Optimal Polyhedra of Multiplicity Greater Than One


For n=15, n=62, n=76, and n=117 the best known solutions to the problem of spreading n points on a sphere so as to maximize the distance between the closest two points, come in pairs. In each case both members of the pair share exactly the same minimal edge length. One member of each pair has some symmetry but has at least one point which can be moved to an alternative position, destroying symmetry but preserving the minimal edge. Each of these possibilites is shown visually below. Unfortunately, the difference between alternative objects is small, so not readily noticed except perhaps when n=15. Nonetheless, each arrangement is a bonafide rigid packing and each packing differs from its alternate form. (Note: The line through each figure represents the z-axis.)


Static Figures

(Click on figure for a larger version.)

n=15 3-symmetry
n=15, 3-symmetry
5.87 KB
n=15 no symmetry
n=15, no symmetry
6.07 KB
n=62, 2-symmetry
n=62, 2-symmetry
9.94 KB
n=62 no symmetry
n=62 no symmetry
10.40 KB
n=76, 2-symmetry
n=76, 2-symmetry
10.69 KB
n=76, no symmetry
n=76, no symmetry
11.01 KB
n=117 3-symmetry
n=117, 3-symmetry
12.26 KB
n=117 no symmetry
n=117, no symmetry
13.14 KB

Figures You Can Rotate With a Mouse

(Click on figure, then try the mouse both without and with the shift key.)

n=15 3-symmetry
n=15, 3-symmetry
Java
n=15 no symmetry
n=15, no symmetry
Java
n=62, 2-symmetry
n=62, 2-symmetry
Java
n=62 no symmetry
n=62, no symmetry
Java
n=76, 2-symmetry
n=76, 2-symmetry
Java
n=76, no symmetry
n=76, no symmetry
Java
n=117 3-symmetry
n=117, 3-symmetry
Java
n=117 no symmetry
n=117, no symmetry
Java

Created by Jim Buddenhagen using:      Data generated by programs of D.A. Kottwitz and myself.

Faces found with the qhull.

Dynamic rotation using the  java applet LiveGraphics3D of Martin Kraus.

Thumbnails and associated html generated by   IrfanView.