Talk:120-cell
Wikipedia or Wikiversity?
editThis 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
editStart 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
editHi @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?
- 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)