### Thomson Problem Polyhedra

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Published on May 7, 2012

#### Description

J.J. Thomson (1904) asked: How do a small number of electrons arrange themselves on the surface of a unit sphere? Solutions have been proposed for cases where the number of electrons varies from 2 to 400 and beyond. However few solutions have been proved to be optimal.

Two types of polyhedra are useful for those working on the problem; the convex hull and its dual. A nice applet from Syracuse University shows virtual versions of these polyhedra.

(Run thomson.phy.syr.edu/thomsonapplet.php
Type n = 11 into the applet and press start - one can then alternate between the convex hull, "mesh", and its dual, "dual".)

It would be nice to print physical versions of these polyhedra and it was suggested to me that qhull - qhull.org/ - outputs these polyhedra in a simple format called OFF. I was able to write a program that coverts OFF format to openSCAD polyhedron commands.

Coordinates for possible solutions to the Thomson problem are available at The Cambridge Cluster Database www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html Solutions for n=10, 11, 13, and 14 were downloaded and converted to polyhedron commands.

Additionally an openSCAD program used to make Christmas ornaments, thing:14337, needed little modification to show the positions of the electrons as stars on the surface of a sphere. Spheres decorated with possible solutions for n = 10, 11, 13 and 14 were calculated.

The following STL files are provided:

Solution_Sphere_N11.stl
Convex_Hull_N11.stl
Convex_Hull_Dual_N11.stl

The image at the left shows printed versions of these files.

The stl files are in Thomson_polyhedra_stl.zip.
All the SCAD files and the FORTRAN program used to make the polyhedron commands are in Thomson_polyhedra_programs.zip.

#### Instructions

Solution_Sphere_N11.stl needs external support. The other two files should be oriented so that a face rests on the build platform. They can be printed with or without a raft.

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