How to Distribute N Points on the Surface of a Sphere ?
Energiesummenprobleme
For -∞ < α < 2, α ≠ 0 let
f = ∑i<j
d(pi,pj)α,
For α = 0 let
f = log ∑i<j
1/d(pi,pj).
We are interested in the equilibrium of the points on respect of this functional.
For α = -1 the question can be interpreted as arrangement of electron
on the sphere each repelling each other by electrostatic forces.
There are many papers dealing with computational resulats for this question.
The papers of mathematicians (e.g. Stolarsky, Beck, Wagner) are mostly
interested in estimates for large N; they also prefer the case α = 1.
[1] Berman, Joel D; Hanes, Kit
Optimizing the Arrangement of Points on the Unit Sphere
Math. Comp. 31 (1977), 1006-1008
[2] Dragnev,P.P.; Legg,D.A; Townsend, D.W.
Discrete Logarithmic Energy on the Sphere
Pac. J. Math. 207 (2002), 345-358
[3] Hardin, R.H.; Sloane, N.J.A.; Smith, W.D.
www.research.att.com/~njas/electrons/
[4] Stolarsky, Kenneth B.
The Sum of Distances of N Points on a Sphere
Pac. J. Math. 57 (1975), 563-573
[5] Wagner, Gerold
On the Product of Distances to a Point Set on a Sphere
Pac. J. Math. 144 (1990) no 2, 389-398
[6] Wagner, Gerold
On Averaging Sets
Monatshefte der Mathematik 111 (1991) no 1, 69-79
Lienhard Wimmer
2004-01-09