Bell-ringing methods as polyhedra

In his popular book "The Bob Caller's companion" Mr. Steve Coleman notes how any bell-ringing method can be represented as a directed graph, with a node for each lead head and an edge for each plain lead or bob lead. He further notes that if the graph can be drawn without any of the edges crossing, then it can be made into a polyhedron, with the edges and nodes of the graph being the edges and nodes of the polyhedron. He gives Grandsire Doubles and Plain Bob Doubles as examples of this.

This left me asking myself the question: how many different polyhedra are there whose nodes and edges map onto the graph of the lead-heads of a ringing method, popular or otherwise? I enumerated the possible cases for lead heads of plain doubles and minor methods and found the following cases, some of which have polyhedral graphs and some of which do not.

Plain Doubles

Leads in Plain course Leads in Bob course lead-end change lead-head code Polyhedron Example (and net)
4 2 125 13524 p Truncated Octahedron Plain Bob (PDF of net)
4 3 1 14253 s Rhombicuboctahedron Union Doubles (PDF of net)
4 3 1 13524 r Rhombicuboctahedron Reverse Bob
4 4 125 14253 q Graph not un-crossable St. Simon

Plain Minor

Leads in Plain course Leads in Bob course lead-end change lead-head code Polyhedron Example (and net)
5 3 12 35264 a rhombicosidodecahedron
with a bit of cheating or a small icosicosidodecahedron
Plain Bob ( PDF of net)
5 5 12 56342 b Graph not un-crossable Double Bob
5 2 12 64523 e Truncated Icosahedron Hereward Bob
5 5 12 42635 f Graph not un-crossable St. Clements
5 5 16 35264 g Graph not un-crossable Single Court
5 2 16 56342 h Truncated Icosahedron Double Court ( PDF of net)
5 5 16 64523 l Graph not un-crossable Lytham Bob
5 3 16 42635 m rhombicosidodecahedron
same cheat as "a" lead-ends or
a small icosicosidodecahedron
Barking Mad Bob

The cheat for "a" and "m" lead-ends is to arrange the lead-ends on the pentagonal faces as a 5-pointed star, so the edges of the pentagon are neither a plain nor a bob lead. I do not know if there is a non-regular method whose graph forms a rhombicosidodecahedron without this trick.

For plain doubles with an extra hunt or slow course bell I note the following cases

Plain doubles with extra hunt or slow course bell

Leads in Plain course Leads in Bob course lead-end change lead-head Polyhedron Example (and net)
3 2 xx xx Truncated Tetrahedron Grandsire (PDF of net)
3 3 xx xx Cuboctahedron Slapton Slow Course(PDF of net)
Photo of polyhedra

All of these polygons are Archimedean solids (except for the small icosicosidodecahedron). Two of them are in Mr. Coleman's book and one (Slapton) almost is: the graph is shown but the polyhedron is not. I have not seen Union or a method like it as a Rhombicuboctahedron anywhere else. I have not seen either of the minor methods as polyhedra either. I believe that there can be no others: can anyone prove me wrong? Emails detailing what I have missed welcome at H dot C dot Pumphrey at ed dot ac dot uk.

If we examine the full list of archimedean solids, we find that for most of the ones not listed above, there is a good reason why their nodes and edges do not form the graph of a ringing method. The solids that work tend to have two distinct types of edge. One type of edge maps to a plain lead, the other to a bob lead. The odd one out is the cuboctahedron: this works because slapton breaks the symmetry of the solid by making three of the triangular faces be plain courses and the other three be bob courses. The square faces become the touch PBPB. This would not work for the icosidodecahedron (the only other Archimedean solid with just one type of edge) because the non-triangular faces have an odd number of edges.

It is clear why the solids with three types of edge would not work: if one sort of edge is plain leads and another sort is bob leads, what is the third sort?

The remaining shapes that don't work would describe methods with 3-lead plain courses, 2-lead bob courses (like Grandsire Doubles) but 8 or 10-lead alternating courses. I suspect that no such method can exist.

Name No. edge types Method or reason why none exists
Truncated tetrahedron 2 Grandsire
Cuboctahedron 1 Slapton
Truncated cube 2 None. Method would have a 3-lead plain course, 2-lead bob course and 8-lead alternating course (PBPBPBPB). I don't think such a method exists.
Truncated octahedron 2 Plain Bob Doubles
Rhombicuboctahedron 2 Union Doubles
Truncated cuboctahedron 3 None. Shape has three distinct types of edge.
Icosidodecahedron 1 None
Truncated dodecahedron 2 None. Method would have 3-lead plain course, 2-lead bob course and 10-lead alternating course (PBPBPBPBPB)
Truncated icosahedron 2 Double Court
rhombicosidodecahedron 2 Plain Bob Minor (with star cheat)
Truncated icosidodecahedron 3 None. The three edges reason again.

I have not considered the snub cube and snub dodecahedron (with their chiral pairs of forms) here: I suspect that they are not symmetrical enough to have form the graph of a method.

Other material on geometry and ringing

The truncated octahedron also turns up in studies of minimus methods, with every row as a vertex and every edge representing a change. There are some java-animated examples here. The earliest discussion of this that I can find is
A.L. Leigh Silver, "Some Musico-Mathematical curiosities", The Mathematical Gazette, Vol. 48, No. 363 pp. 1-17 (1964)

John Harrison has written an extended discussion of these ideas which provides several further references.

Thanks to Charles Walmsley for pointing me at the small icosicosidodecahedron and its friends.