Martin's Polyhedra. Martin Trump has an excellent polyhedra recipe. He places the vertices at random on an imaginary sphere then computes the forces that repulsion and inverse square law would generate. Then applies this force to the vertices to get a new position. After iteration the vertices reach a stable position. Another routine creates faces. I recommend Martin's Polyhedra Page for a full description.
As an attempt to classify these polyhedra I have encoded the length of each edge as its color. Some of these polyhedra appear in the applet opposite. You can find them by selecting vertice numbers from the choice box and rotate them by dragging the mouse over them.
I certainly did not come up with a simple classification but lots of different series. 8,16,32 & 48 have 2 opposite square faces they look like extra rows of triangles have been inserted to create the next member of the series. 7,12,17,27 are a 5 sided bipyramid with the 2 conical sections separated by increasing rows of triangles. 24,48a &72 appear to be expansions of a cube by a similar method. 32a is a beatifull stellated dodecahedron 72a seems to be a snub dodecahedron with the 5 sided faces stellated. Most of the smaller ones(less than 32) are in a series of some sort.
The higher values over 40 are rarely in these series but they show many symmetries. 29 has a rotational symmetry where every diameter from the axis of rotation is a mirror. It takes time to spot but this sort of symmetry seems to be quite common in the higher vertice numbers. The dual function in the applet is handy to help see symmetries. Unchecking the faces button in the view tile makes the polhedra a line drawing which can be handy for spotting symmetries.
I have written a full polyhedra generator using Martin's method. Be warned it is not a graceful program
click here to visit it
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