On Jul 15, 2015, at 9:17 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
The rotational symmetry group of a dodecahedron is well known to be A_5.
One way to see this is by noticing that precisely five cubes can be formed by taking 8 of the 20 vertices of the dodecahedron; the rotations of the dodecahedron induce even permutations on those cubes:
https://en.wikipedia.org/wiki/Compound_of_five_cubes
Then the Penrose involution gives an odd permutation, upgrading the symmetry group to S_5.
That’s not quite right. The Penrose involution (sqrt(5) <-> -sqrt(5)) does not have the effect of giving you a bigger group, rather, it gives a geometrical interpretation of the unique outer automorphism of the original symmetry group, A_5. Try mapping the “golden coordinates” of the regular icosahedron ([tau,1,0] , …) and you will see that the involution maps the icosahedron into the stellated icosahedron. In particular, pentagonal faces map to pentagrams. That in turn tells you that 2pi/5 rotations get mapped to 4pi/5 rotations about the same axis (this is also the outer automorphism of the 5-element cyclic group, the symmetry of the planar Penrose pattern). Little did we know that the only thing that separates the headquarters of the US Defense Department and ritual satanic worship is a minus sign. -Veit