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WikiProject Mathematics (Rated Start-class, Low-priority)
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 Field: Geometry


Wow, that's eldritch. I wanna see it from the center! —Tamfang 00:26, 24 July 2006 (UTC)

icosahedra by 20 tetrahedra[edit]

The claim that "...icosahedron formed by 20 tetrahedron cells meeting at a point." is misleading. the 20 tetrahedrons, although very close, are not equalateral. the edge from the center of the icosahedron is smaller than the other edges.

It is correct. 20 regular tetrahedra CAN meet at a point (and close) when they can be folded into the fourth dimension. Tom Ruen 22:43, 1 March 2007 (UTC)
Think of the analogous situation in 3D with the icosahedron: there, you have 5 equilateral triangles that meet at a vertex. You cannot accomplish this in 2D, but if you fold the triangles into 3D, then it is possible for them to meet at a vertex and be equilateral at the same time. In fact, it is this folding that makes the icosahedron closed. Similarly, in 20 regular tetrahedra in 3D do not exactly meet at a point—but they can be made to do so by folding slightly into 4D. This causes a curvature in 4D which allows the polytope to be closed and bounded, since otherwise it would be a tiling, not a convex polytope. Hope this helps to clarify.—Tetracube (talk) 17:00, 28 May 2008 (UTC)
"Me too!"... Saying "20 regular tetrahedra cannot meet at a point and form an icosahedron" is like saying "four triangles cannot meet at a point and form a square". The latter is true in a flat plane, but not true in three dimensions - nor on the surface of a sphere. Likewise, the former is true in 'flat' 3D space, but not true in flat 4D space - nor in curved (specifically, spherical) 3D space. mike40033 (talk) 06:46, 29 May 2008 (UTC)

4th dimension[edit]

How is it that it is possible to display the general drift of a polychoron by displaying it in only 2 dimensions? Because the 4th dimension is confusing like that. —Preceding unsigned comment added by Lighted Match (talkcontribs) 02:04, 28 May 2008 (UTC)

Yes, it's tough. Doesn't mean we can't try, eh? —Tamfang (talk) 09:01, 28 May 2008 (UTC)


It is also called a tetraplex and or polytetrahedron due to it being bounded by tetrahedral cells.

This new sentence has two features offensive to conservative grammarians. Due is an adjective, and adjectives apply to nouns, not verb clauses. Letting that go, since this use of due is not likely to go away, the name is not due to it but to its being, note the possessive. —Tamfang (talk) 01:14, 14 September 2008 (UTC)

I don't have an opinion on grammar, but it shouldn't be implied that the 5-cell or 16-cell are called tetraplexes as well for having tetrahedral cells, since they are not called this. Tom Ruen (talk) 01:54, 14 September 2008 (UTC)


Is this the regular polytope with the highest number of faces (above dimension 2)? (talk) 22:31, 12 November 2008 (UTC)

Nope, you could easily get a regular polytope with a larger number of facets/faces with a cross polytope of a sufficiently high dimension (a 10-dimensional cross polytope has 1024 facets). A 301-dimensional hypercube would have 602 facets, and a 600-dimensional simplex would have 601 facets. And you can just keep going.—Tetracube (talk) 23:53, 12 November 2008 (UTC)

New images for consideration[edit]

Please consider projections with a bit more color.

Cell600Petrie.svg Cell600 3D.png Cell600SquareSym.png

Okay, what do the colors mean? —Tamfang (talk) 20:48, 2 February 2010 (UTC)
As described in the image description, they are allocated based on the position of the vertices. Another option is to allocate based on overlap counts. If the color scheme seems objectionable, there are 50 gradients to choose from. Jgmoxness (talk) 02:43, 3 February 2010 (UTC)

Not an image here but a video I took part in creating showing 3D cross-sections of 600-cell. See . I have source code for creating this. Let me know if you'd like to use it. Rnicomaths (talk) 22:56, 2 November 2015 (UTC)

It's nice. Can you make an animated gif of it? Tom Ruen (talk) 10:16, 3 November 2015 (UTC)
I've never made an animated GIF before but I'll have a look into it soon.Rnicomaths (talk) 08:09, 4 November 2015 (UTC)
I use PolyView to generate animated Gifs, like here [1]. it can batch convert files to GIF, and then combine them into a single animation. Tom Ruen (talk) 10:26, 4 November 2015 (UTC)
Ok, I could not get polyview to download for me on chrome, instead I have to create this. Nt entirely sure on the caption so let me know of any suggestions.Rnicomaths (talk) 08:28, 5 November 2015 (UTC)
An example of 3D cross-sections of a 600-cell as it translates through the z-axis (or'fourth dimension') at a particular angle.
By the way, I have some similar videos for other regular 4d polytopes (although the pentatope seems to bug up). I am also able to change colours of faces and rotation/translation of the polytope if needed. As well as similar code for 2d cross-sections of 3d shapes.Rnicomaths (talk) 08:34, 5 November 2015 (UTC)

This seems wrong: In geometry, the 600-cell (or hexacosichoron). Should a 600-cell be called a hexahectachoron?[edit]

This seems wrong: In geometry, the 600-cell (or hexacosichoron)

Since icosi means 20, should hexacosi mean 6x20 = 120, then an hexacosichoron would be a 120-cell.

Should a 600-cell be called a hexahectachoron? —Preceding unsigned comment added by Tentacles (talkcontribs) 15:44, 6 June 2010 (UTC)

It's hexa-cosi, not hex-icosi. Apparently –cosi– is a combining form meaning 'hundred'. (Ancient number systems don't always make the most sense in the world.) —Tamfang (talk) 17:41, 6 June 2010 (UTC)

Proof 30-tet Boerdijk–Coxeter helix circumnavigates the 600-cell.[edit]

This proof has two parts, a heuristic part and an analytical part. It is also leveraged off the analysis of R.W.Gray's Tetrahelix page referenced on the BC Wiki page.

Heuristic part

1. The center axis of the BC Helix is a straight line, i.e., no torsion or curvature.
2. Locally follow the path of the axis of a BC Helix through the tets in the 600-cell.
3. Since this path is a geodesic (from #1 above) in the 600-cell it must close on itself and, since the 600-cell exists, there can be no leftover fraction of a tet.

So the BC Helix circumnavigates the 600-cell with an integral number of tets.

Analytical part (instead of numbers below you can (painfully) use the analytical radicals from Gray's page if you prefer. But since the heuristic part above demonstrates an integral number of tets, a small round off error is still sufficient. )

1. When a vector traverses a hyperplane boundary between two facets of a polytope the turning angle θ in Euclidean space is
\cos\theta = \sin^2\alpha  - \cos^2\alpha \; \cos\phi\

where α is the angle of incidence of the vector to the normal of the boundary, and Φ is the dihedral angle between the facets/cells.

2. The dihedral angle between tets in the 600-cell is ~164.48°.
3. The angle of incidence of the BC axis to an internal tet surface is ~39.23°. You can derive this by taking the BC helix vertices from R.W.Gray's page, taking the cross product of two sides of an internal triangle face to get the normal vector, then take the dot product with the axis to get the angle of incidence.
4. Plugging 39.23° and 164.48° into the equation of 1 above we get
\cos\theta = 0.978
So θ = 12.0°, which is 1/30 of a circle.

QED Cloudswrest (talk)

Your analytical proof only shows that θ is close to 12.0°, since you're working with approximtions. I guess you're combining that with knowledge that θ = 360/n to conclude n is 12. But it might be fun to figure out the exact dihedral angle between tets in the 600-cells, and the exact angle of incidence of the BC axis to an internal tet.
In fact, there is probably some easier proof. But it's nice to understand the geometry of this thing in many ways. John Baez (talk) 23:42, 3 September 2015 (UTC)
The angle of incidence is sin-1(sqrt(2/5)) = ~39.23. Cloudswrest (talk) 01:45, 4 September 2015 (UTC)

Hello Cloudswrest,

Perhaps you are responding to my comment in the talk page of the "Boerdijk-Coxeter helix" page, perhaps not. There I asked for a reference for the claim that these helices can be used to realize a simplicial version of the Hopf fibration. The claim is true (not too hard to verify via GAP, for example) and interesting. I checked the references on the page though and none contained a proof as far as I could tell. (Or mention of the result.) It is important to me to know whether this is an original claim made here by an editor or whether this appears in the literature. If the latter, a reference would be useful. If the former, I would like to know whether such editors have a proof.

To be clear, I am not interested in any wikipedia policy ramifications of an admission that this is an original claim. I am simply interested in correct attribution of this result and whether any published proof exists. (talk) 12:01, 20 September 2015 (UTC)

I've made cursory online searches for references but have been unable to find any. The initial claim was added by editor "Sam Nead" on 19:02, 2 October 2012‎ to the "discrete examples" section of the Hopf fibration article (said section since deleted by an ever vigilant Wikipedian). This prompted some discussion here , whereupon I realized that these rings must be Boerdijk-Coxeter helices bent in the fourth dimension, and from there editor Tom Ruen and I fleshed it out with drawings and examples on this page, the Hopf fibration page and the Boerdijk-Coxeter helix page. Cloudswrest (talk) 14:48, 20 September 2015 (UTC)
Excellent, thanks. (talk) 17:20, 20 September 2015 (UTC)