[r-t] Star polyhedra

Philip Saddleton pabs at cantab.net
Fri Apr 11 06:27:09 UTC 2008

Philip Saddleton wrote:
> The icosahedron: since there is an odd number of edges at each vertex, 
> one must be non-directional. Give the two faces meeting at this edge the 
> same colour. Since no two adjcent edges are of this type, a face has one 
> or zero such edges, and there are 12 faces of this colour, three 
> adjacent to each vertex. Give the same colour to the faces that share a 
> vertex, but not an edge with these two. Extend to give twelve faces. 
> Colour the remaining eight faces as two sets of four, such that no faces 
> in the same set share a vertex. The rotation group is A4, and has a 
> representation as permutations of one of the sets of four. Double Bob 
> Doubles, with p=5432, b=3542, s=5324.
Or if you prefer, make the 123 lh the plain lead (Pensthorpe Bob), and 
have a 125 extreme.

Using only two of the three lead types, we get a different polyhedron:

- omit the extreme, and pairs of triangles merge to give squares 
- omit the bob, and four triangles become a hexagon (truncated tetrahedron)

So three for the price of one.

Now, the dodecahedron: there is a subgroup of the automorphism group 
with 20 elements, (the Thurstans part ends), but this is the stabilizer 
of two opposite faces, and hence not vertex transitive.


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