imagine a regular dodecahedron, and its group of rotations. this group is "simple" of order 60, hence isomorphic to the group A5, of even permutations of 5 elements.(adsbygoogle = window.adsbygoogle || []).push({});

i.e. each face is preserved by 5 rotations, yet can be translated to 12 different faces, hence there are 60 motions in all.

can one see how the rotations of a dodecahedron do indeed permute faithfully some collection of 5 objects associated to that solid?

Note that these rotations certainly permute the 12 faces, the 20 vertices, and the 30 edges, hence also the 6 axes joining centers of opposite faces, and the 15 axes joining opposite edges, and the 10 axes joining opposite pairs of vertices, but what collection of 5 objects is permuted?

It helps to look at a picture.

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# A group theoretic puzzle in solid geometry

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