Uniform Polyhedra Pictures
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Uniform polyhedra have regular faces and equivalent vertices. They include the regular polyhedra (known to Plato) and the semi-regular polyhedra (known to Archimedes). By allowing star-shaped regular polygons for faces many others can be obtained. Some of these were known to Kepler. Others were found in the 1880's and in the 1930's.
All were eventually found. The complete list first appeared in the Coxeter, Longuet-Higgins, and Miller paper: Uniform Polyhedra, Phil. Trans. Royal Soc. London, Ser. A, 246 (1953), 401-409.
In 1993 Ziv Har'El published a very nice paper "Uniform Solution for Uniform Polyhedra". It appeared in Geometriae Dedicata, 47 (1993), 57-110. At the same time (if not earlier) Ziv Har'El released his C code implementation of his algorithms and executables for various platforms. These programs generate data and pictures for all of the uniform polyhedra and their duals. All this and much more is available at his site on the web. Be sure to check out his Java Demo to see rotating uniform polyhedra and their duals. Or rotate them yourself using the Virtual Reality Models (VRML) at his site. For this you may need to get a VRML viewer. A free one for Windows users is Cosmo Player, which you can get here.
The DOS executable, kaleido.exe still runs fine under Windows and will generate pictures on your screen. Download it from the kaleido link at Ziv Har'El's site. Then, for example, enter 'kaleido -g -s' in a 'DOS box' to view images of uniform polyhedra on your screen.
Roman Maeder ported Kaleido to Mathematica. For much information on uniform polyhedra and pictures (including animated gif's) check out his web site.
Russell Towle wrote programs to convert the uniform polyhedra created using Roman Maeder's Mathematica notebook to POV-Ray #include files. POV-Ray is a freeware ray tracing program. Using Russell Towle's include files with POV-Ray anyone can generate high resolution pictures (from any viewpoint) of uniform polyhedra. That is how I created the uniform polyhedra pictures below. There are 75 in all, plus 5 infinite families (not shown), related to prisms and antiprisms. The duals are not shown below. See Ziv Har'El or Roman Maeder's web pages (links above) for pictures including the duals.
Thumbnails and associated html created by IrfanView.