Uniform Polyhedra


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.


Uniform Polyhedra Pictures

tetrahedron
Uniform01.jpg
39.70 KB
truncated tetrahedron
Uniform02.jpg
40.33 KB
octahemioctahedron
Uniform03.jpg
44.67 KB
tetrahemihexahedron
Uniform04.jpg
43.34 KB
octahedron
Uniform05.jpg
42.15 KB
cube
Uniform06.jpg
40.00 KB
cuboctahedron
Uniform07.jpg
43.31 KB
truncated octahedron
Uniform08.jpg
42.30 KB
truncated cube
Uniform09.jpg
41.56 KB
rhombicuboctahedron
Uniform10.jpg
45.04 KB
truncated cuboctahedron
Uniform11.jpg
44.35 KB
snub cube
Uniform12.jpg
47.84 KB
small cubicuboctahedron
Uniform13.jpg
46.85 KB
great cubicuboctahedron
Uniform14.jpg
60.10 KB
cubohemioctahedron
Uniform15.jpg
44.12 KB
cubitruncated cuboctahedron
Uniform16.jpg
52.50 KB
great rhombicuboctahedron
Uniform17.jpg
62.85 KB
small rhombihexahedron
Uniform18.jpg
47.37 KB
stellated truncated hexahedron
Uniform19.jpg
60.17 KB
great truncated cuboctahedron
Uniform20.jpg
56.54 KB
great rhombihexahedron
Uniform21.jpg
59.33 KB
icosahedron
Uniform22.jpg
44.93 KB
dodecahedron
Uniform23.jpg
42.47 KB
icosidodecahedron
Uniform24.jpg
46.34 KB
truncated icosahedron
Uniform25.jpg
48.19 KB
truncated dodecahedron
Uniform26.jpg
45.45 KB
rhombicosidodecahedron
Uniform27.jpg
51.25 KB
truncated icosidodechedon
Uniform28.jpg
52.82 KB
snub dodecahedron
Uniform29.jpg
52.92 KB
small ditrigonal icosidodecahedron
Uniform30.jpg
56.45 KB
small icosicosidodecahedron
Uniform31.jpg
54.35 KB
small snub icosicosidodecahedron
Uniform32.jpg
59.61 KB
small dodecicosidodecahedron
Uniform33.jpg
52.74 KB
small stellated dodecahedron
Uniform34.jpg
59.86 KB
great dodecahedron
Uniform35.jpg
47.48 KB
dodecadodecahedron
Uniform36.jpg
54.11 KB
truncated great dodecahedron
Uniform37.jpg
51.63 KB
rhombidodecadodecahedron
Uniform38.jpg
58.82 KB
small rhombidodecahedron
Uniform39.jpg
52.29 KB
snub dodecadodecahedron
Uniform40.jpg
61.43 KB
ditrigonal dodecadodecahedron
Uniform41.jpg
56.31 KB
great ditrigonal dodecicosidodecahedron
Uniform42.jpg
58.02 KB
small ditrigonal dodecicosidodecahedron
Uniform43.jpg
54.83 KB
icosidodecadodecahedron
Uniform44.jpg
59.51 KB
icositruncated dodecadodecahedron
Uniform45.jpg
57.43 KB
snub icosidodecadodecahedron
Uniform46.jpg
62.86 KB
great ditrigonal icosidodecahedron
Uniform47.jpg
56.42 KB
great icosicosidodecahedron
Uniform48.jpg
59.42 KB
small icosihemidodecahedron
Uniform49.jpg
47.98 KB
small dodecicosahedron
Uniform50.jpg
55.19 KB
small dodecahemidodecahedron
Uniform51.jpg
46.96 KB
great stellated dodecahedron
Uniform52.jpg
68.99 KB
great icosahedron
Uniform53.jpg
63.85 KB
great icosidodecahedron
Uniform54.jpg
62.34 KB
great truncated icosahedron
Uniform55.jpg
60.42 KB
rhombicosahedron
Uniform56.jpg
58.76 KB
great snub icosidodecahedron
Uniform57.jpg
66.93 KB
small stellated truncated dodecahedron
Uniform58.jpg
62.13 KB
truncated dodecadodecahedron
Uniform59.jpg
60.13 KB
inverted snub dodecadodecahedron
Uniform60.jpg
65.88 KB
great dodecicosidodecahedron
Uniform61.jpg
70.63 KB
small dodecahemicosahedron
Uniform62.jpg
53.44 KB
great dodecicosahedron
Uniform63.jpg
57.86 KB
great snub dodecicosidodecahedron
Uniform64.jpg
74.68 KB
great dodecahemicosahedron
Uniform65.jpg
54.28 KB
great stellated truncated dodecahedron
Uniform66.jpg
70.85 KB
great rhombicosidodecahedron
Uniform67.jpg
72.47 KB
great truncated icosidodecahedron
Uniform68.jpg
66.16 KB
great inverted snub icosidodecahedron
Uniform69.jpg
78.48 KB
great dodecahemidodecahedron
Uniform70.jpg
68.21 KB
great icosihemidodecahedron
Uniform71.jpg
64.15 KB
small retrosnub icosicosidodecahedron
Uniform72.jpg
95.32 KB
great rhombidodecahedron
Uniform73.jpg
71.03 KB
great retrosnub icosidodecahedron
Uniform74.jpg
91.60 KB
great dirhombicosidodecahedron
Uniform75.jpg
76.01 KB

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