![]() ![]() ![]() ![]() 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. Dodecahedron with 5 tetrahedra, Visual Insight.So, the compound of five tetrahedra has dissolved into group theory, with each figure becoming a group, and the big group acting to permute its own subgroups!Īll this is just the start of a longer story about compounds of Platonic solids: Indeed, it acts to give all the even permutations-after all, it’s So, to the eyes of a group theorist, the tetrahedra in our compound of five tetrahedra are just the subgroups of acts on itself by conjugation, and this action permutes these 5 subgroups. So, these symmetries form a subgroup of that is isomorphic to There are exactly 5 such subgroups-one for each tetrahedron. If we pick any tetrahedron in our compound of five tetrahedra, its rotational symmetries give rotational symmetries of the dodecahedron. On the other hand, the rotational symmetry group of the tetrahedron is since any rotation gives an even permutation of the 4 vertices of the tetrahedron. So, by the marvel of mathematical reasoning, the rotational symmetry group of the dodecahedron must be the alternating group Furthermore, knowing this permutation, we can tell what rotation we did. We can only get even permutations this way, but we can get any even permutation. To see how this works, we need to think about symmetries.Īny rotational symmetry of the dodecahedron acts to permute the tetrahedra in our compound of five tetrahedra. And there’s a way of partitioning the vertices of the 600-cell into 5 sets, each being the vertices of a 24-cell! So, we get a ‘compound of five 24-cells’. Amazingly, the 4-dimensional version arises from studying the symmetries of the 3-dimensional thing you see above! The symmetries of the tetrahedron gives a 4-dimensional regular polytope called the ’24-cell’, while the symmetries of the dodecahedron give one called the ‘600-cell’. I want to tell you about a 4-dimensional version of the same thing. The result is the compound of five tetrahedra. ![]()
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