[r-t] Methods as polyhedra

[Mark Davies]

Peter King writes,

> For royal & above there are just 3 calling positions (so 180 vertices).
> These are certainly complicated polyhedra, but there is a huge amount of
> symemtry so it can't be that hard to construct.

Ah, but it's the symmetry which limits the physical possibilities. To start
with, you can't map a search space to a polyhedron if the edges of the
network cross; I suspect this is the case with the "Royal and up
tenors-together nodes" although I don't know for sure.

Then, you have to find a vertex-transitive polyhedron with the properties
you need. Certainly vertex-transitive polyhedra with regular faces seem to
be pretty few in number - apart from the prisms and anti-prisms, which are
infinite but by definition seem to contain infeasibly large circuits, we
appear to have a relatively small set; a quick glance at
<http://en.wikipedia.org/wiki/List_of_uniform_polyhedra> gives 120 as the
maximum number of vertices of any uniform non-prismatic polyhedron (examples
being the great rhombicosidodecahedron and the icositruncated
dodecadodecahedron). There isn't one with 180 vertices, let alone 240.

Are there any ringing search spaces which don't have crossing edges but also
don't map to a uniform polyhedron? Or are these two things one and the same?

MBD