Uniform Polyhedron

The uniform polyhedra are Polyhedra with identical Vertices. Coxeteret al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meet at an Edge, and this was subsequently proven. (However, when any Even number of faces may meet, there are 76 polyhedra.)If the five pentagonal Prisms are included, the number rises to 80.

The Vertices of a uniform polyhedron all lie on a Sphere whose center is theirCentroid. The Vertices joined to anotherVertex lie on a Circle.

Source code and binary programs for generating and viewing the uniformpolyhedra are also available at http://www.math.technion.ac.il/~rl/kaleido/. The following depictions of the polyhedra were produced by R. Maeder'sUniformPolyhedra.m package for Mathematica (Wolfram Research, Champaign, IL). Due to a limitationin Mathematica's renderer, uniform polyhedra 69, 72, 74, and 75 cannot be displayed using this package.

	Name	Dual Polyhedron	Wythoff Symbol
1	Tetrahedron	Tetrahedron
2	Truncated Tetrahedron	Triakis Tetrahedron
3	Octahemioctahedron	Octahemioctacron
4	Tetrahemihexahedron	Tetrahemihexacron
5	Octahedron	Cube
6	Cube	Octahedron
7	Cuboctahedron	Rhombic Dodecahedron
8	Truncated Octahedron	Tetrakis Hexahedron
9	Truncated Cube	Triakis Octahedron
10	Small Rhombicuboctahedron	Deltoidal Icositetrahedron
11	Truncated Cuboctahedron	Disdyakis Dodecahedron
12	Snub Cube	Pentagonal Icositetrahedron
13	Small Cubicuboctahedron	Small Hexacronic Icositetrahedron
14	Great Cubicuboctahedron	Great Hexacronic Icositetrahedron
15	Cubohemioctahedron	Hexahemioctahedron
16	Cubitruncated Cuboctahedron	Tetradyakis Hexahedron
17	Great Rhombicuboctahedron	Great Deltoidal Icositetrahedron
18	Small Rhombihexahedron	Small Rhombihexacron
19	Stellated Truncated Hexahedron	Great Triakis Octahedron
20	Great Truncated Cuboctahedron	Great Disdyakis Dodecahedron
21	Great Rhombihexahedron	Great Rhombihexacron
22	Icosahedron	Dodecahedron
23	Dodecahedron	Icosahedron
24	Icosidodecahedron	Rhombic Triacontahedron
25	Truncated Icosahedron	Pentakis Dodecahedron
26	Truncated Dodecahedron	Triakis Icosahedron
27	Small Rhombicosidodecahedron	Deltoidal Hexecontahedron
28	Truncated Icosidodecahedron	Disdyakis Triacontahedron
29	Snub Dodecahedron	Pentagonal Hexecontahedron
30	Small Ditrigonal Icosidodecahedron	Small Triambic Icosahedron
31	Small Icosicosidodecahedron	Small Icosacronic Hexecontahedron
32	Small Snub Icosicosidodecahedron	Small Hexagonal Hexecontahedron
33	Small Dodecicosidodecahedron	Small Dodecacronic Hexecontahedron
34	Small Stellated Dodecahedron	Great Dodecahedron
35	Great Dodecahedron	Small Stellated Dodecahedron
36	Dodecadodecahedron	Medial Rhombic Triacontahedron
37	Truncated Great Dodecahedron	Small Stellapentakis Dodecahedron
38	Rhombidodecadodecahedron	Medial Deltoidal Hexecontahedron
39	Small Rhombidodecahedron	Small Rhombidodecacron
40	Snub Dodecadodecahedron	Medial Pentagonal Hexecontahedron
41	Ditrigonal Dodecadodecahedron	Medial Triambic Icosahedron
42	Great Ditrigonal Dodecicosidodecahedron	Great Ditrigonal Dodecacronic Hexecontahedron
43	Small Ditrigonal Dodecicosidodecahedron	Small Ditrigonal Dodecacronic Hexecontahedron
44	Icosidodecadodecahedron	Medial Icosacronic Hexecontahedron
45	Icositruncated Dodecadodecahedron	Tridyakis Icosahedron
46	Snub Icosidodecadodecahedron	Medial Hexagonal Hexecontahedron
47	Great Ditrigonal Icosidodecahedron	Great Triambic Icosahedron
48	Great Icosicosidodecahedron	Great Icosacronic Hexecontahedron
49	Small Icosihemidodecahedron	Small Icosihemidodecacron
50	Small Dodecicosahedron	Small Dodecicosacron
51	Small Dodecahemidodecahedron	Small Dodecahemidodecacron
52	Great Stellated Dodecahedron	Great Icosahedron
53	Great Icosahedron	Great Stellated Dodecahedron
54	Great Icosidodecahedron	Great Rhombic Triacontahedron
55	Great Truncated Icosahedron	Great Stellapentakis Dodecahedron
56	Rhombicosahedron	Rhombicosacron
57	Great Snub Icosidodecahedron	Great Pentagonal Hexecontahedron
58	Small Stellated Truncated Dodecahedron	Great Pentakis Dodecahedron
59	Truncated Dodecadodecahedron	Medial Disdyakis Triacontahedron
60	Inverted Snub Dodecadodecahedron	Medial Inverted Pentagonal Hexecontahedron
61	Great Dodecicosidodecahedron	Great Dodecacronic Hexecontahedron
62	Small Dodecahemicosahedron	Small Dodecahemicosacron
63	Great Dodecicosahedron	Great Dodecicosacron
64	Great Snub Dodecicosidodecahedron	Great Hexagonal Hexecontahedron
65	Great Dodecahemicosahedron	Great Dodecahemicosacron
66	Great Stellated Truncated Dodecahedron	Great Triakis Icosahedron
67	Great Rhombicosidodecahedron	Great Deltoidal Hexecontahedron
68	Great Truncated Icosidodecahedron	Great Disdyakis Triacontahedron
69	Great Inverted Snub Icosidodecahedron	Great Inverted Pentagonal Hexecontahedron
70	Great Dodecahemidodecahedron	Great Dodecahemidodecacron
71	Great Icosihemidodecahedron	Great Icosihemidodecacron
72	Small Retrosnub Icosicosidodecahedron	Small Hexagrammic Hexecontahedron
73	Great Rhombidodecahedron	Great Rhombidodecacron
74	Great Retrosnub Icosidodecahedron	Great Pentagrammic Hexecontahedron
75	Great Dirhombicosidodecahedron	Great Dirhombicosidodecacron	5/2
76	Pentagonal Prism	Pentagonal Dipyramid
77	Pentagonal Antiprism	Pentagonal Deltahedron
78	Pentagrammic Prism	Pentagrammic Dipyramid
79	Pentagrammic Antiprism	Pentagrammic Deltahedron
80	Pentagrammic Crossed Antiprism	Pentagrammic Concave Deltahedron

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 136, 1987.

Bulatov, V. ``Compounds of Uniform Polyhedra.''http://www.physics.orst.edu/~bulatov/polyhedra/uniform_compounds/.

Bulatov, V. ``Dual Uniform Polyhedra.'' http://www.physics.orst.edu/~bulatov/polyhedra/dual/.

Bulatov, V. ``Uniform Polyhedra.'' http://www.physics.orst.edu/~bulatov/polyhedra/uniform/.

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.'' Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.

Har'El, Z. ``Uniform Solution for Uniform Polyhedra.'' Geometriae Dedicata 47, 57-110, 1993.

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Har'El, Z. ``Eighty Uniform Polyhedra Generated by Kaleido.'' http://www.math.technion.ac.il/~rl/kaleido/poly.html.

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Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, pp. 1-10 and 98, 1989.

Zalgaller, V. Convex Polyhedra with Regular Faces. New York: Consultants Bureau, 1969.

Ziegler, G. M. Lectures on Polytopes. Berlin: Springer-Verlag, 1995.

• Astroid Pedal Curve
• Astroid Radial Curve
• Asymptosy
• Asymptote
• Asymptotic
• Asymptotic Curve
• Asymptotic Direction
• Asymptotic Notation
• Asymptotic Series
• Atiyah-Singer Index Theorem
• Atkin-Goldwasser-Kilian-Morain Certificate
• Atomic Statement
• Attraction Basin
• Attractor
• Auction
• Augend
• Augmented Amicable Pair
• Augmented Dodecahedron
• Augmented Hexagonal Prism
• Augmented Pentagonal Prism
• Augmented Polyhedron
• Augmented Sphenocorona
• Augmented Triangular Prism
• Augmented Tridiminished Icosahedron
• Augmented Truncated Cube