List of polygons, polyhedra and polytopes

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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.

Polytope elements[edit]

Polygon (2-polytope)[edit]

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

Polyhedron (3-polytope)[edit]

  • 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

Polychoron (4-polytope)[edit]

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

Polyteron (5-polytope)[edit]

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

Other[edit]

Two dimensional[edit]

Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendecagon
Dodecagon
Triskaidecagon
Tetradecagon
Pentadecagon
Hexadecagon
Heptadecagon
Octadecagon
Enneadecagon
Icosagon
Triacontagon
Gnomon
Golygon
Star polygons

Families[edit]

Tilings[edit]

List of uniform tilings Uniform tilings in hyperbolic plane

Archimedean tiling

Other[edit]

Three dimensional[edit]

Three-dimensional space

Regular[edit]

Regular polyhedron

Tetrahedron
Pentahedron
Hexahedron
Heptahedron
Octahedron
Enneahedron
Decahedron
Dodecahedron

Archimedean solids[edit]

Archimedean solid

Prisms and antiprisms[edit]

Prism
Antiprism

Catalan solids(13)[edit]

Catalan solid

Bipyramids and Trapezohedron[edit]

Uniform star polyhedra[edit]

Uniform star polyhedron

Uniform prismatic star polyhedra[edit]

Prismatic uniform polyhedron

Johnson solids[edit]

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[edit]

Honeycombs[edit]

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

Other[edit]

Regular and uniform compound polyhedra[edit]

Polyhedral compound and Uniform polyhedron compound

Four dimensions[edit]

Four-dimensional space

Polychoron – general term for a four dimensional polytope

Regular convex polychoron
Abstract regular polytope
Schläfli–Hess polychoron (Regular star polychoron)
Uniform polychoron
Prismatic uniform polychoron
Uniform antiprismatic prism

Honeycombs[edit]

Five dimensions[edit]

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

Honeycombs[edit]

Six dimensions[edit]

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

Honeycombs[edit]

Seven dimensions[edit]

Seven-dimensional space, uniform 7-polytope

Honeycombs[edit]

Eight dimension[edit]

Eight-dimensional space, uniform 8-polytope

Honeycombs[edit]

Nine dimensions[edit]

9-polytope

Hyperbolic honeycombs[edit]

Ten dimensions[edit]

10-polytope

Dimensional families[edit]

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

Geometric operators[edit]

See also[edit]