Searching for images of "origami five tetrahedra" produces lots of nice results. Note also that the icosadodecahedron has 6 great-circle like paths on it, which get permuted by its rotations. This shows that A_5 can be embedded in S_6 in a nontrivial way --- which has to do with the outer automorphism of S_6 :-) - Cris On Jul 16, 2015, at 11:13 AM, Dan Asimov <dasimov@earthlink.net> wrote:
[I don't know from the Penrose involution, but:]
Isn't it a bit easier to see the equivalence of the rotational symmetry group of a dodecahedron with A_5 by embedding the compound of 5 tetrahedra in its 20 vertices? Then its 60 rotations correspond to the even permutations of the tetrahedra.
—Dan
On Jul 15, 2015, at 6:17 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 <https://en.wikipedia.org/wiki/Compound_of_five_cubes>
Then the Penrose involution gives an odd permutation, upgrading the symmetry group to S_5.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/