[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
http://xweb.geos.ed.ac.uk/~hcp/bells
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
More information about the ringing-theory
mailing list