From: parendt@nmt.edu (Paul Arendt)
Subject: Re: What exactly _is_ a soccer ball?
Date: 26 May 1999 20:43:43 GMT
Newsgroups: sci.math
Keywords: Cayley graph of symmetry group of icosahedron
In article <374c4e1f.12312405@news.prosurfr.com>,
John Savard wrote:
>cq315@FreeNet.Carleton.CA (Hank Walker) wrote, in part:
>
>>Are soccer balls for real? Pentagons and hexagons
>>happily tesselating on a sphere...
>
>Essentially, it has the same symmetry as a dodecahedron, and that's
>why there are 12 pentagons on it. The 20 hexagons are one way to fill
>the space between the pentagons if you shrink them and rotate them 180
>degrees.
Actually, in one sense it *is* the symmetry group of the dodecahedron/
icosahedron. If you draw the (discrete) group of symmetries as a graph,
where vertices are group elements and edges are generators, then the
resulting lattice is that of a soccer ball. (Do you see which
subgroups the hexagons and pentagons with the identity at one vertex
correspond to?) All you'd have to do to make the correspondence
complete is give each edge the same length, and embed the thing in
R^3.