Talk:Pentachoron

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Mathematics rating:	Start Class	Low Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Please add useful comments here--Cronholm144 08:50, 24 May 2007 (UTC)

If I'm not mistaken, this is the only figure in either three or four dimensional space in which you can have five equidistant points around a central vertex. This should be noted If it is correct.

This property is true of all in the simplex family - (n+1)-points being equidistant in n-space. Tom Ruen 00:26, 16 November 2006 (UTC)

The symmetry group is listed as A₄. I don't think this can be true and should be A₅ instead. In general, I believe that the symmetry group of the n-dimenstional simplex is A_n+1.HannsEwald 01:51, 13 November 2007 (UTC)

A₄ actually refers to the Coxeter group A₄ not the alternating group. The Coxeter group A₄ is isomorphic to the symmetric group S₅ and the subgroup of proper rotations is indeed isomorphic to the alternating group A₅ as you say. Yes, the notation is unfortunate. -- Fropuff 04:42, 13 November 2007 (UTC)

Thanks for the quick explanation. I suppose then we've got to deal with some inconsistencies: in Tetrahedron the symmetry group is named as T_d, while in the Tetrahedral symmetry it is referred to as A₄, which in this context must refer to the alternating group.

While I have to admit unfamiliarity with Coexter groups, it seems to me that we would not lose anything if we stayed with the symmetric group when discussing symmetry groups of the n-simplex, especially if my generalization above is correct.HannsEwald 11:36, 13 November 2007 (UTC)

The advantage of using Coxeter groups is that the symmetry group of every regular polytope is a finite Coxeter group. While the symmetric and alternating groups are sufficient to discuss the rotational symmetries of the 3-dimensional regular polytopes the same is not true in higher dimensions, particularly 4. Also, the Coxeter groups have a direct geometric interpretation since the finite ones are usually defined as groups generated by reflections in Euclidean space.

The symmetry group of the n-simplex is the Coxeter group A_n for all n. This group is isomorphic to S_n+1 as you say, but I find it more helpful to think of the group as a Coxeter group rather than a symmetric group. -- Fropuff 16:43, 13 November 2007 (UTC)