[r-t] Methods as polyhedra

Hugh Pumphrey hcp at staffmail.ed.ac.uk
Fri Apr 4 16:23:02 UTC 2008

Dear ringing theorists,

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 a ringing method, 
popular or otherwise?

I know of the following cases, which are all plain-hunting doubles methods.

PCL    BCL   Polyhedron             Example method
3       2   Truncated Tetrahedron    Grandsire
3       3   Cuboctahedron            Slapton Slow Course
4       2   Truncated Octahedron     Plain Bob
4       3   Rhombicuboctahedron      Union Doubles

PCL and BCL are the number of leads in a plain course and a bob course. 
Nets for the listed methods are available at 

The only method in this list that Mr. Coleman does not mention is Union, 
although he does not give the polyhedron for Slapton.

That leaves me with these questions:

(*) Are there any "method-polyhedra" that I have missed?
(*) Are there any written articles on this? (So far, my assult on the 
literature has not turned up anything apart from Mr. Coleman's book.)

Yours geometrically

Hugh Pumphrey

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