Remarks on Algorithms

general properties of the polyhedra:

search for the best point-arrangement

If the polyhedron is given the first step is to optimize it preserving its type. Afterwards we change the combinatorial type searching the combinatorial type of the best point-arrangement.

how to construct useful point-sets on the sphere

simple polyhedra

Each polyhedron containing a non-triangle-facet can be made triangular by drawing diagonal; therefore it is interesting to know the simple polyhedra, i.e. the polyhedra with only triangular facets.

The question how many different simple polyhedral types of with N vertices there are and how to construct them is quite old. Max Brückner constructed a complete liste for N = 4,...,10, which had been reproofed for serveral times. Gunnar Brinkmann wrote a programm which constructs for given N a complete list of simple polyhedral types of with N vertices.

Because the number of polyhedral types of with N vertices grows very fast such complete lists are only useful for quite small N. If vertices of degree 3 and 4 can be excluded it is possible to use these lists up to N=72.

For most evaluations the convex hull of the best point-arrangement has only vertices of degree 5 and degree 6. Thus it is interesting also to study "spherical cluster", e.g. using an algorithm of G.Brinkmann and A.Dress.