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单词 Uniform Polyhedron
释义

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.

NameDual PolyhedronWythoff Symbol
1TetrahedronTetrahedron
2Truncated TetrahedronTriakis Tetrahedron
3OctahemioctahedronOctahemioctacron
4TetrahemihexahedronTetrahemihexacron
5OctahedronCube
6CubeOctahedron
7CuboctahedronRhombic Dodecahedron
8Truncated OctahedronTetrakis Hexahedron
9Truncated CubeTriakis Octahedron
10Small RhombicuboctahedronDeltoidal Icositetrahedron
11Truncated CuboctahedronDisdyakis Dodecahedron
12Snub CubePentagonal Icositetrahedron
13Small CubicuboctahedronSmall Hexacronic Icositetrahedron
14Great CubicuboctahedronGreat Hexacronic Icositetrahedron
15CubohemioctahedronHexahemioctahedron
16Cubitruncated CuboctahedronTetradyakis Hexahedron
17Great RhombicuboctahedronGreat Deltoidal Icositetrahedron
18Small RhombihexahedronSmall Rhombihexacron
19Stellated Truncated HexahedronGreat Triakis Octahedron
20Great Truncated CuboctahedronGreat Disdyakis Dodecahedron
21Great RhombihexahedronGreat Rhombihexacron
22IcosahedronDodecahedron
23DodecahedronIcosahedron
24IcosidodecahedronRhombic Triacontahedron
25Truncated IcosahedronPentakis Dodecahedron
26Truncated DodecahedronTriakis Icosahedron
27Small RhombicosidodecahedronDeltoidal Hexecontahedron
28Truncated IcosidodecahedronDisdyakis Triacontahedron
29Snub DodecahedronPentagonal Hexecontahedron
30Small Ditrigonal IcosidodecahedronSmall Triambic Icosahedron
31Small IcosicosidodecahedronSmall Icosacronic Hexecontahedron
32Small Snub IcosicosidodecahedronSmall Hexagonal Hexecontahedron
33Small DodecicosidodecahedronSmall Dodecacronic Hexecontahedron
34Small Stellated DodecahedronGreat Dodecahedron
35Great DodecahedronSmall Stellated Dodecahedron
36DodecadodecahedronMedial Rhombic Triacontahedron
37Truncated Great DodecahedronSmall Stellapentakis Dodecahedron
38RhombidodecadodecahedronMedial Deltoidal Hexecontahedron
39Small RhombidodecahedronSmall Rhombidodecacron
40Snub DodecadodecahedronMedial Pentagonal Hexecontahedron
41Ditrigonal DodecadodecahedronMedial Triambic Icosahedron
42Great Ditrigonal DodecicosidodecahedronGreat Ditrigonal Dodecacronic Hexecontahedron
43Small Ditrigonal DodecicosidodecahedronSmall Ditrigonal Dodecacronic Hexecontahedron
44IcosidodecadodecahedronMedial Icosacronic Hexecontahedron
45Icositruncated DodecadodecahedronTridyakis Icosahedron
46Snub IcosidodecadodecahedronMedial Hexagonal Hexecontahedron
47Great Ditrigonal IcosidodecahedronGreat Triambic Icosahedron
48Great IcosicosidodecahedronGreat Icosacronic Hexecontahedron
49Small IcosihemidodecahedronSmall Icosihemidodecacron
50Small DodecicosahedronSmall Dodecicosacron
51Small DodecahemidodecahedronSmall Dodecahemidodecacron
52Great Stellated DodecahedronGreat Icosahedron
53Great IcosahedronGreat Stellated Dodecahedron
54Great IcosidodecahedronGreat Rhombic Triacontahedron
55Great Truncated IcosahedronGreat Stellapentakis Dodecahedron
56RhombicosahedronRhombicosacron
57Great Snub IcosidodecahedronGreat Pentagonal Hexecontahedron
58Small Stellated Truncated DodecahedronGreat Pentakis Dodecahedron
59Truncated DodecadodecahedronMedial Disdyakis Triacontahedron
60Inverted Snub DodecadodecahedronMedial Inverted Pentagonal Hexecontahedron
61Great DodecicosidodecahedronGreat Dodecacronic Hexecontahedron
62Small DodecahemicosahedronSmall Dodecahemicosacron
63Great DodecicosahedronGreat Dodecicosacron
64Great Snub DodecicosidodecahedronGreat Hexagonal Hexecontahedron
65Great DodecahemicosahedronGreat Dodecahemicosacron
66Great Stellated Truncated DodecahedronGreat Triakis Icosahedron
67Great RhombicosidodecahedronGreat Deltoidal Hexecontahedron
68Great Truncated IcosidodecahedronGreat Disdyakis Triacontahedron
69Great Inverted Snub IcosidodecahedronGreat Inverted Pentagonal Hexecontahedron
70Great DodecahemidodecahedronGreat Dodecahemidodecacron
71Great IcosihemidodecahedronGreat Icosihemidodecacron
72Small Retrosnub IcosicosidodecahedronSmall Hexagrammic Hexecontahedron
73Great RhombidodecahedronGreat Rhombidodecacron
74Great Retrosnub IcosidodecahedronGreat Pentagrammic Hexecontahedron
75Great DirhombicosidodecahedronGreat Dirhombicosidodecacron5/2
76Pentagonal PrismPentagonal Dipyramid
77Pentagonal AntiprismPentagonal Deltahedron
78Pentagrammic PrismPentagrammic Dipyramid
79Pentagrammic AntiprismPentagrammic Deltahedron
80Pentagrammic Crossed AntiprismPentagrammic Concave Deltahedron

See also Archimedean Solid, Augmented Polyhedron, Johnson Solid, Kepler-Poinsot Solid, PlatonicSolid, Polyhedron, Vertex Figure, Wythoff Symbol
References

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.

Har'El, Z. ``Kaleido.'' http://www.math.technion.ac.il/~rl/kaleido/.

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

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

Hume, A. ``Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals.'' Computing Science Tech. Rept. No. 130. Murray Hill, NJ: AT&T Bell Lab., 1986.

Hume, A. Information files on polyhedra. http://netlib.bell-labs.com/netlib/polyhedra/.

Johnson, N. W. ``Convex Polyhedra with Regular Faces.'' Canad. J. Math. 18, 169-200, 1966.

Maeder, R. E. ``Uniform Polyhedra.'' Mathematica J. 3, 1993. ftp://ftp.inf.ethz.ch/doc/papers/ti/scs/unipoly.ps.gz.

Maeder, R. E. Polyhedra.m and PolyhedraExamples Mathematica notebooks. http://www.inf.ethz.ch/department/TI/rm/programs.html.

Maeder, R. E. ``The Uniform Polyhedra.'' http://www.inf.ethz.ch/department/TI/rm/unipoly/.

Skilling, J. ``The Complete Set of Uniform Polyhedron.'' Phil. Trans. Roy. Soc. London, Ser. A 278, 111-136, 1975.

Virtual Image. ``The Uniform Polyhedra CD-ROM.'' http://ourworld.compuserve.com/homepages/vir_image/html/uniformpolyhedra.html.

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.


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