## Discrete Hopf fibrations.

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Discrete Hopf fibrations.

This is a section from the Hopf fibration article on Wikipedia that I and Tom Ruen wrote that has since been deleted due to "lack of sources" and/or "original work". The original article can be found here https://en.wikipedia.org/wiki/User:Cloudswrest/Regular_polychoric_rings

Three of the six regular 4-polytopes, the 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 ring in the 120-cell
120-cell_rings.jpg
120 cell ring.

One ring in the 600-cell
600-cell_tet_ring.png
600 cell ring.
Last edited by cloudswrest on Fri Oct 30, 2015 11:29 pm, edited 1 time in total.
cloudswrest
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### Re: Discrete Hopf fibrations.

There is a clifford group for each of the 3d symmetries, and this is the last shared symmetry between the regular complex polytopes and the mirror groups.

Thus 8= o3o3o4x = {2,2} 24 = x3o4o3o = [3,3] 48 = o3x4x3o = [3,4] and 120 = o3o3o5x = [3,5] generally 2p = xPo2oPx = [2,P]. These are the poincare cube, octahedron, trunc cube, and dodecahedron. I demonstrated to JHConway, that the poincare dodecahedron can be replaced by a pentagonal tegum, or cluster of five tetrahedra.

Coxeter describes a lot of this in , regular complex polytopes (1971)', but the text is fiercefully cryptic.
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### Re: Discrete Hopf fibrations.

Hmm. This is the sort of thing that Wikipedia would benefit from keeping. I'm sure I've seen the polychoric rings before, and either those images or other images showing the same construction, but I can't for the life of me remember where. I think if someone can find some sources for the material, it would be deemed worthy of staying.

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### Re: Discrete Hopf fibrations.

The "Discrete examples" section in the Hopf fibration article lasted for about four years until Nov 2014 when an influential editor got annoyed with Tom Ruen's pictures (calling them "cruft" and, paraphrasing, "polluting the purity of the article") and deleted the section for lack of adequate references. You can read "talk" debates on this issue

here: https://en.wikipedia.org/wiki/Talk:Hopf_fibration#Discrete_examples

and here: https://en.wikipedia.org/wiki/Talk:600-cell#Proof_30-tet_Boerdijk.E2.80.93Coxeter_helix_circumnavigates_the_600-cell.
cloudswrest
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### Re: Discrete Hopf fibrations.

With the hop fibulation etc.

The space of great arrows in 4D is a bi-spheric prism, the individual spheres correspond to a left and right Hopf fibulation. So by way of quarterions, the great circles represent l,X,r where X is a point, l and r are left and right unit quarterions. It is rather akin to the complex multiplication a.X giving rise to rotation and the cyclotomic groups over a.

So there are regular groups for [2,2], [2,p], [3,3], [3,4], [3,5]. giving respectively, the tesseract, a bi-polygonal prism, a 24choron, an octagonny, and a dodecahedron. The various subgroups of this are directly related to the various regular complex polytopes in Coxeter (1971). The local jargon here is swirlibob, and i believe it is Polyhedrondude who did a net of {5,3,3} based on the ten sets of dodecahedra.

When one considers l.X or X.r, one gets a hopf fibulation, which is discrete, (in that the values of l are finite), when the members of l form a 3D symmetry. John Conway (incorrectly) asserted that because one can find r by a reflection of l, that this does not represent the A_n symmertry that gives from using left and right hopf fibulations.

The rotational summetries [3,3,5], [3+,4,3+], [3,2,3] and CI correspond to the groups AA5, AA4, AA3, AA2, corresponding to an even swap of pairs of letters or numbers in square of 5, 4, 3, 2 of each eg (1 2 3 4 5) ** (A, B, C, D, E).
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### Re: Discrete Hopf fibrations.

I was thinking about the "tiger cage" yesterday in relation to these discrete Hopf fibrations. It occurred to me that the cage is just the intersection of two orthogonal, opposite chiral fibers. For example you can build it out of a subnet of tets from the 600-cell. Here is an "unrolled" picture of 100 tets forming a clifford torus boundary in the 600-cell.
100 tets
100_tets.jpg (66.06 KiB) Viewed 24726 times
Pick any two diagonal orthogonal rows, wrap them around and they join again at the opposite pole, forming a tiger cage. You can also build it out of a tesseract using six of the eight cells.

-CW
cloudswrest
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### Re: Discrete Hopf fibrations.

The general group derived from {2,p} makes said cage any size.

[3,3,5] itself contains something like 50 centred bi-triangular cages.
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### Re: Discrete Hopf fibrations.

cloudswrest wrote:I was thinking about the "tiger cage" yesterday in relation to these discrete Hopf fibrations. It occurred to me that the cage is just the intersection of two orthogonal, opposite chiral fibers.

Yes, in a way, the tiger cage is another type discrete hopf fibration, in toroidal form. It's interesting to see the two great circles intersecting at 90 degrees, from the 45 degree intersection with a 3-plane. When the minor ring diameter of the tiger is set to 0, you get the Clifford torus.

Tiger cage, with an aspect ratio of 40:

and, passing through a 3-plane at the 45 degree angle:

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### Re: Discrete Hopf fibrations.

That's sad, this info is actually very relevant to polytope research and has a solid mathematical foundation. I have found that information very useful, myself. I sympathize with you about deletionist Wikipedia editors... many years ago I gave up contributing to Wikipedia because some reputable editors were (and apparently, still are) too extreme in their interpretation of what's relevant and what's not, especially on issues that are not always as black and white as they would like to think. But at least they (so far) can't touch what you put on your user page, so at least the info is still out there... (hopefully Google will find it and index it for people who have actual interest in the subject beyond just keeping Wikipedia "pure" according to their own agenda).
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