# 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.)

## Figures You Can Rotate With a Mouse

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

**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.**