Remarks on Algorithms
general properties of the polyhedra:
- all facets are positive orientated.
- each point is contained at polygon defined by its neighboured vertices
- Die Kanten schneiden sich nur in den Eckpunkten des Mosaiks.
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
- random points: Choose N points in the unit cube and
project them onto the sphere.
- spiralmethod according to Kuilaars-Saff
- box method according to Gerold Wagner:
Teile die Kugel in n gleichgroße Teile (i.d.R. Vierecke) und
wähle aus jedem Teil per Zufall einen Punkt aus.
- polyhedral constructions
- double-pyramide Ein Punkt befindet sich auf dem Nordpol,
ein anderer auf dem Südpol; die übrigen Punkte
befinden sich auf dem Äquator.
- antiprismatic double-pyramide:
auf die Deck- und Grundfläche eines Antiprismas
werden kongruente, gerade Pyramiden aufgesetzt.
- prismatic cluster:
Auf einer Anzahl von Breitenkreisen befinden sich je 6 Ecken;
ergänzt werden die Polyder durch geeignete Kappen am Nord-
bzw. Südpol.
- icosahedral symmetry see: Ian Stewart
- best point-arrangement of an other evaluation, e.g. problem of Thomson
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.