User:Tomruen/Regular polychoric rings

Three of the six regular 4-polytopes – 8-cell (tesseract), 24-cell, and 120-cell – can each be partitioned into disjoint great circle (regular polygon) rings of cells forming discrete Hopf fibrations of these polytopes. The tesseract partitions into two interlocking rings of four cubes each. The 24-cell partitions into four rings of six octahedra each. The 120-cell partitions into twelve rings of ten dodecahedra each. The 24-cell also contains a fibration of six rings of four octahedra each stacked end to end at their vertices.

The 600-cell partitions into 20 rings of 30 tetrahedra each in a very interesting, quasi-periodic chain called the Boerdijk–Coxeter helix. When superimposed onto the 3-sphere curvature it becomes periodic with a period of 10 vertices, encompassing all 30 cells. Note, the "fiber" in this case is the center axis of the Boerdijk–Coxeter helix, not the decagon edge, which is the same as the 12-10 fibration of the dual 120-cell.

One face-stacked ring of 6 octahedra in 24-cell	One Coxeter helix ring of 30 tetrahedra of a 600-cell	Two face-stacked rings of 10 dodecahedra of a 120-cell

In addition, the 16-cell partitions into two 8-tetrahedron chains, four edges long, and the 5-cell partitions into a single degenerate 5-tetrahedron chain.

5-cell	16-cell
Coxeter helix rings as polyhedral nets

Table

The above fibrations all map to the following specific tilings of the 2-sphere.

S3	S2	Rings	Cells/ring	Cell Stacking	4-polytope projection
{3,3,5}	{3,5}	20	30 {3,3}	Boerdijk–Coxeter helix
{5,3,3}	{5,3}	12	10 {5,3}	face stacking
{3,4,3}	{3,3}	4	6 {3,4}	face stacking
{3,4,3}	{4,3}	6	4 {3,4}	vertex stacking
{3,3,4}	{n,2}	2	8 {3,3}	Boerdijk–Coxeter helix
{3,3,4}	{3,3}	4	4 {3,3}	edge stacking
{4,3,3}	{n,2}	2	4 {4,3}	face stacking
{3,3,3}	Sphere	1	5 {3,3}	Boerdijk–Coxeter helix