# Talk:24-cell

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Field: Geometry

## Analogous construction in 3-space

I am questioning this statement: "An analogous construction in 3-space gives the rhombic dodecahedron, which, however, is not regular." To me, the construction yields a stellated cube. And where would the irregularity of the rhombus come from anyway?

The analagous construction in 3-space would be 8 vertices of the form (±½,±½,±½), and 6 vertices obtained from (0,0,±1) by permuting coordinates, which gives a rhombic dodecahedron. --Zundark 13 Mar 2005
A stellated cube is a rhombic dodecahedron (provided, of course, the apices lie half an edge's length above the cube face). The rhombus is irregular in the sense that it is not a regular polygon (not all vertices are equivalent).—Tetracube 21:52, 19 February 2007 (UTC)

## Facets

Can someone tell me how many 24-cells (how many polychlora), how many ridges (how many triangular faces), and how many edges meet at any given vertex in a tessellation of regular 24-cells?

I question the portion of the following statement in italics: "The tessellation [of Euclidean 4-space] by regular 24-cells can be described in terms of the F4 lattice: the facets are centered at those points with even norm squared (the D4 sublattice), and the vertices form the set of all points with odd norm squared. Each facet has 24 neighbors and there are 8 facets meeting at any given vertex." If one of the 24-cells is centered at the origin, then the point (0, 0, 0, 0) is not at the center of a facet (border cell) of the 24-cell even though it's Euclidean norm squared (02 + 02 + 02 + 02) is 0, which is even. Other points in the F4 lattace with an even Euclidean norm squared such as permutations of (±1, ±1, 0, 0) seem logical as centers of other 24-cells in the tessellation, so I think it may be the 24-cells themselves, and not their octahedral facets, that are centered at those points with an even norm squared.  : Kevin Lamoreau 17:34, 14 August 2005 (UTC)

In the quoted statement facet refers to "facets of the tesselation", not "facets of the 24-cells". The facets of the 24-cells are usually called cells. I agree the terminology is confusing. Perhaps 4-facet would be more precise. -- Fropuff 04:02, 7 November 2005 (UTC)

## Orientable?

A recent edit added 'orientable' to 'Properties'. I'm just wondering if this is at all necessary, since all regular polytopes are orientable by definition. It seems redundant.—Tetracube 16:28, 9 February 2006 (UTC)

Looks like change was: 02:39, January 23, 2006 Shawn81 (Properties: orientable)

Orientable is trivially true for convex polytopes with convex cells, so it does seem redundant. Tom Ruen 22:36, 9 February 2006 (UTC)

## Terminology: mutually orthogonal planes?

I'm wondering if the term mutually orthogonal planes, as used in the caption of the animation of the 24-cell's 3D projection, is accurate. I know what it's referring to: two planes that intersect only in a point, each with basis vectors orthogonal to the basis vectors of the other plane. However, mutually orthogonal planes could be misconstrued as planes that intersect in a line but are otherwise perpendicular. This is possible in 3D, but the kind of setup being referred to in the image label is only possible in 4D or higher. Perhaps a better term might be linearly independent planes? Although one still needs to somehow convey the fact that they are orthogonal as well.—Tetracube 21:50, 19 February 2007 (UTC)

In the case where the two rotation planes intersect along a line, the rotations would no longer be independent, and the order in which they were applied would matter. Therefore, to make this caption clearer, perhaps the fact that the rotations themselves are independent would be useful? — JasonHise 18:43, 21 February 2007 (UTC)
Hmm, that is an interesting observation. You're right, it's the rotations that are independent. Anyway, I erased what I was about to reply after I did a little poking around and discovered that Wikipedia does have a term for it: double rotations. It's sad that it appears in an article with such an obscure title as "SO(4)", but at least it's there. For consistency, I think we should just refer to our rotations as "double rotations", and link to that section. This should clear things up.—Tetracube 05:03, 22 February 2007 (UTC)
Sounds good, I'll update the captions on the four pages that I have images for so far. — JasonHise 05:44, 22 February 2007 (UTC)
Cool, thanks. I wonder if it might be better to link "double rotation" to SO(4)#Geometry_of_4D_rotations, though, because the double rotation section seems out-of-context when you jump to it from another article. Just a thought.—Tetracube 05:46, 23 February 2007 (UTC)
That sounds reasonable, I'll fix it. — JasonHise 02:04, 24 February 2007 (UTC)

## 3 sphere inscribes a the 24-cell with Vertices(±1, ±1, 0, 0)

1st of all it has to be every 24 has a inscribed sphere and a shpere that inscribes it. 2nd wouldn't it be a 4-Sphere? (sry for bad english...)-- Ackermiv (talk) 11:10, 3 February 2010 (UTC)

Every uniform polytope has a circumscribed sphere (whose surface contains each vertex); every regular polytope also has an inscribed sphere (whose surface is tangent to each facet). Does this help?
"n-sphere" conventionally means a sphere whose surface has n dimensions; this is because the surface itself can be considered a non-Euclidean space of n dimensions. —Tamfang (talk) 07:35, 12 February 2010 (UTC)

but it has a good analog in 2 dimensions, the Hexagon. It can also be achieve by reducing the faces of a Tesseract, a.k.a. Octachoron.

I don't understand this passage. —Tamfang (talk) 04:56, 25 April 2011 (UTC)

Most surely OR, but my interpretation would seeing the self-dual hexagon --> hexagonal tiling by 3 hexagons per vertex {6} --> {6,3}, AND the self-dual 24-cell and the 24-cell honeycomb by 3 24-cells per face, {3,4,3} --> {3,4,3,3}. Both of these are special families that don't exist with the same sort of symmetries in other dimensions (G2 and F4). Another relation might be that D4 (one symmetry of 24-cell can be triple-folded into G2! Tom Ruen (talk) 05:46, 25 April 2011 (UTC)

## Should the Template Convex regular polychora default to uncollapsed?

When I got to the end of the article, wanted to navigate to the other convex polychora as many visitors to this page would want to I imagine. Anyway I missed the collapsable navbar at the bottom and didn't think to go back to the top of the page, instead found them by other methods and added a link to the "see also" which someone rightly reverted.

So anyway spotted the navbar this time around, but it would be easier to see if it was uncollapsed by default. Also means you can get to any of the other convex regular polychora with a single click. Also it doesn't take up much space on the page even when uncollapsed.

Just an idea. Great article BTW :) Robert Walker (talk) 11:20, 21 July 2012 (UTC)