[r-t] Methods as polyhedra

[King, Peter R]

I remembered the name of the solid - it is a cuboctahedron.  

> -----Original Message-----
> From: ringing-theory-bounces at bellringers.net 
> [mailto:ringing-theory-bounces at bellringers.net] On Behalf Of 
> King, Peter R
> Sent: 09 April 2008 15:53
> To: ringing-theory at bellringers.net
> Subject: Re: [r-t] Methods as polyhedra
> 
> You can also represent touches as polyhedra. For simplicity, 
> if you just allow bobs at W & H then there are 12 coursing 
> orders. 3 H's or 3 W's can be represented by triangles. 
> Attached is a picture of the flattened out polyhedron 
> resulting (I think this is called a Schlegel diagram).
> I'm not entirely sure what the polyhedron is called, it has 
> 12 vertices (the coursing orders), 8 triangles (4 lots of 3Hs 
> and 4 lots of 3Ws) and
> 6 squares (blocks of WHWH). So I think it is a truncated 
> cube, but I think it has some special name which I can't 
> remember. The Hs are the blue traingles and the Ws the arrows.
> 
> Then the problem of composition is to find closed paths 
> through the vertices. For example on the second picture the 
> red line is 3(H2W).
> Presumably some kind of search algorithm related to the 
> travelling salesman problem could be used to select such 
> paths. In the presence of falseness vsiting certain vertices 
> would then preclude other vertices from the proposed path, 
> but I haven't thought through how to implement this.
> 
> Also I don't think the full polyhedron for tenors together 
> compositions should be that complicated. There are 60 in 
> course coursing orders which are the vertices. The faces 
> would then be triangles (3Hs 3 Ws or 3Ms) and pentagons (5 
> Bs) and other simple polygons for WHWH, MHMH etc. I'm sure 
> some bright spark out there could construct it, perhaqps it 
> has already been done.
> 
> PRK
> 
> > -----Original Message-----
> > From: ringing-theory-bounces at bellringers.net
> > [mailto:ringing-theory-bounces at bellringers.net] On Behalf Of Mark 
> > Davies
> > Sent: 08 April 2008 22:19
> > To: ringing-theory at bellringers.net
> > Subject: [r-t] Methods as polyhedra
> > 
> > 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??
> > 
> > MBD
> > 
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