User:Dc.samizdat/Radially equilateral polychora

Author: David Brooks Christie

Plan
Envision this paper as collecting only my own (in some cases possibly original) observations, as concisely as possible, without the requirement of accessibility - not as an enyclopedic survey of the known properties of any polychora; for those see e.g. the 24-cell article to which I have made contributions.

Definition of radially equilateral polytopes (original terminology). 24-cell and 8-cell (tesseract) as the 4-dimensional instances. Unique that there are two in 4th dimension. Uniqueness of their succession.

The central canonical apex vertex (terminology).

24-cell as analogue of cubctahedron/rhombic dodecahedron, as symmetric union of the geometries of all the 0-4-dimensional polytopes except the {5}s, and the

The (three lower symmetry forms of the one) radially equilateral 4-honeycomb.

Hypercubic vertex chords of the 24-cell, and the unique radially equilateral hypercube. Intersection of vertex planes (cell faces) only at vertex chords (cell edges).

The orthogonal great circle polygons of the 24-cell as disjoint vertex sets (because of the way orthogonal planes intersect in 4-space as in 2-space).

The 3-fold greater amount of space in 4-space than in 3-space (the three sets of orthogonal axes). The chiral (isoclinic?) rotations among the 3 orientations.

The $\sqrt{3}$ chord of the 24-cell, the third great circle polygon, and the central tetrahedron as a degenerate (in the same sense as a honeycomb) 5-cell.

The central tetrahedron as 3/4-transimplex, vs. the 3/4-transcube (the 4-orthoplex); see also Trans polytopes.

The 3-fold inscribed 4-polytopes in the 24-cell, as isoclinic (or single?) rotations through π/? and π/? respectively.

Visualization of cell boundaries of interior 4-polytopes (in 24-cell) as polychoric rings (Hopf fibrations), their overlaps and interstices (not original; credit Cloudswrest).

Radially equilateral polytopes
Hexagon

Cuboctahedron and Tetrahedral-octahedral honeycomb, Bucky Fuller's Vector Equilibrium and Isotropic Vector Matrix

8-cell and 24-cell - two in dimension 4!

Is there a 5-space analog of the cubctahedron/rhombic dodecahedron dual that is radially equilateral? (Is 5-demicube radially equilateral?)

Radially equilateral space
In 3D, there is just one regular honeycomb, {4,3,4}, and it is not radially equilateral. In 4D, there are three regular honeycombs, but they are all forms of the same lattice, and it is radially equilateral. 4-space is radially equilateral, as 2-space, 3-space and 5-space and higher are not.

Because $\sqrt{4}$ is 2, four dimensional space itself is radially equilateral. The vertex positions of unit-edge-length 24-cells, 8-cells, and 16-cells anywhere in the honeycomb have coordinate values which are always integers or half-integers. If the edge length is taken to be 2 instead of 1, all vertex coordinates become integers, and the hamming distance between any two vertices (their separation measured rectilinearly, along the orthogonal edges of the lattice) is always the same integer as their geodesic distance. This is not true in 2-space, 3-space, or 5-space, where $\sqrt{n}$ is irrational.

