[r-t] Methods as polyhedra

[Mark Davies]

Peter King writes,

> You can also represent touches as polyhedra.

You mean, as paths along the edges of polyhedra? Yes, that's what we've been
doing. The polyhedron as a whole represents the "search space" - i.e. all
the lead ends or coursing order nodes that can be reached (vertices) plus
the calls that join them (edges). That's what Hugh meant when he said put a
method onto a polyhedron; it is really the search space of the method plus
calls.

> There are 60 in course coursing orders which are the vertices.

Not exactly - in Minor, there are 60 in-course leadheads, however if you are
looking at tenors-together compositions on high numbers, then there are
always 60 in-course coursing orders, but there are many more vertices
required to represent a composition. Unless you are only dealing with one
calling position (e.g. 3H your only touch!) you have to look at individual
parts of the course.

For example, the section from W to H in coursing order 53246 is distinct
from that between H and W in 53246, and so more than one vertex is required
to represent 53246. You have to split the course into individual nodes. For
a simple case with calls at M, W and H you may have just three nodes per
course, but that is still 180 vertices not 60. If allowing Befores or 
dealing with an nth's place method, there are more.

And sadly, I don't think representing the search space as a polyhedron or
connected graph gives you any new composition search algorithms - tree
searches are still all you have.

MBD