How to Distribute N Points on the Surface of a Sphere ?

Isoperimetrisches Problem

The question which polyhedron of given surface has largest volume had been studied thoroughly by L. Lindelöf and H. Minkowski. They proofed that the best polyhedron is circumscribe the sphere and contact it the barycenter of its facets.
Thus this problem asks for the smallest volume of the tangential polyhedron. That means:

m: sum (l1)
D: volume of the tetrahedron defined by
the sphere's center, the point on the sphere and
two neighboured vertices of the tangential polyhedron.

The best polyhedron of a given combinatorical type fullfills a local criterion:

Each point on the sphere is the centroid of the polyhedron's facet it belongs to.

The problem ist solved for N = 4,5,6,12; a list of "good" polyhedra is due to M. Goldberg.

Literatur:

[1] Fejes-Tóth,Lászlo:
Lagerungen in der Ebene, auf der Kugel und im Raum
Springer, Berlin-Heidelberg, 1956, 1972
[2] Goldberg, Michael:
The Isoperimetric Problem for Polyhedra
Tôhoku Math. J. 40 (1935), 226-236
[3] Lindelöf, L.:
Propriétés générales des polyèdres qui, sous une étendue superficielle donnée, renferment le plus grand volume.
Bull.Acad.Sci.St.Petersburg 14 (1869), 257-269
Math. Ann 2 (1870). 150-159
[4] Minkowski, Hermann:
Volumen und Oberfläche
Math. Ann. 57 (1903), 447-495
[5] Steinitz, Eduard:
Über isoperimetrische Probleme bei konvexen Polyedern I,II
J. reine angew. Math. 158 (1927), 129-153
J. reine angew. Math. 159 (1928), 133-143
[4] Sucksdorff, C.-G.:
Détermination du Pentaèdre de Volume donné
Journal de Mathématiques Pures et Appliquées, Sér II (1857), 91-109


Lienhard Wimmer
2009-01-30