![]() ![]() But it would still be nice to find a human-readable proof. I believe neither of these references has a proof! David Roberson has verified it using Sage, as explained in his comment on a previous post. The disparaging remark in the second footnote to Coxeter 4, p. 231) was right when he said the 120 vertices of ’s in ten different ways. Coxeter claims it’s true in the footnote in Section 14.3 of his Regular Polytopes, in which he apologizes for criticizing someone else who earlier claimed it was true: It’s easy to get ahold of 10, so the hard part is proving there are no more. ![]() This leads to another question: how many ways can we fit a compound of five 24-cells into a 600-cell? So, we get a compound of five 24-cells, whose vertices are those of the 600-cell. Each coset gives the vertices of a 24-cell inscribed in the 600-cell, and there are 5 of these cosets, all disjoint. And it works!Īnd it’s easy: just take a subgroup of, and consider the cosets of this subgroup. Next, since 120 = 24 × 5, you can try to partition the 600-cell’s vertices into the vertices of five 24-cells. So it has a double cover, the binary tetrahedral groupĪnd since we can see as the unit quaternions, the elements of the binary tetrahedral group are the vertices of a 4d polytope! This polytope obviously has 24 vertices, twice the number of elements in -but less obviously, it also has 24 octahedral faces, so it’s called the 24-cell:Įach way of making into a subgroup of gives a way of making the binary tetrahedral group into a subgroup of the binary dodecahedral group … and thus a way of inscribing the 24-cell in the 600-cell! The rotational symmetry group of a tetrahedron is contained in the group of rotations in 3d space: Just as we can inscribe a compound of five tetrahedra in the dodecahedron, we can inscribe a compound of five 24-cells in a 600-cell! īut only recently did I notice how this story generalizes to four dimensions.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |