at least in 2- and 3-D cases known as Wyuthoff construction. In the 2-D case, this construction is based on the so-called Schwarz triangles associated with a fundamental region of tessellation in the 2-D case and so-called Goursat tetrahedrons in the 3-D case. A natural generalization is that in n-dimensional one has n-simplexes. One would have what topologists call triangulation, which is very special in the sense that it utilizes the symmetries of the tessellation. These very special simplices are also consistent with the number theoretical constraints in the angles between n-1-faces correspond to angles defined by the roots of unity.

In the 2-D case, the angles between the edges of the fundamental triangle are rational multiples of π so that the cosines and sines of the angle are algebraic numbers, which are natural for a tessellation whose points in natural coordinates (momenta) have components that are numbers in an algebraic extension of rationals. In the 2-D case, the fundamental triangle is obtained by drawing from center points of the 2-D unit cell, say a regular polygon, connecting it to its vertices. In the 3-D case, the same is done for the 3-D unit cells of the fundamental region. Note that the tessellation can have several different types of unit cells and this is indeed true in the case of icosatetrahedral tessellations.

2-dimensional case

In the 2-D case, the angles between the edges of the triangle are given as (1/p,1/r,1/s)-multiples of π. p, r, and s are the orders of discrete rotation groups assignable to the vertices. They are generated by the reflections si with respect to edges of the triangle in one-to-one correspondence with opposite vertices. They satisfy the conditions si2=1 as reflections and the reflections si and si+j, j>1, commute and si si+1 generates a rotation with respect to the third vertex of the triangles with order determined by one of the numbers p, r, s. The conditions can be summed up to si2=1 and (si sj)mij=1, mij=2 for j ≠ i+/-1 and mij> 2 for j=i+/-1.

The conditions can be expressed in a concise way by using Coxeter-Dynkin diagrams having 3 vertices connected by edges. For mij=2, there is no edge, and for mij> 2, there is an edge and a number telling the order of the cyclic group in question.

All these 3 spaces are constant curvature spaces with positive, vanishing, or negative curvature, which is reflected as properties of the angle sum of the geodesic Schwartz triangle (note that these spaces also occur in cosmology). In the spherical case, the sum is larger than π and one has 1/p+1/r+1/s≥1. In the Euclidean case, the sum of the angles of the Schwarz triangle is π, which gives the condition 1/p+1/r+1/s=1. In the hyperbolic case, the angle sum is smaller than π and one has 1/p+1/r+1/s/le;1. Note that in the hyperbolic plane, the angles of infinitely sized Schwartz triangle can vanish (ideal triangle).

For the 2-sphere, these conditions give only Platonic solids as regular (vertex- and face-transitive) tessellation (no overlap between triangles). For the plane, the non-compactness implies that the conditions are not so restrictive as for the sphere. The most symmetric tessellations are regular tessellations: they involve only one kind of polygon and are vertex-, edge-, and face-transitive. For the Euclidean plane, there are regular tessellations by triangles, squares, and hexagons. If one weakens the transitivity conditions to say vertex-transitivity, more tessellations are possible and involve different kinds of regular polygons.

The Wikipedia article about the uniform tilings of the hyperbolic plane gives a good overall view of the uniform tessellations of the hyperbolic plane. For the hyperbolic tessellations, the conditions are the least restrictive. Intuitively, this is due to the fact that the angle sum can be small, and this allows small angles between edges and more degree of freedom at vertices. For a spherical tessellation, the situation is just the opposite. Uniform tilings of hyperbolic plane H2 are by definition vertex-transitive and have a constant distance between neighboring vertices. This condition is physically natural and would correspond to mechanical equilibrium in which vertices are connected by springs of the same string tension. Each symmetry (p, r, s) allows 7 uniform tilings characterized by Wythoff symbol or Coxeter diagram. These tiling, in general, contain several kinds of geodesic polygons. Families with r=2 (right triangle) contain hyperbolic regular tilings.

The 3-dimensional case

There is a Wikipedia article about the uniform tessellations/honeycombs in the 3-D case, obtained by Wyuthoff construction, is a generalization from the 2-D case. Schwarz triangle is replaced with Goursat tetrahedron, and reflections are now in tetrahedral faces opposite to the vertices of the tetrahedron so that there are 4 reflections si satisfying si2=1 and (si sj)mij=1, mij=2 for j ≠ i +/-1. The cyclic subgroups act as rotations of faces meeting at the edges, and the angles defining the cyclic groups are dihedral angles. There are 9 compact Coxeter groups, and they define uniform tessellations with a finite fundamental domain. What is interesting is that the cyclic subgroups involved do not have order larger than 5.

The conditions are expressible in terms of Coxeter-Dynkin diagram with 4 vertices. The 2-D conditions are satisfied for the Schwarz triangles defining the faces of the tetrahedron. Besides the angle parameters defining the triangular phases of the tetrahedron, there are angle parameters defining the angles between the faces. All these angles are rational multiples of π and define subgroups of the symmetries of the tessellation. What is so beautiful is that the construction is generalized to higher dimensions and is recursive/hierarchical.

The hyperbolic character of the geometry allows Schwarz triangles and Goursat tetrahedra which in Euclidian case would not be possible due to the condition that the edges have the same length and faces have the same area.

Could hyperbolic, Euclidean, and spherical tessellations be realized in TGD space-time

An interesting question is whether the hyperbolic, Euclidean, and spherical tessellations could be realized in the TGD framework as induced 3-D geometry or rather, as slicing of space-time surface by time parameter such that each slice represents hyperbolic, Euclidean or spherical geometry locally allowing the tessellation.

Hyperbolic tessellations can be realized on the cosmic time constant hyperboloids and Euclidean tessellations on the Minkowski time constant hyperplanes of M4 and possibly partially on 3-surfaces which have hyperbolic 3-space as M4 projection.

The question boils down to a construction of a model of Robertson-Walker cosmology for which the induced metric of a=constant 3-surface is that of H3, E3, or S3 corresponding to the cosmologies with subcritical, critical and overcritical mass densities. The metric of H3 is proportional to a2 scale factor. The simplest ansatz is a geodesic circle at geodesic sphere S2⊂CP2 with metric ds2= -R22-sin2(θ)dΦ2. The ansatz (sin(θ)=a/a0,Φ=f(r)) gives in Robertson-Walker coordinates the induced metric

ds2= [1- R2 (dθ/da)2] da2 -a2 (1/(1+r2)+ (R/a0)2(df/dr)2) dr2 + r22

This gives the flat metric of E3 if the condition

(df/dr)2= (a0/R)2 r2/(1+r2)

This condition is satisfied for all values of r.

For S3 metric one obtains the condition

(df/dr)2= (a0/R)2 2r2/(1-r4)

r=1 corresponds to singularity. For r=1, one has rM= ar= a, which gives t= 21/2a. One can construct the S3 by gluing together the hemispheres corresponding to the 2 roots for df/dr so that it seems that one obtains the tessellations. The divergence of df/dr tells that the half-spheres become orthogonal to H3 at the gluing points.

For both E3 and S3 option, the component gaa of the induced metric is equal to

gaa= 1-(R/a0)2 1/(1-(a/a0)2)

gaa diverges at a=a0 so that the cosmic time would run infinitely fast. gaa changes sign for a=a0 so that for a>a0 the signature of the induced metric becomes Euclidean. Unless one allows Euclidean signature in long scales, one must assume a0. Note that the action defined as the sum of Kähler action and volume action. If S2 corresponds to the homologically trivially geodesic sphere of CP2, the action reduces to volume action for these surfaces. The densities of Noether currents for volume action vanish at a= a0 since they are proportional to the factor (gaa)1/2gaa and thus approach to zero like [1-(a/a0)2]1/2. This is true also for the contribution of Kähler action present for homologically non-trivial geodesic sphere of CP2. Very probably, this surface is not a minimal surface although the volume is finite. This is suggested by the fact that the volume element increases in comparison to hyperbolic volume element giving rise to minimal volume increases as the parameter a increases.

See the chapter More about TGD and Cosmology.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

No comments: