Talk:120-cell

Wikipedia or Wikiversity?

This is very cool! What distinguishes this as a learning resource, suitable for Wikiversity, rather than an encyclopedia entry suitable for Wikipedia? Can you describe learning objectives and include student assignments? Does this teach students how to do something rather than describe something? Thanks! --Lbeaumont (discuss • contribs) 17:31, 28 November 2019 (UTC)

There is already an (earlier) version of this article on Wikipedia; this version is derived from it (its complete history was copied over to Wikiversity). This Wikiversity version is an in-progress update to the Wikipedia article, which will be merged back into the Wikipedia article when complete. I'm doing it that way because I don't want to edit the Wikipedia version incrementally, and I don't want to publish my updates on Wikipedia until I have completely sourced all the material I am adding (tracked down good references for it in the literature) and perhaps gotten some review / help from other editors. Dc.samizdat (discuss • contribs) 16:47, 18 February 2021 (UTC)

Active research

Start a topic on this page that is relevant to the 120-cell article but not present in it yet, or that is incompletely or wrongly described. Point out what is missing in the article, question its findings (perhaps some things in the article are wrong), or ask a question about something you don't understand, in the article or about its subject generally.

Your question does not have to be inspired or especially deep for you to ask it by starting a topic on this page. If there is already a topic here that is somehow related to it, it may be best to ask your question as a reply to that topic, but this Discuss page doesn't have to be perfectly organized; it is only important that a conversation develops. Try to answer a question on this page, or make a suggestion about how to approach that question. Participate and contribute.

Simple questions are not stupid, and they are usually helpful, because the people who contributed the text of the article are often the least likely to understand how it fails to describe the subject adequately to someone who isn't already an expert in it. Anyone who provides thoughtful feedback is a participant in this research.

All scientific discovery begins by asking a question, and often the most naive questions turn out to be the most illuminating, and lead to original discoveries. Dc.samizdat (discuss • contribs) 18:52, 26 May 2024 (UTC)

A question about your Hull #8 with 60 vertices in the 120-cell

Hi @Jgmoxness,

Many thanks for your quaternion-based contributions (and your corrections to my erroneous contributions) to the 120-cell Wikipedia article. I find myself especially interested in your "Hull #8 with 60 vertices" in your Concentric Hulls illustration. Apparently it is a non-uniform rhombicosidodecahedron. Can you tell me more about this particular section of the 120-cell? Most of all, I would like to know its incidence in the 120-cell: in how many distinct ways can you slice a 60-point section of this shape out of the 600-point 120-cell? Dc.samizdat (discuss • contribs) 07:30, 27 January 2025 (UTC)

Your welcome - working through these edits helps me understand my own stuff better - so thank you! Your work on the 4-polytopes in WP is impressive and your question is a good one. Yet, I don't have an answer since I haven't pondered it - but will try to help if I can.

You may have determined this already given it is the hull with Norm=√8 but see the attached for one (of 4) possible sets of orthogonal 3D (xyz, xyw, xzw, yzw) projection basis options (I used the imaginary part of the quaternions or yzw).

There are:

12 vertices from the 24-cell ({0, 0, ±2, ±2})

24 vertices from the first snub-24-cell row ([0, ±φ^−1, ±φ, ±√5])

24 vertices from the second snub-24-cell ([0, ±φ^−2, ±1, ±φ^2])

Not sure if this is sufficient to determine the full incidence...

BTW - I just produced a Powerpoint that relates to how H4 (8, 16, 24, 600-cell) embeds into E8 (with a 3rd projection basis vector that gives a 2D Petrie projection on one pair of 3D cubic faces and the 2D orthonormal shadow of the 600 cell's Pentakis Icosidodecahedron on another pair of faces). I would like to hear your opinions on it - see the link to that in this post. Jgmoxness (discuss • contribs) 00:29, 28 January 2025 (UTC)

Thanks for your quick reply! It will be useful to have the coordinates you provided for a Hull #8.

Am I correct in my belief that Hull #8 is a central section of the 120-cell? It appears to me that hulls #1 - #7 are off-center sections, which occur in parallel pairs on either side of a central Hull #8 section. Coxeter gives them as sections 1 - 15 of {5, 3, 3} beginning with a cell (on p 299 of Regular Polytopes). Sections 1 and 15 are the same smallest section (simply the 120-cell’s dodecahedral cell) and section 8 is the largest-radius central section (bisecting the 3-sphere, like the 4D analog of an equator or great circle on an ordinary sphere). Coxeter lists sections 6 and 10 as a pair of uniform rhombicosidodecahedra, but he does not identify section 8 as a non-uniform rhombicosidodecahedron — he apparently never visualized Moxness’s Hull #8! The coordinates he gives for it match the ones you just gave me, though. So he found it first, but you were the first person to see it!

My understanding of hulls is that they are 3D sections, flat 3D hyperplane slices through the 4-polytope, and that there is a complete parallel stack (of 15 of them in this case) spindled on each axis of the 4-polytope. In this case of the 120-cell sections beginning with a cell (your Hulls #1 - #8), there is a full stack of 15 parallel sections on each of the 60 axes connecting a pair of antipodal cell centers. Therefore the incidence of each kind of section is 60 * 2 = 120 for the off-center sections, which occur in parallel pairs — and indeed, the 120-cell has 120 dodecahedral cells as its sections 1 and 15 (Hull #1). There is only one central section (Hull #8) on each axis, so its incidence is only 60.

If I have got the above right, there is a typo in the caption for your Concentric Hulls Illustration in the 120-cell article. Only the caption below the illustration contains an error -- the labels on the hulls in the image are fine. But the caption below disagrees with the image for Hull #8. The caption states that Hulls #6 and #8 are pairs of rhombicosidodecahedrons, which is true of Hull #6 but not of Hull #8 — if it is the single central section, it is just one 60-vertex rhombicosidodecahedron. I don’t know who wrote this buggy caption for your image, it may even have been me! If you agree that it is an error, I will fix it.

Explain to me, if you will, the “Overall Hull” in your illustration. Is it also a section, but through some other hyperplane than Hulls #1 - #8? Coxeter also lists a set of 30 sections for {5,3,3} beginning with a vertex — is it one of those? Coxeter gives their coordinates, which you might be able to recognize. See pp 300-301 of Regular Polytopes. The central section beginning with a vertex is his section 15 of 30 (with 54 vertices). All the others occur in parallel pairs, spindled on the 300 axes connecting antipodal vertices. So their incidence is 600, except for the central section of which there are only 300.

I didn’t know the 600-cell contained a Pentakis Icosidodecahedron — fascinating. I did know it has some isosceles triangles in it (see my Golden Chords illustration in the 600-cell article). I will try to grok your powerpoint, with interest. Though I am hopelessly out of my depth above dimension 4! Dc.samizdat (discuss • contribs) 22:23, 28 January 2025 (UTC)

Re: my question about the "Overall Hull" in your illustration, I see now that it is not in fact one of the 30 sections of {5,3,3} beginning with a vertex. As your caption indicates, it is a Chamfered dodecahedron with 80 vertices, while the largest section beginning with a vertex in Coxeter's list has only 54 vertices. "Chamfered" means "edge-truncated", so perhaps your "Overall Hull" is a section of the 120-cell beginning with an edge? Coxeter's Regular Polytopes doesn't have a list of those, unfortunately! Does it make sense that your "Overall Hull" is the 1st section of the 120-cell beginning with an edge? Dc.samizdat (discuss • contribs) 21:49, 28 January 2025 (UTC)

Your description of Coxeter's sections vs. my projected hulls with one of the 4 xyzw to 0 is correct (section 8 is the hull 8 (largest) and 1-7 hulls are pairs of sections 1-7 (combined with or doubling the vertex counts with 15-9, respectively). Coxeter's "sections" for the 600-cell are somewhat different though, as there are only 7 sections (3+1+3) in the dissection shown in the 600-cell article (based on generating from the T quaternions).

Coxeter also describes the 7 sections (3+1+3) shown in the 600-cell article, on page 298. They are the sections starting with a vertex, and he includes the two antipodal vertices as degenerate sections, so he lists 9 sections (4+1+4) in his table.Dc.samizdat (discuss • contribs) 03:51, 29 January 2025 (UTC)

As I had not really used Coxeter's Regular Polytopes book as a reference, it turns out what he is describing in the {3,3,5} 600-cell & {5,3,3} 120-cell table is the (alternate) 600-cell generated from T' - which has 15 sections (in 8 hulls).

This Coxeter table on page 299 is the sections starting with a cell, of {3,3,5} and {5,3,3}. So apparently your (alternate) T’ projections reveal the sections starting with a cell.Dc.samizdat (discuss • contribs) 03:51, 29 January 2025 (UTC)

I've now generated both of these for WP as linked here and here. Please add to the article as you like. You may want to replace that older blue-hue version which is IMO prettier but has less technical detail.

It is also interesting that the alternate 600-cell he describes in that table has rows of 4 vertex tetrahedons, which are really left/right chiral pairs (e.g. section 1 & 15 or 2 & 14) which form cube hulls in my projections. All the other shapes there too (I think), except the central one, are formed from chiral pairs as well. This may be new news to others, but I've been visualizing these for years w/o really thinking it was novel. It would take some digging to determine how novel it really is.

Yes I noticed that your polyhedron images in this set of 15 600-cell sections starting with a cell do not exactly match Coxeter’s. I see that they are different because your projections superimpose both polyhedra of the pair as one object, and because they are chiral pairs (inside-out versions of each other, but the same indistinguishable object except for orientation) they combine to make a compound object (so two tetrahedra become a cube for example). Of course there really is no cube anywhere, that is an artifact of the superimposition performed by the projection — the two tetrahedra are a long way apart from each other, in parallel hyperplanes, completely orthogonal to each other on opposite sides of the 600-cell. But your visualization proves they are in inside-out orientation to each other, and that is indeed a fundamental observation. I have seen it mentioned before, for example in descriptions of how cells turn themselves inside out during an isoclinic rotation as they change places with their antipodal cell, but once again your imaging is probably the first time the phenomenon has actually been seen.

Of course your two sets of 600-cell hulls (from T and T’) still leave us with the question of what exactly the 4 T hulls are. They are not the sections starting with a vertex (p 298), and they are not the T’ sections starting with a cell (p 299). I notice in your image that their tallyList has 5 entries, the first being “2”, as if the first section is a polyhedron with only 2 vertices: an edge. So these hulls are probably the sections of the 600-cell starting with an edge.

I wonder what the “Overall Hull” of this set is — it is the Petrakis icosidodecahedron, but what section is it? It has to be a flat 3D section of the 600-cell (starting with something!). It is not an imaginary composite object created by the projection, is it? I presume there are actual Petrakis icosidodecahedra in the 600-cell, with all their vertices lying on the same sphere. I read on Wikipedia that the Petrakis icosidodecahedron is constructed by the “kis” operation, which is a sort of inverse of vertex truncation: it sucks out a pyramid on each pentagon face, creating a new apex vertex. So it seems plausible that this Petrakis icosidodecahedon “Overall Hull” is a section of the 600-cell starting with a face. Dc.samizdat (discuss • contribs) 03:51, 29 January 2025 (UTC)

As for being the first person "to see" hull #8, that is a cool distinction. More interesting (IMHO) is the visualization of the other distinct 120-cell object (J') constructed in a similar way as J, but from the base of the alternate (dual) 24-cell (T=D4) instead of T'. The "overall hull" of J' is a solid that has yet to be categorized at all! (maybe a Johnson solid, not sure - call it a Moxness solid for now ; -). Since it would be WP OR, I am reluctant to incorporate it into a WP article. Publishing a paper on it would be something for the future, but you saw it here (and on WP) first.

That nondescript (as in never-before-described) polyhedron in the 120-cell is very cool indeed. Do you have counts for its vertices and faces? This is a perfect example of WP:OR which should not be added to the Wikipedia 120-cell article, but could be added immediately to the Wikiversity 120-cell article — the expandable version of the Wikipedia article that I put there for researchers and students to work with. You should edit it! Even if no one notices your discovery there (no one much has yet discovered my Polyscheme project), it would cement your precedence of discovery forever, in the revision history of the Wikiversity article. Think about it.

And of course, we should try to figure out exactly which section of the 120-cell it is. Assuming it is a real section, and not an artifact of superimposition by projection. Possibly a deeper section starting with an edge, like the chamfered dodecahedron?

The answer is it is the vertex-first description of {5,3,3} in the form of projected hulls vs. sections. The center section (#15) is hull 16 (as he starts numbering with 0 when they are a point). The counts match exactly (hulls 1-15 are doubled section counts 0-14 & 30-16). (discuss • contribs) 07:41, 29 January 2025 (UTC)

This is fun. Dc.samizdat (discuss • contribs) 03:51, 29 January 2025 (UTC)

Good catch on the caption for hulls 6 & 8 not being "pairs of" - now corrected.

The "overall hull" only shows the actual outer hull without any opacity and color from the underlying (i.e. smaller norm interior) solids that help form it (aka. the "combined hulls"). In the case of the (J) 120-cell, one can see that hull 6,7, and 8 form the 3D solid of the Truncated Rhombic Triacontahedron or Chamfered Dodecahedron. The 600-cell's outer hull of the Pentakis Icosidodecahedron is easier to see when showing it with the overall and combined hulls, as in this E8=H4+H4ϕ visualization.

I also build animations of building up the hulls to show the interior structure - kind-a-fun in a geeky sort of way. Jgmoxness (discuss • contribs) 00:31, 29 January 2025 (UTC)

I hadn't seen this email before I quickly sent my draft that had been waiting for me to complete some errands today.

I don't have an issue with these emails being published (but will take care in my writing style to be less informal ; -)

As for the overall hull concept, as I mentioned in my last email - it is really a "projection" (not a "section", where orientation of the object against the plane is important) as Coxeter describes the two ways of visualizing these 4D objects.

But although Coxeter only gives lists of two sets of sections as hulls (starting from a vertex, and starting from a cell), he was well aware that there are two other ways to cut sections: starting from an edge, and starting from a face. He just didn’t work those out. You have, I suspect, at least for some cases of them. I suspect that all your “Overall hulls” are real polyhedra, lying in just one flat 3D hyperplane sliced through the 4-polytope, and not composite fictions of the projection. I may well be wrong, but we can surely find out the truth of the matter. Dc.samizdat (discuss • contribs) 3:50, 29 January 2025 (UTC)

I prefer projection (using computer) to interactively orient and turn in 4D space. I've found a bit of Mathematica code to generate / interact with those, but have also developed my own with a little more flexibility. But unless you have Wolfram's Mathematica, PDF and SVG to WP is the best alternative. (discuss • contribs) 01:13, 29 January 2025 (UTC)

Ok, I don't (yet) have a physical copy of the book (I found a PDF copy with different page numbering and other issues) so let me find what you're seeing while I get a physical copy.

Yes, there are two (4D antipodal) vertices projected in 3D to the origin (aka. degenerate) in the T based projection of the 600-cell, but my algorithm doesn't visualize them as a hull (obviously).

You are correct in terms of the description of the "sections" (for Flatlanders, as he puts it, by MOVING the object through a stationary plane or conversely moving a plane through the object) and taking 3D snapshots at locations that have a vertex in the moving plane. Neither of the sections nor the projections are any more or less real representations of the whole 4D object.

I am "projecting" the whole object into one frame and viewing the 3D sections (individually or all at once as a combined hull or overall hull with hidden smaller norm internal vertices). A hull is simply the vertices with the same 3D distance from the origin after having set all the 4th dimension x values set to 0. All 4D object distances from the origin are of course the same (one 4D hull of a given norm or scale).

Let me create some other visualizations that will demonstrate all this for you (e.g. 4D projections like this one https://commons.wikimedia.org/wiki/File:Cell600Cmp.ogv or static / animated sectioning as Coxeter does it).

The "overall hull" of the 600-cell being a Petrakis icosidodecahedron and the 120-cell having a chamfered dodecahedron is simply what you see as a convex hull if you ignore (i.e. project to 0) one of the 4 dimensions of the object when you look at it (it doesn't matter which dimension you choose to ignore) and ignore all vertices inside the convex hull. (discuss • contribs) 07:41, 29 January 2025 (UTC)

Moxness's 60-point cell-first Hull #8 rhombicosidodecahedron

Orthogonal projections of the 120-cell by Moxness using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th section starting with a cell.

Moxness's 60-point Hull #8 is a non-uniform rhombicosidodecahedron, Coxeter's 8th section of the 120-cell beginning with a cell $8_{3}$. Dc.samizdat (discuss • contribs) 19:58, 15 February 2025 (UTC)

It is the central section beginning with a cell, lying in an equatorial hyperplane that bisects the 3-sphere.

Since there are 60 pairs of antipodal cells there are 60 such sections, which intersect at the center of the 120-cell, and elsewhere as follows.

Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The {12} vertices also form an irregular great dodecagon with 6 120-cell edges that alternate with 6 #4 chords (44.5°). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 such central planes. The 5-cell faces do not lie in central planes.

The 60 rhombicosidodecahedra are non-disjoint, with each 120-cell vertex occurring in 6 rhombicosidodecahedra.

Each pair of rhombicosidodecahedra intersects in a central plane containing 12 vertices: an irregular great {12} dodecagon of alternating #1 and #4 chords. Three rhombicosidodecahedra meet at each chord in the {12} central plane, but the plane itself occurs in four rhombicosidodecahedra.

The 120-cell contains 200 such irregular great {12} dodecagons, with 4 intersecting at each vertex.

10 irregular great {12} dodecagons occur in each Hull #8 rhombicosidodecahedron, with 2 intersecting at each vertex. Each rhombicosidodecahedron shares a {12} central plane with 10 other rhombicosidodecahedra. Groups of eleven rhombicosidodecahedra share central planes pairwise.

Dc.samizdat (discuss • contribs) 21:01, 17 February 2025 (UTC)

The major chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon of the 120-cell.

The Hull #8 chords are a subset of the 120-cell's 30 distinct numbered chord lengths. The chord lengths given below are for a √2-radius 120-cell.

The short edge is the pentagon face edge: chord #1 of 15.5° arc, length $1/\phi ^{2}\approx 0.382$.

The little pentagon faces of Hull #8 are 120-cell faces.

The long edge is the minor chord between chords #3~4 of 41.4° arc, length 1.

The 20 long edge triangle faces of Hull #8 are found only inside the 120-cell. They are not 5-cell faces, or 16-cell faces, or 24-cell faces, which are all larger, or 600-cell faces, which are smaller.

The 20 triangle faces are disjoint, separated from each other by rectangles and pentagons. Each triangle face is joined to three other triangle faces by a minor chord #6~6~7 of 75.5° arc, length √3. This is √3 x #3~4, the rhombicosidodecahedron long edge.

This rhombicosidodecahedron is the convex hull of 5 truncated tetrahedra, 12-point Archimedean polyhedra with 4 of these triangle faces, opposing 4 hexagon faces of length √3.

The rectangle face diagonal is the dodecahedral cell diameter: chord #4 of 44.5° arc, length ${\sqrt {3}}/\phi \approx 1.071$. It lies on the same {12} great circle as the 120-cell edges (and the 5-cell edges, see below). Chord #4 × 𝜙 is the minor chord #6~6~7 of length √3, above, which is a deeper chord parallel to chord #4.

An invisible dodecahedral cell fits between two pentagon faces, touching them with its opposite edges. The two pentagon faces belong to dodecahedral cells that are only edge-bonded (not face-bonded) to the invisible dodecahedral cell between them.

This rhombicosidodecahedron is also the convex hull of 5 cuboctahedra, the central section of the 600-cell beginning with a cell $8_{3}^{J}$. The rhombicosidodecahedron contains all the #5, #7, #10 and #15 chords of those 5 inscribed cuboctahedra.

The #7 chord of 90° arc has length 2. It is the edge chord of a great square {4} central plane. It is the only chord in this rhombicosidodecahedron which does not lie in a {12} central plane. Dc.samizdat (discuss • contribs) 22:21, 7 March 2025 (UTC)

The 10 irregular great {12} dodecagons have alternating #1 and #4 chord edges, and these other chords which are arc-sums of them:

Arcs #1 + #4 = the 8-cell and 24-cell edge: chord #5 of 60° arc, length √2.

Arcs #4 + #1 + #4 = the 5-cell edge: chord #8 of 104.5° arc, length √5.

The 60 5-cell edge chords form 20 triangular 5-cell faces inscribed in the Hull #8. (These faces do not lie in the 10 irregular great {12} dodecagon central planes, only their edges do.)

The 20 5-cell faces are completely disjoint; they do not meet at an edge or even at vertex.

Only two 5-cell edge chords meet at each Hull #8 vertex. Four 5-cell edges meet at each vertex of the 5-cell (in a tetrahedral vertex figure), but only two of them are chords of the same Hull #8 (edges of the same inscribed 5-cell face). The other two 5-cell edges at the vertex belong to other Hull #8s. Each 5-cell edge belongs to three Hull #8s, and each 5-cell face belongs to just one Hull #8. Each 5-cell face has an edge in each of three distinct {12} central planes. Dc.samizdat (discuss • contribs) 22:40, 7 March 2025 (UTC)

The Hull #8 rhombicosidodecahedron does not have any whole 5-cell tetrahedral cells inscribed in it, only 5-cell faces, from 20 completely disjoint 5-cells.

There are 60 Hull #8 rhombicosidodecahedra in the 120-cell. They are non-disjoint, with each vertex occurring in 6 rhombicosidodecahedra. The hulls meet 6 around each vertex, and 3 around each edge, but they do not meet face-to-face, as cells do. Each pentagon face, each triangle face, and each rectangle face belongs to just one Hull #8.

The rhombicosidodecahedra do not share interior 5-cell faces either. Each 5-cell face is inscribed in just one Hull #8. However, each of these interior 5-cell faces is shared between 2 face-bonded tetrahedral cells, in just one 5-cell.

Arcs (#8 = #4 + #1 + #4) + #1 = chord #10 of 120° arc, length √6.

{30/11} triacontagram

Arcs #1 + (#8 = #4 + #1 + #4) + #1 = chord #11 of 135.5° arc, length $\phi ^{2}\approx 2.618$. This chord is #7 × 𝜙, where #7 is the 90° arc great square edge chord.

Groups of eleven rhombicosidodecahedra share central planes pairwise. An isoclinic rotation takes these rhombicosidodecahedra to each other, in a circuit of 11 rotational displacements of the #11 chord length. The rotation's circular isocline of eleven #11 chords is a {30/11} skew polygram.

Arcs #11 + #4 = chord #15 of 180°, the 120-cell diameter.

Dc.samizdat (discuss • contribs) 01:38, 17 February 2025 (UTC)

Moxness's 80-point cell-first Overall hull

Lower left in Moxness's illustration, above, an irregular chamfered dodecahedron with 12 pentagon and 30 irregular hexagon faces. 80 vertices, 60 short edges and 60 long edges. This Overall hull is isomorphic to the 60-point Hull #8 rhombicosidodecahedron central section (lower right), with 20 additional vertices added at the centers of the Hull #8 triangle faces. Both the 60-point and 80-point polyhedra can be seen as central sections of the 120-cell, in the cell-first equatorial hyperplane. But the 80-point polyhedron does not actually occur there in the 120-cell, because the extra 20 vertices are not 120-cell vertices. The 60-point Hull #8 rhombicosidodecahedron is a simplified section of the 120-cell, defined by Coxeter as the vertex set intersected by a sectioning hyperplane. The 80-point Overall hull polyhedron is an unsimplified section that includes 20 additional points where the sectioning hyperplane intersects a 120-cell edge. The 120-cell contains 60 Hull #8 sections (60-vertex polyhedra), but it contains 0 Overall hull sections (80-vertex polyhedra lying in one 3-dimensional hyperplane).

However, the 120-cell does contain instances of the Overall hull 80-vertex set, just not as sections confined to a 3-dimensional hyperplane. The convex hull of the compound of 16 5-cells is an 80-vertex polychoron that is Moxness's 80-vertex Overall hull. That 80-point Overall Hull vertex set is not found anywhere in the 120-cell as a flat polyhedron, only as a polychoron. 60 of that polychoron are found in the 120-cell, as inscribed 80-point 4-polytopes, non-disjoint compounds of 16 the 120-cell's 120 inscribed 5-cells, with each of the 600 vertices shared by 8 80-point polychorons. We might describe these 4-polytopes as skew polyhedra, since they lie skew in 4 dimensions, in the same sense that a 5-point 5-cell is a skew pentagon. In fact each 80-point polychoron is several distinct 80-point skew polyhedra, the same way a 5 cell is several distinct skew Petrie pentagons.Dc.samizdat (discuss • contribs) 21:01, 7 April 2025 (UTC)

On either side of the 60-point Hull #8 central section, there is a non-central section Hull #7, parallel to the central section and to each other, at a distance of one 120-cell edge apart from each other (6° of arc above or below the central hyperplane). Notice that Moxness's Hull #7 is a regular dodecahedron. Those two 20-point Hull #7 dodecahedra would be coincident if their corresponding vertices were not a 120-cell edge length apart in the 4th dimension. A point on the edge of each of those 20 connecting edges is intersected by the central section hyperplane, and therefore the point on the edge is a vertex of the unsimplified 80-point Overall hull central section, although it is not a vertex of the 120-cell. Dc.samizdat (discuss • contribs) 02:09, 27 March 2025 (UTC)

The point on the edge intersected by the sectioning hyperplane divides the edge into golden sections. The edge length isn't the orthogonal distance between Hull #7 planes. The orthogonal distance is 12° of arc, but the edge length is 15.5° of arc, because the connecting edges are not orthogonal to the parallel hyperplanes.Dc.samizdat (discuss • contribs) 04:07, 27 March 2025 (UTC)

I think the statement:

The 120-cell contains 60 Hull #8 polyhedra (60-vertex sets), but it contains 0 Overall hull polyhedra (80-vertex sets in one 3-dimensional hyperplane).

is not quite correct. If you look at the "combined hull" you see the "overall hull" is made up of the 20 points from the section 7 & 9 dodecahedrons (with two 120 cell vertex overlap per location). That means the 80-vertex chamfered dodecahedron's 80 vertices are the 60 hull #8 and 20 from either section 7 or 9 or some desired combination.

They are not "where the sectioning hyperplane intersects a 120-cell edge" - they are from the sections above. Jgmoxness (discuss • contribs) 02:26, 8 April 2025 (UTC)

Yes, that's right, the 20 extra vertices are really there, and they are 120-cell vertices, but they are off the hyperplane containing the other 60 vertices. My point is just that your 80-vertex object does not exist as a flat 3D polyhedron in the 120-cell. You can find your 80-vertex object as an unsimplified section, which means as a 3D polyhedron lying in a 3D section hyperplane. But in that case its 20 extra vertices are not 120-cell vertices, they are edge intersections of the sectioning hyperplane. You can also find your 80-vertex polytope in the 120-cell with all 80 vertices as actual 120-cell vertices, but in that case, all 80 vertices cannot lie in the same 3D hyperplane. 20 of them are above or below the hyperplane containing the other 60 vertices. So the 80-vertex object is not a 3D polyhedron, like your other cell-first section hulls #1 - #8 which really are 3D polyhedra lying just in one hyperplane. The 80-vertex object lies on the 3-sphere, not the 2-sphere, an 80-vertex 4-polytope occupying all 4 dimensions, not a 3D polyhedron. Your chamfered dodecahedron rendering is a projection of that 4-polytope's outer hull to a 3-dimensional polyhedron, a shadow. The actual 80-vertex object only exists as a 4-polytope, there is no actual 80-vertex flat 3-polytope in the 120-cell, your image is a shadow of a 4-polytope. But the 80-vertex 4-polytope does exist in the 120-cell.

Your chamfered dodecahedron is the Outer hull of an 80-vertex 4-polytope. It is just the outer hull, so its 30 hexagons and 12 pentagons are only some of its faces. It is a 4-polytope, so it has 3-polytope cells. Each pentagon face is a face of one of the 120-cell's dodecahedral cells. Each irregular hexagon face is a section of another of the 120-cell's dodecahedral cells. The hexagon's long edges are pentagon face diagonals, and the hexagon's 3 diameters are dodecahedral cell diameters: the irregular hexagon face is a central section of a dodecahedron cell. Since this chamfered dodecahedron overall hull is a 4-polytope, its pentagon and hexagon faces must be faces of polyhedrons, the cells of the 4-polytope. Exactly what shape are those cells? The overall hull is a 42-cell 4-polytope, inscribed in the 120-cell. I think there are 60 of them in the 120-cell, with each of the 600 vertices shared by 8 42-cells. Dc.samizdat (discuss • contribs) 19:23, 10 April 2025 (UTC)

Sections 2 and 28 beginning with a vertex

Section 2 is the third section beginning with a vertex (the second non-point section). Section 0 is the vertex itself, and section 1 is the 120-cell vertex figure, a tetrahedron whose edges are 120-cell edges (the #1 chord).

Section 2 is a 12-vertex semi-regular truncated tetrahedron with two edge lengths, the #1 and #2 chords.

It's worth noticing that another name for a truncated tetrahedron is truncated cuboctahedron. The cuboctahedron has four central planes that contain great hexagons, inclined at 60° to each other. Bisecting a cuboctahedron at one of those hexagon planes produces two truncated tetrahedra. The 24-cell has 16 of those central great hexagons (of edge length 1). But section 2 of the 120-cell does not contain those great hexagons. Its hexagons are not great hexagons in central planes of the 120-cell, and they are not regular hexagons.

Section 2 has four triangle faces with edge length #2 (arc 25.2°) ~= 0.437. Its semi-regular hexagon faces have alternating edges: three #2 edges, and three 120-cell edges of length #1 (arc 15.5°) ~= 0.270.

In every section beginning with a vertex this much is true: in the #n section, the #n chord is an edge of the section polyhedron: the only edge length if the polyhedron is regular, or one of two edge lengths if the polyhedron is semi-regular.

The irregular hexagon has two chord lengths: #3 (arc 36°) ~= 0.618 and #3~4 (arc 41.4°) ~= 0.707.

Section 2 and section 28 are congruent polyhedra, but they lie in "inside out" spatial orientations to each other. They are exactly the same congruent polyhedra, not enantiamorphous like a pair of shoes, so their chirality is only a matter of their orientation in their environment, not an intrinsic property. It is not usual to call them chiral objects because that terminology means enantiamorphous objects. They are spatially separated identical objects that are contextually entangled in their environment in a chiral relationship. They are not concentric and do not overlap in space, but if you project them both at the same time one behind the other along the w axis so that they appear to be superimposed, they form a non-convex compound polyhedron. It must be emphasized that no such 24-vertex compound polyhedron exists anywhere in the 120-cell: it is a mere shadow of two truncated tetrahedra, which are located in two different places on opposite sides of the 120-cell. Nevertheless, it well illustrates the chiral orientation relationship of the section 2 and 28 truncated tetrahedra.

Even more "shadowlike" is the convex hull of this imaginary 24-vertex polyhedron, which forms a non-uniform rhombicuboctahedron. This misleading projection is much less helpful as an illustration of the two truncated tetrahedra, or their special spatial-orientation chiral relationship. The rhombicuboctahedron is actually misinformation, since it posesses square and rectangle faces, and those faces simply do not exist in the actual 120-cell. There is no actual polyhedron with square or rectangle faces to be found here! Although there is this projection which "shows" them in these sections, those quadrilateral faces are illusory: artifacts of the projection method. Worse, the actual faces (triangles and hexagons) are not visible in the convex hull, which in this case is a projection that hides more than it reveals. That said, orthogonal projection to a convex hull is a hugely successful method for isolating polyhedra within 4-polytopes. It is singularly revealing, accurate, and deeply insightful, when applied to almost all sections of the uniform 4-polytopes. However, without enhancement to cull one of the section pairs, it fails entirely for those section pairs in the 120-cell that have the unusual chiral spatial-orientation relationship. Dc.samizdat (discuss • contribs) 22:48, 26 February 2025 (UTC)

Rotation polygon notation

For the 120-cell's irregular dodecagon central polygon, write:

$\{12_{1|4}\}$

as a 120-cell-specific shorthand for the more general notation:

$\{12_{\{30/1\}|\{30/4\}}\}$

This denotes a semi-regular polygon with alternating edges of two different lengths, the #1 and #4 chords of the triacontagon.

The 120-cell's {12} central polygon has chords #1, #4, #5, #8, #10, #11 and #15, so it has many equivalent representations, such as:

$2\{6_{1|8}\}_{4}$

denoting 2 irregular hexagons with alternating edges of #1 and #8 chords. These truncated triangles are inscribed in opposing positions, with their corresponding vertices separated by a #4 chord.

Here is another construction of the very same irregular dodecagon polygon, as two regular hexagons rotated with respect to each other, such that their corresponding vertices are separated by a #1 chord:

$2{\{6_{5}\}}_{1}$

The regular hexagons with #5 edges naturally have regular triangles with #10 edges inscribed in them, so this is yet another construction:

$4\{3_{10}\}_{1|4}$

showing that the corresponding vertices of the triangles are separated by the alternating #1 and #4 chords.

Numbered chord subscripts may be specified in either order without ambiguity, but sometimes the order is significant, as described below.

Isoclinic rotations may be represented as a triplet of three sets of Clifford parallel central planes:

$(P_{\text{left}},P_{\text{isocline}},P_{\text{right}})$

The rotation takes the left invariant planes to the right invariant planes in parallel, each vertex moving over a chord of the isocline planes. Left and right planes are isoclinic (separated by two equal rotation angles). They contain Clifford parallel (disjoint) great circle polygons. Isocline planes intersect both the left and right planes, and the isocline itself lies over a skew polygon in which each successive chord is from a distinct plane. The left (right) planes tilt sideways (like coins flipping) to reach the right (left) planes, while simultaneously rotating themselves (like wheels). The tilt rotation angle equals the wheel rotation angle (the rotation is isoclinic). Each vertex moves over a chord of the isocline polygon (the isocline chord), while simultaneously moving over a chord within the tilting left (right) plane (the rotation edge chord). In rotation plane specification subscripts $P_{a|b}$, the significant chord is the one which appears leftmost ($a$), as that is the chord traversed by the vertex.

For example, the characteristic rotation of the 24-cell is invariant in 4 of its 16 great hexagon planes:

$(4{\{6_{5}\}},8{\{3_{10}\}},4{\{6_{5}\}})$

The significant new information conveyed by this triplet, not conveyed by its three terms separately, is that in the isoclinic rotation in invariant planes rotating great circle hexagons of #5 edge chords, the vertices move through 4-space helically on geodesic circles of #10 isocline chords. The triplet describes a class of rotation, the 24-cell's characteristic rotation over its edges, that is uniquely determined by its #5 rotation edge chord and its #10 isocline chord. All the other information about the rotation class, that may be expressed by the triplet's terms or their equivalent terms in equivalent triplets, is implied. (Some additional information must be provided in order to specify a discrete rotational displacement of this class.)

For example, the characteristic isoclinic rotation of the 120-cell, along its edges the {30/1} chord, takes place in these invariant {12} central planes. It includes this rotational displacement, in which each vertex is displaced over an isocline {30/4} chord to another vertex 44.5° away (the opposite vertex of a dodecahedral cell).

$(\{12_{1|4}\},\{12_{4|1}\},\{12_{1|4}\})$

In this example, the left, isocline, and right planes are all the same kind of irregular {12} central plane. The left and right planes rotate (like wheels) by a #1 chord, while each vertex simultaneously traverses a #4 chord in the isocline plane, as the left plane tilts sideways (like a coin flipping) into the the right plane's original location. There are 200 irregular {12} great dodecagons in the 120-cell, and an isoclinic rotation spins (and tilts) all of them at once.

Our second example is the characteristic isoclinic rotation of the 5-cell. In the 120-cell it rotates all 120 5-cells in parallel, along 5-cell edges of 104.5° arc, over the {30/8} chord. The #8 chord also lies in the irregular {12} central planes, so this rotation takes place in the same invariant planes as our first example. The only difference is the angular distances of the displacement:

$(\{12_{8|1}\},\{12_{8|1}\},\{12_{8|1}\})$

In the 5-cell (uniquely) the isocline chord is the same chord as the edge chord, so all three angular distances are the same #8 chord. In other isoclinic rotations of regular 4-polytopes, the isocline chord is longer, a diagonal product of the left and right chords. In any isoclinic (equi-angled) rotation, the left and right chords are the same length.

Our third example takes place in another kind of left and right invariant planes:

$(\{10_{3}\},\{12_{5|1}\},\{10_{3}\})$

This is the characteristic isoclinic rotation of the 600-cell, in decagonal invariant planes. The left and right polygons are regular {10} decagons with #3 chord edges of 36°. There are 720 regular great decagons in the 120-cell, and an isoclinc rotation spins (and tilts) all of them at once.

The isocline chord in this rotation, however, is the #5 chord, which lies in a different kind of central plane: the same irregular {12} planes found in the preceding examples.

Dc.samizdat (discuss • contribs) 23:53, 2 March 2025 (UTC)