[r-t] Methods as polyhedra

Mark Davies mark at snowtiger.net
Tue Apr 8 21:18:49 UTC 2008

Hugh Pumphrey:

> After making the original post I discovered that double court minor also
> makes a truncated isosahedron.

Truncated icosahedron, presumably!

It's interesting that all of these you've named so far are Archimedean
solids, presumably this is because these solid have the property that all
vertices radiate the same number of edges: in ringing terms, every leadhead
has the same number of connections via plains and calls. Can we construct
methods to match other such solids? According to Wikipedia, ones with
sensible numbers of vertices (factorials or factorials/2) are:

Truncated cube - 3 edges per vertex, 24 vertices - but has an 8-sided face,
is this too long? Would need PBPBPBPB to be a touch.

Snub cube - 5 edges per vertex, 24 vertices, each vertex has two 3-sided
faces and one 4-sided face. So would need e.g. a plain course of 4 leads and
a bob and a single course of 3 leads each, where a bob and a single are
reverse transpositions. Is this possible? Sounds unlikely. If it is, do the
two chiral forms give rise to two different but related methods?

Truncated dodecahedron - 3 edges per vertex, 60 vertices - contains 10-sided
faces, so e.g. PB*5 must be a touch. Hmm.

Rhombicosidodecahedron - 4 edges per vertex, 60 vertices, faces of 3, 4 and
5 sides. Sounds all right. Method?

Truncated rhombicosidodecahedron - 3 edges per vertex, 120 vertices, faces
of 4, 6 and 10 sides. Hmm again.

Snub dodecahedron - 5 edges per vertex, 60 vertices, faces of 3 and 5 sides.
Would need two types of call, both in-course, so not very sensible.

If none of these work, why not? Do any non-convex polyhedra with regular
vertices work? What about the Great disnub dirhombidodecahedron??


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