drmike wrote:I think you can put any number of coils around a center and have the field go to zero in the exact center (or if you get fancy, a couple of places). You don't have spherical symmetry though unless you have 4, 6 or 12 coils.
The main things which Dr. B. stated was necessary for the core polyhedron for a polywell are:
1. A high degree of spherical symmetry, and
2. Polyhedron vertices which have an even number of edges.
The first gets you the symmetry, the second assures that the faces of the polyhedron can be 2-colored -- each face is either N or S, and adjacent faces are different.
The edges surrounding an N-labeled face correspond to a coil in the polywell, and the edges surrounding an S-labeled face correspond to segments of several coils in the polywell.
An easy way to get appropriate polyhedra is to take normally very symmetric polyhedra which might not have the second property and "rectify" it -- truncate each vertex to the midpoint of each edge, yielding a resultant polyhedron which has as many faces as the original had faces and vertices, as many vertices as the original had edges, and as many edges needed to make the Euler characteristic V-E+F = 2. Any convex polyhedron can be rectified, and any rectified polyhedron has vertices of order 4, with new and original faces alternating around each vertex. Rectified convex polyhedrons are exactly what you need for a polywell.
Rectified convex polyhedrons are also exactly what you get if you make a polywell by putting a coil on each face of a polyhedron. The center of each original face corresponds to what we've been calling a "center cusp", the center of each new face corresponds to what we've been calling a "corner cusp", and the vertex of the rectified polyhedron corresponds to what we've been calling an "edge cusp" or a "line cusp".
The platonic solids are a good example of very symmetric polyhedra, and work well for this.
A tetrahedron (V=4,F=4,E=6) rectifies into an octahedron (V=6,F=8,E=12). So with 4 coils you get a polywell with octahedral symmetry.
A cube (V=8, F=6, E=12) rectifies into a "cuboctahedron", which has V=12, F=14, E=24. Six of the faces are square, eight are triangular. Six coils in a cube gives you a WB-7, which has cuboctahedral symmetry.
A cube and an octahedron are dual of each other -- you get from the other by replacing vertices with faces or vice versa. Dual polyhedra rectify to the same polyhedra, so a rectified octahedron is also a cuboctahedron. So with eight coils you also get a polywell with cuboctahedral symmetry, not so different than from a six-coil polywell.
A dodecahedron has V=20, F=12, and E=30. Rectifying it gives a structure called an Icosidodecahedron, which has V=30, F=32, and E=60.
It is a different structure than a cuboctahedron, and is a reasonable alternative to investigate. It has 12 pentagonal faces and 20 triangular faces, and only takes 12 coils to investigate.
An icosahedron has V=12, F=20, and E=30, and is the dual to the dodecahedron. Rectification also yields an icosidodecahedron. You can make one with 20 coils.
While there are some differences between the two cubocahedral arrangements due to the fact that the coils are not square/triangular but are round (and similar for the differences between the to icosidodecahedral arrangements), it is unlikely that, big picture wise, the difference is sufficient to make one significantly better or worse than the other. Comparing 6 v 8 coils, or 12 v 20 coils is probably best done after we know the answer to "will it work?"
A "buckyball" is a highly symmetric shape, corresponding to a truncated icosahedron. It doesn't have property 2 above, but a rectified truncated icosahedron would. But then we quickly run into issues of diminishing returns: A rectified truncated icosahedral polywell would require 32 coils, likely of two different sizes. Compared to a 6-coil "cube" or a 12-coil "dodecahedron", I doubt the improvement would justify the complexity.