Uniform Polyhedra (80)

Compiled by Paul Bourke
October 2004

Based upon: Zvi Har'El. Uniform solution for uniform polyhedra. Geometriae Dedicata, 47, 1993.
Geometry is PovRay format, and three VRML formats: spheres/cylinders, edges, faces.


01

Tetrahedron

Vertices: 4

Edges: 6

Faces: 4

Wythoff symbol: 3|2 3

Symmetry Group: {3, 3, 3}

Vertex Configuration: tetrahedral

Dual: tetrahedron

02

Truncated tetrahedron

Vertices: 12

Edges: 18

Faces: 8

Wythoff symbol: 2 3|3

Symmetry Group: tetrahedral

Vertex Configuration: {6, 6, 3}

Dual: triakis tetrahedron

03

Octahemioctahedron

Vertices: 12

Edges: 24

Faces: 12

Wythoff symbol: 3/2 3|3

Symmetry Group: tetrahedral

Vertex Configuration: {6, 3/2, 6, 3}

Dual: octahemioctacron

04

Tetrahemihexahedron

Vertices: 6

Edges: 12

Faces: 7

Wythoff symbol: 3/2 3|2

Symmetry Group: tetrahedral

Vertex Configuration: {4, 3/2, 4, 3}

Dual: tetrahemihexacron

05

Octahedron

Vertices: 6

Edges: 12

Faces: 8

Wythoff symbol: 4|2 3

Symmetry Group: octahedral

Vertex Configuration: {3, 3, 3, 3}

Dual: cube

06

Cube

Vertices: 8

Edges: 12

Faces: 6

Wythoff symbol: 3|2 4

Symmetry Group: octahedral

Vertex Configuration: {4, 4, 4}

Dual: octahedron

07

Cuboctahedron

Vertices: 12

Edges: 24

Faces: 14

Wythoff symbol: 2|3 4

Symmetry Group: octahedral

Vertex Configuration: {3, 4, 3, 4}

Dual: rhombic dodecahedron

08

Truncated octahedron

Vertices: 24

Edges: 36

Faces: 14

Wythoff symbol: 2 4|3

Symmetry Group: octahedral

Vertex Configuration: {6, 6, 4}

Dual: tetrakis hexahedron

09

Truncated cube

Vertices: 24

Edges: 36

Faces: 14

Wythoff symbol: 2 3|4

Symmetry Group: octahedral

Vertex Configuration: {8, 8, 3}

Dual: triakis octahedron

10

Rhombicuboctahedron

Vertices: 24

Edges: 48

Faces: 26

Wythoff symbol: 3 4|2

Symmetry Group: octahedral

Vertex Configuration: {4, 3, 4, 4}

Dual: deltoidal icositetrahedron

11

Truncated cuboctahedron

Vertices: 48

Edges: 72

Faces: 26

Wythoff symbol: 2 3 4|

Symmetry Group: octahedral

Vertex Configuration: {4, 6, 8}

Dual: disdyakis dodecahedron

12

Snub cube

Vertices: 24

Edges: 60

Faces: 38

Wythoff symbol: |2 3 4

Symmetry Group: octahedral

Vertex Configuration: {3, 3, 3, 3, 4}

Dual: pentagonal icositetrahedron

13

Small cubicuboctahedron

Vertices: 24

Edges: 48

Faces: 20

Wythoff symbol: 3/2 4|4

Symmetry Group: octahedral

Vertex Configuration: {8, 3/2, 8, 4}

Dual: small hexacronic icositetrahedron

14

Great cubicuboctahedron

Vertices: 24

Edges: 48

Faces: 20

Wythoff symbol: 3 4|4/3

Symmetry Group: octahedral

Vertex Configuration: {8/3, 3, 8/3, 4}

Dual: great hexacronic icositetrahedron

15

Cubohemioctahedron

Vertices: 12

Edges: 24

Faces: 10

Wythoff symbol: 4/3 4|3

Symmetry Group: octahedral

Vertex Configuration: {6, 4/3, 6, 4}

Dual: hexahemioctacron

16

Cubitruncated cuboctahedron

Vertices: 48

Edges: 72

Faces: 20

Wythoff symbol: 4/3 3 4|

Symmetry Group: octahedral

Vertex Configuration: {8/3, 6, 8}

Dual: tetradyakis hexahedron

17

Great rhombicuboctahedron

Vertices: 24

Edges: 48

Faces: 26

Wythoff symbol: 3/2 4|2

Symmetry Group: octahedral

Vertex Configuration: {4, 3/2, 4, 4}

Dual: great deltoidal icositetrahedron

18

Small rhombihexahedron

Vertices: 24

Edges: 48

Faces: 18

Wythoff symbol: 3/2 2 4|

Symmetry Group: octahedral

Vertex Configuration: {8, 4, 8/7, 4/3}

Dual: small rhombihexacron

19

Stellated truncated hexahedron

Vertices: 24

Edges: 36

Faces: 14

Wythoff symbol: 2 3|4/3

Symmetry Group: octahedral

Vertex Configuration: {8/3, 8/3, 3}

Dual: great triakis octahedron

20

Great truncated cuboctahedron

Vertices: 48

Edges: 72

Faces: 26

Wythoff symbol: 4/3 2 3|

Symmetry Group: octahedral

Vertex Configuration: {8/3, 4, 6}

Dual: great disdyakis dodecahedron

21

Great truncated cuboctahedron

Vertices: 24

Edges: 48

Faces: 18

Wythoff symbol: 4/3 3/2 2|

Symmetry Group: octahedral

Vertex Configuration: {4, 8/3, 4/3, 8/5}

Dual: great rhombihexacron

22

Icosahedron

Vertices: 12

Edges: 30

Faces: 20

Wythoff symbol: 5|2 3

Symmetry Group: icosahedral

Vertex Configuration: {3, 3, 3, 3, 3}

Dual: dodecahedron

23

Dodecahedron

Vertices: 20

Edges: 30

Faces: 12

Wythoff symbol: 3|2 5

Symmetry Group: icosahedral

Vertex Configuration: {5, 5, 5}

Dual: icosahedron

24

Icosidodecahedron

Vertices: 30

Edges: 60

Faces: 32

Wythoff symbol: 2|3 5

Symmetry Group: icosahedral

Vertex Configuration: {3, 5, 3, 5}

Dual: rhombic triacontahedron

25

Truncated icosahedron

Vertices: 60

Edges: 90

Faces: 32

Wythoff symbol: 2 5|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 6, 5}

Dual: pentakis dodecahedron

26

Truncated dodecahedron

Vertices: 60

Edges: 90

Faces: 32

Wythoff symbol: 2 3|5

Symmetry Group: icosahedral

Vertex Configuration: {10, 10, 3}

Dual: triakis icosahedron

27

Rhombicosidodecahedron

Vertices: 60

Edges: 120

Faces: 62

Wythoff symbol: 3 5|2

Symmetry Group: icosahedral

Vertex Configuration: {4, 3, 4, 5}

Dual: deltoidal hexecontahedron

28

Truncated icosidodechedon

Vertices: 120

Edges: 180

Faces: 62

Wythoff symbol: 2 3 5|

Symmetry Group: icosahedral

Vertex Configuration: {4, 6, 10}

Dual: disdyakis triacontahedron

29

Snub dodecahedron

Vertices: 60

Edges: 150

Faces: 92

Wythoff symbol: |2 3 5

Symmetry Group: icosahedral

Vertex Configuration: {3, 3, 3, 3, 5}

Dual: pentagonal hexecontahedron

30

Small ditrigonal icosidodecahedron

Vertices: 20

Edges: 60

Faces: 32

Wythoff symbol: 3|5/2 3

Symmetry Group: icosahedral

Vertex Configuration: {5/2, 3, 5/2, 3, 5/2, 3}

Dual: small triambic icosahedron

31

Small icosicosidodecahedron

Vertices: 60

Edges: 120

Faces: 52

Wythoff symbol: 5/2 3|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 5/2, 6, 3}

Dual: small icosacronic hexecontahedron

32

Small snub icosicosidodecahedron

Vertices: 60

Edges: 180

Faces: 112

Wythoff symbol: |5/2 3 3

Symmetry Group: icosahedral

Vertex Configuration: {3, 5/2, 3, 3, 3, 3}

Dual: small hexagonal hexecontahedron

33

Small dodecicosidodecahedron

Vertices: 60

Edges: 120

Faces: 44

Wythoff symbol: 3/2 5|5

Symmetry Group: icosahedral

Vertex Configuration: {10, 3/2, 10, 5}

Dual: small dodecacronic hexecontahedron

34

Small stellated dodecahedron

Vertices: 12

Edges: 30

Faces: 12

Wythoff symbol: 5|2 5/2

Symmetry Group: icosahedral

Vertex Configuration: {5/2, 5/2, 5/2, 5/2, 5/2}

Dual: great dodecahedron

35

Great dodecahedron

Vertices: 12

Edges: 30

Faces: 12

Wythoff symbol: 5/2|2 5

Symmetry Group: icosahedral

Vertex Configuration: {5, 5, 5, 5, 5}/2

Dual: small stellated dodecahedron

36

Dodecadodecahedron

Vertices: 30

Edges: 60

Faces: 24

Wythoff symbol: 2|5/2 5

Symmetry Group: icosahedral

Vertex Configuration: {5/2, 5, 5/2, 5}

Dual: medial rhombic triacontahedron

37

Truncated great dodecahedron

Vertices: 60

Edges: 90

Faces: 24

Wythoff symbol: 2 5/2|5

Symmetry Group: icosahedral

Vertex Configuration: {10, 10, 5/2}

Dual: small stellapentakis dodecahedron

38

Rhombidodecadodecahedron

Vertices: 60

Edges: 120

Faces: 54

Wythoff symbol: 5/2 5|2

Symmetry Group: icosahedral

Vertex Configuration: {4, 5/2, 4, 5}

Dual: medial deltoidal hexecontahedron

39

Small rhombidodecahedron

Vertices: 60

Edges: 120

Faces: 42

Wythoff symbol: 2 5/2 5|

Symmetry Group: icosahedral

Vertex Configuration: {10, 4, 10/9, 4/3}

Dual: small rhombidodecacron

40

Snub dodecadodecahedron

Vertices: 60

Edges: 150

Faces: 84

Wythoff symbol: |2 5/2 5

Symmetry Group: icosahedral

Vertex Configuration: {3, 3, 5/2, 3, 5}

Dual: medial pentagonal hexecontahedron

41

Ditrigonal dodecadodecahedron

Vertices: 20

Edges: 60

Faces: 24

Wythoff symbol: 3|5/3 5

Symmetry Group: icosahedral

Vertex Configuration: {5/3, 5, 5/3, 5, 5/3, 5}

Dual: medial triambic icosahedron

42

Great ditrigonal dodecicosidodecahedron

Vertices: 60

Edges: 120

Faces: 44

Wythoff symbol: 3 5|5/3

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 3, 10/3, 5}

Dual: great ditrigonal dodecacronic hexecontahedron

43

Small ditrigonal dodecicosidodecahedron

Vertices: 60

Edges: 120

Faces: 44

Wythoff symbol: 5/3 3|5

Symmetry Group: icosahedral

Vertex Configuration: {10, 5/3, 10, 3}

Dual: small ditrigonal dodecacronic hexecontahedron

44

Icosidodecadodecahedron

Vertices: 60

Edges: 120

Faces: 44

Wythoff symbol: 5/3 5|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 5/3, 6, 5}

Dual: medial icosacronic hexecontahedron

45

Icositruncated dodecadodecahedron

Vertices: 120

Edges: 180

Faces: 44

Wythoff symbol: 5/3 3 5|

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 6, 10}

Dual: tridyakis icosahedron

46

Snub icosidodecadodecahedron

Vertices: 60

Edges: 180

Faces: 104

Wythoff symbol: |5/3 3 5

Symmetry Group: icosahedral

Vertex Configuration: {3, 5/3, 3, 3, 3, 5}

Dual: medial hexagonal hexecontahedron

47

Great ditrigonal icosidodecahedron

Vertices: 20

Edges: 60

Faces: 32

Wythoff symbol: 3/2|3 5

Symmetry Group: icosahedral

Vertex Configuration: {3, 5, 3, 5, 3, 5}/2

Dual: great triambic icosahedron

48

Great icosicosidodecahedron

Vertices: 60

Edges: 120

Faces: 52

Wythoff symbol: 3/2 5|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 3/2, 6, 5}

Dual: great icosacronic hexecontahedron

49

Small icosihemidodecahedron

Vertices: 30

Edges: 60

Faces: 26

Wythoff symbol: 3/2 3|5

Symmetry Group: icosahedral

Vertex Configuration: {10, 3/2, 10, 3}

Dual: small icosihemidodecacron

50

Small dodecicosahedron

Vertices: 60

Edges: 120

Faces: 32

Wythoff symbol: 3/2 3 5|

Symmetry Group: icosahedral

Vertex Configuration: {10, 6, 10/9, 6/5}

Dual: small dodecicosacron

51

Small dodecahemidodecahedron

Vertices: 30

Edges: 60

Faces: 18

Wythoff symbol: 5/4 5|5

Symmetry Group: icosahedral

Vertex Configuration: {10, 5/4, 10, 5}

Dual: small dodecahemidodecacron

52

Great stellated dodecahedron

Vertices: 20

Edges: 30

Faces: 12

Wythoff symbol: 3|2 5/2

Symmetry Group: icosahedral

Vertex Configuration: {5/2, 5/2, 5/2}

Dual: great icosahedron

53

Great icosahedron

Vertices: 12

Edges: 30

Faces: 20

Wythoff symbol: 5/2|2 3

Symmetry Group: icosahedral

Vertex Configuration: {3, 3, 3, 3, 3}/2

Dual: great stellated dodecahedron

54

Great icosidodecahedron

Vertices: 30

Edges: 60

Faces: 32

Wythoff symbol: 2|5/2 3

Symmetry Group: icosahedral

Vertex Configuration: {5/2, 3, 5/2, 3}

Dual: great rhombic triacontahedron

55

Great truncated icosahedron

Vertices: 60

Edges: 90

Faces: 32

Wythoff symbol: 2 5/2|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 6, 5/2}

Dual: great stellapentakis dodecahedron

56

Rhombicosahedron

Vertices: 60

Edges: 120

Faces: 50

Wythoff symbol: 2 5/2 3|

Symmetry Group: icosahedral

Vertex Configuration: {6, 4, 6/5, 4/3}

Dual: rhombicosacron

57

Great snub icosidodecahedron

Vertices: 60

Edges: 150

Faces: 92

Wythoff symbol: |2 5/2 3

Symmetry Group: icosahedral

Vertex Configuration: {3, 3, 5/2, 3, 3}

Dual: great pentagonal hexecontahedron

58

Small stellated truncated dodecahedron

Vertices: 60

Edges: 90

Faces: 24

Wythoff symbol: 2 5|5/3

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 10/3, 5}

Dual: great pentakis dodecahedron

59

Truncated dodecadodecahedron

Vertices: 120

Edges: 180

Faces: 54

Wythoff symbol: 5/3 2 5|

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 4, 10}

Dual: medial disdyakis triacontahedron

60

Inverted snub dodecadodecahedron

Vertices: 60

Edges: 150

Faces: 84

Wythoff symbol: |5/3 2 5

Symmetry Group: icosahedral

Vertex Configuration: {3, 5/3, 3, 3, 5}

Dual: medial inverted pentagonal hexecontahedron

61

Great dodecicosidodecahedron

Vertices: 60

Edges: 120

Faces: 44

Wythoff symbol: 5/2 3|5/3

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 5/2, 10/3, 3}

Dual: great dodecacronic hexecontahedron

62

Small dodecahemicosahedron

Vertices: 30

Edges: 60

Faces: 22

Wythoff symbol: 5/3 5/2|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 5/3, 6, 5/2}

Dual: small dodecahemicosacron

63

Great dodecicosahedron

Vertices: 60

Edges: 120

Faces: 32

Wythoff symbol: 5/3 5/2 3|

Symmetry Group: icosahedral

Vertex Configuration: {6, 10/3, 6/5, 10/7}

Dual: great dodecicosacron

64

Great snub dodecicosidodecahedron

Vertices: 60

Edges: 180

Faces: 104

Wythoff symbol: |5/3 5/2 3

Symmetry Group: icosahedral

Vertex Configuration: {3, 5/3, 3, 5/2, 3, 3}

Dual: great hexagonal hexecontahedron

65

Great dodecahemicosahedron

Vertices: 30

Edges: 60

Faces: 22

Wythoff symbol: 5/4 5|3

Symmetry Group: icosahedral

Vertex Configuration: {6, 5/4, 6, 5}

Dual: great dodecahemicosacron

66

Great stellated truncated dodecahedron

Vertices: 60

Edges: 90

Faces: 32

Wythoff symbol: 2 3|5/3

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 10/3, 3}

Dual: great triakis icosahedron

67

Great rhombicosidodecahedron

Vertices: 60

Edges: 120

Faces: 62

Wythoff symbol: 5/3 3|2

Symmetry Group: icosahedral

Vertex Configuration: {4, 5/3, 4, 3}

Dual: great deltoidal hexecontahedron

68

Great truncated icosidodecahedron

Vertices: 120

Edges: 180

Faces: 62

Wythoff symbol: 5/3 2 3|

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 4, 6}

Dual: great disdyakis triacontahedron

69

Great inverted snub icosidodecahedron

Vertices: 60

Edges: 150

Faces: 92

Wythoff symbol: |5/3 2 3

Symmetry Group: icosahedral

Vertex Configuration: {3, 5/3, 3, 3, 3}

Dual: great inverted pentagonal hexecontahedron

70

Great dodecahemidodecahedron

Vertices: 30

Edges: 60

Faces: 18

Wythoff symbol: 5/3 5/2|5/3

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 5/3, 10/3, 5/2}

Dual: great dodecahemidodecacron

71

Great icosihemidodecahedron

Vertices: 30

Edges: 60

Faces: 26

Wythoff symbol: 3/2 3|5/3

Symmetry Group: icosahedral

Vertex Configuration: {10/3, 3/2, 10/3, 3}

Dual: great icosihemidodecacron

72

Small retrosnub icosicosidodecahedron

Vertices: 60

Edges: 180

Faces: 112

Wythoff symbol: |3/2 3/2 5/2

Symmetry Group: icosahedral

Vertex Configuration: {3, 3/2, 3, 3/2, 3, 5/2}

Dual: small hexagrammic hexecontahedron

73

Great rhombidodecahedron

Vertices: 60

Edges: 120

Faces: 42

Wythoff symbol: 3/2 5/3 2|

Symmetry Group: icosahedral

Vertex Configuration: {4, 10/3, 4/3, 10/7}

Dual: great rhombidodecacron

74

Great retrosnub icosidodecahedron

Vertices: 60

Edges: 150

Faces: 92

Wythoff symbol: |3/2 5/3 2

Symmetry Group: icosahedral

Vertex Configuration: {3, 3/2, 3, 5/3, 3}

Dual: great pentagrammic hexecontahedron

75

Great dirhombicosidodecahedron

Vertices: 60

Edges: 240

Faces: 124

Wythoff symbol: |3/2 5/3 3 5/2

Symmetry Group: icosahedral

Vertex Configuration: {4, 5/3, 4, 3, 4, 5/2, 4, 3/2}

Dual: great dirhombicosidodecacron

76

Pentagonal prism

Vertices: 10

Edges: 15

Faces: 7

Wythoff symbol: 2 5|2

Symmetry Group: dihedral

Vertex Configuration: {4, 4, 5}

Dual: pentagonal dipyramid

77

Pentagonal antiprism

Vertices: 10

Edges: 20

Faces: 12

Wythoff symbol: |2 2 5

Symmetry Group: dihedral

Vertex Configuration: {3, 3, 3, 5}

Dual: pentagonal trapezohedron

78

Pentagrammic prism

Vertices: 10

Edges: 15

Faces: 7

Wythoff symbol: 2 5/2|2

Symmetry Group: dihedral

Vertex Configuration: {4, 4, 5/2}

Dual: pentagrammic dipyramid

79

Pentagrammic antiprism

Vertices: 10

Edges: 20

Faces: 12

Wythoff symbol: |2 2 5/2

Symmetry Group: dihedral

Vertex Configuration: {3, 3, 3, 5/2}

Dual: pentagrammic deltahedron

80

Pentagrammic crossed antiprism

Vertices: 10

Edges: 20

Faces: 12

Wythoff symbol: |2 2 5/3

Symmetry Group: dihedral

Vertex Configuration: {3, 3, 3, 5/3}

Dual: pentagrammic concave deltahedron

Represented in Second Life