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List of polygons, polyhedra and polytopes

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List of polygons, polyhedra and polytopes

A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.

Contents

  • Polytope elements 1
    • Polygon (2-polytope) 1.1
    • Polyhedron (3-polytope) 1.2
    • 4-polytope 1.3
    • 5-polytope 1.4
    • Other 1.5
  • Two dimensional 2
    • Families 2.1
    • Tilings 2.2
    • Other 2.3
  • Three dimensional 3
    • Regular 3.1
    • Archimedean solids 3.2
    • Prisms and antiprisms 3.3
    • Catalan solids 3.4
    • Bipyramids and Trapezohedron 3.5
    • Uniform star polyhedra 3.6
    • Uniform prismatic star polyhedra 3.7
    • Johnson solids 3.8
    • Dual uniform star polyhedra 3.9
    • Honeycombs 3.10
    • Other 3.11
    • Regular and uniform compound polyhedra 3.12
  • Four dimensions 4
    • Honeycombs 4.1
  • Five dimensions 5
    • Honeycombs 5.1
  • Six dimensions 6
    • Honeycombs 6.1
  • Seven dimensions 7
    • Honeycombs 7.1
  • Eight dimension 8
    • Honeycombs 8.1
  • Nine dimensions 9
    • Hyperbolic honeycombs 9.1
  • Ten dimensions 10
  • Dimensional families 11
  • Geometric operators 12
  • See also 13

Polytope elements

Polygon (2-polytope)

  • Vertex
  • Edge the facet or (n−1)-face of the polygon

Polyhedron (3-polytope)

  • Vertex the peak or (n−3)-face of the polyhedron
  • Edge the ridge or (n−2)-face of the polyhedron
  • Face the facet or (n−1)-face of the polyhedron

4-polytope

  • Vertex
  • Edge the peak or (n−3)-face of the 4-polytope
  • Face the ridge or (n−2)-face of the 4-polytope
  • Cell the facet or (n−1)-face of the 4-polytope

5-polytope

  • Vertex the (n−5)-face of the 5-polytope
  • Edge the (n−4)-face of the 5-polytope
  • Face the peak or (n−3)-face of the 5-polytope
  • Cell the ridge or (n−2)-face of the 5-polytope
  • Hypercell or Teron the facet or (n−1)-face of the 5-polytope

Other

Two dimensional

Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendecagon
Dodecagon
Triskaidecagon
Tetradecagon
Pentadecagon
Hexadecagon
Heptadecagon
Octadecagon
Enneadecagon
Icosagon
Icositetragon
Triacontagon
Tetracontagon
257-gon
Chiliagon
Myriagon
65537-gon
Megagon
Gnomon
Golygon
Star polygons

Families

Tilings

List of uniform tilings Uniform tilings in hyperbolic plane

Archimedean tiling

Other

Three dimensional

Three-dimensional space

Regular

Regular polyhedron

Tetrahedron
Pentahedron
Hexahedron
Heptahedron
Octahedron
Enneahedron
Decahedron
Dodecahedron

Archimedean solids

Archimedean solid

Prisms and antiprisms

Prism
Antiprism

Catalan solids

Catalan solid

Bipyramids and Trapezohedron

Uniform star polyhedra

Uniform star polyhedron

Uniform prismatic star polyhedra

Prismatic uniform polyhedron

Johnson solids

Johnson solid
  1. Augmented dodecahedron
  2. Augmented hexagonal prism
  3. Augmented pentagonal prism
  4. Augmented sphenocorona
  5. Augmented triangular prism
  6. Augmented tridiminished icosahedron
  7. Augmented truncated cube
  8. Augmented truncated dodecahedron
  9. Augmented truncated tetrahedron
  10. Biaugmented pentagonal prism
  11. Biaugmented triangular prism
  12. Biaugmented truncated cube
  13. Bigyrate diminished rhombicosidodecahedron
  14. Bilunabirotunda
  15. Diminished rhombicosidodecahedron
  16. Disphenocingulum
  17. Elongated pentagonal bipyramid
  18. Elongated pentagonal cupola
  19. Elongated pentagonal gyrobicupola
  20. Elongated pentagonal gyrobirotunda
  21. Elongated pentagonal gyrocupolarotunda
  22. Elongated pentagonal orthobicupola
  23. Elongated pentagonal orthobirotunda
  24. Elongated pentagonal orthocupolarotunda
  25. Elongated pentagonal pyramid
  26. Elongated pentagonal rotunda
  27. Elongated square bipyramid
  28. Elongated square cupola
  29. Elongated square gyrobicupola
  30. Elongated square pyramid
  31. Elongated triangular bipyramid
  32. Elongated triangular cupola
  33. Elongated triangular gyrobicupola
  34. Elongated triangular orthobicupola
  35. Elongated triangular pyramid
  36. Gyrate bidiminished rhombicosidodecahedron
  37. Gyrate rhombicosidodecahedron
  38. Gyrobifastigium
  39. Gyroelongated pentagonal bicupola
  40. Gyroelongated pentagonal birotunda
  41. Gyroelongated pentagonal cupola
  42. Gyroelongated pentagonal cupolarotunda
  43. Gyroelongated pentagonal pyramid
  44. Gyroelongated pentagonal rotunda
  45. Gyroelongated square bicupola
  46. Gyroelongated square bipyramid
  47. Gyroelongated square cupola
  48. Gyroelongated square pyramid
  49. Gyroelongated triangular bicupola
  50. Gyroelongated triangular cupola
  51. Hebesphenomegacorona
  52. Metabiaugmented dodecahedron
  53. Metabiaugmented hexagonal prism
  54. Metabiaugmented truncated dodecahedron
  55. Metabidiminished icosahedron
  56. Metabidiminished rhombicosidodecahedron
  57. Metabigyrate rhombicosidodecahedron
  58. Metagyrate diminished rhombicosidodecahedron
  59. Parabiaugmented dodecahedron
  60. Parabiaugmented hexagonal prism
  61. Parabiaugmented truncated dodecahedron
  62. Parabidiminished rhombicosidodecahedron
  63. Parabigyrate rhombicosidodecahedron
  64. Paragyrate diminished rhombicosidodecahedron
  65. Pentagonal bipyramid
  66. Pentagonal cupola
  67. Pentagonal gyrobicupola
  68. Pentagonal gyrocupolarotunda
  69. Pentagonal orthobicupola
  70. Pentagonal orthobirotunda
  71. Pentagonal orthocupolarotunda
  72. Pentagonal pyramid
  73. Pentagonal rotunda
  74. Snub disphenoid
  75. Snub square antiprism
  76. Sphenocorona
  77. Sphenomegacorona
  78. Square cupola
  79. Square gyrobicupola
  80. Square orthobicupola
  81. Square pyramid
  82. Triangular bipyramid
  83. Triangular cupola
  84. Triangular hebesphenorotunda
  85. Triangular orthobicupola
  86. Triaugmented dodecahedron
  87. Triaugmented hexagonal prism
  88. Triaugmented triangular prism
  89. Triaugmented truncated dodecahedron
  90. Tridiminished icosahedron
  91. Tridiminished rhombicosidodecahedron
  92. Trigyrate rhombicosidodecahedron

Dual uniform star polyhedra

Honeycombs

Convex uniform honeycomb
Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space

Other

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound

Four dimensions

Four-dimensional space

4-polytope – general term for a four dimensional polytope

Regular 4-polytope
Abstract regular polytope
Regular star 4-polytope
Uniform 4-polytope
Prismatic uniform 4-polytope
Uniform antiprismatic prism

Honeycombs

Five dimensions

Five-dimensional space, 5-polytope and uniform 5-polytope
Prismatic uniform 5-polytope

Honeycombs

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope

Honeycombs

Seven dimensions

Seven-dimensional space, uniform 7-polytope

Honeycombs

Eight dimension

Eight-dimensional space, uniform 8-polytope

Honeycombs

Nine dimensions

9-polytope

Hyperbolic honeycombs

Ten dimensions

10-polytope

Dimensional families

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

Geometric operators

See also

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