Wikipedia

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.

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 (polygons)

Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendecagon
Dodecagon
Triskaidecagon
Tetradecagon
Pentadecagon
Hexadecagon
Heptadecagon
Octadecagon
Enneadecagon
Icosagon
Icosihenagon
Icosidigon
Icositrigon
Icositetragon
Icosipentagon
Icosihexagon
Icosiheptagon
Icosioctagon
Icosienneagon
Triacontagon
Tetracontagon
257-gon
hectogon
Chiliagon
Myriagon
65537-gon
Megagon
Apeirogon

Star polygons

Families

Tilings

List of uniform tilings Uniform tilings in hyperbolic plane

Archimedean tiling

Three dimensional (polyhedra)

Three-dimensional space

Regular

Regular polyhedron

Tetrahedron
  • Disphenoid
Pentahedron
Hexahedron
Heptahedron
Octahedron
Enneahedron
  • Octagonal pyramid, Heptagonal prism
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

  • Great complex icosidodecahedron
  • Great deltoidal hexecontahedron
  • Great deltoidal icositetrahedron
  • Great dirhombicosidodecacron
  • Great dirhombicosidodecahedron
  • Great disdyakis dodecahedron
  • Great disdyakis triacontahedron
  • Great disnub dirhombidodecacron
  • Great ditrigonal dodecacronic hexecontahedron
  • Great dodecacronic hexecontahedron
  • Great dodecahemicosacron
  • Great dodecicosacron
  • Great hexacronic icositetrahedron
  • Great hexagonal hexecontahedron
  • Great icosacronic hexecontahedron
  • Great icosihemidodecacron
  • Great inverted pentagonal hexecontahedron
  • Great pentagonal hexecontahedron
  • Great pentagrammic hexecontahedron
  • Great pentakis dodecahedron
  • Great rhombic triacontahedron
  • Great rhombidodecacron
  • Great rhombihexacron
  • Great stellapentakis dodecahedron
  • Great triakis icosahedron
  • Great triakis octahedron
  • Great triambic icosahedron
  • Medial deltoidal hexecontahedron
  • Medial disdyakis triacontahedron
  • Medial hexagonal hexecontahedron
  • Medial icosacronic hexecontahedron
  • Medial inverted pentagonal hexecontahedron
  • Medial pentagonal hexecontahedron
  • Medial rhombic triacontahedron
  • Hexahemioctacron
  • Hemipolyhedron
  • Octahemioctacron
  • Rhombicosacron
  • Small complex icosidodecahedron
  • Small ditrigonal dodecacronic hexecontahedron
  • Small dodecacronic hexecontahedron
  • Small dodecahemicosacron
  • Small dodecahemidodecacron
  • Small dodecicosacron
  • Small hexacronic icositetrahedron
  • Small hexagonal hexecontahedron
  • Small hexagrammic hexecontahedron
  • Small icosacronic hexecontahedron
  • Small icosihemidodecacron
  • Small rhombidodecacron
  • Small rhombihexacron
  • Small stellapentakis dodecahedron
  • Small triambic icosahedron
  • Tetrahemihexacron

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
  • Compound of cube and octahedron
  • Compound of dodecahedron and icosahedron
  • Compound of eight octahedra with rotational freedom
  • Compound of eight triangular prisms
  • Compound of five cubes
  • Compound of five cuboctahedra
  • Compound of five cubohemioctahedra
  • Compound of five great cubicuboctahedra
  • Compound of five great dodecahedra
  • Compound of five great icosahedra
  • Compound of five great rhombihexahedra
  • Compound of five icosahedra
  • Compound of five octahedra
  • Compound of five octahemioctahedra
  • Compound of five small cubicuboctahedra
  • Compound of five small rhombicuboctahedra
  • Compound of five small rhombihexahedra
  • Compound of five small stellated dodecahedra
  • Compound of five stellated truncated cubes
  • Compound of five tetrahedra
  • Compound of five tetrahemihexahedra
  • Compound of five truncated cubes
  • Compound of five truncated tetrahedra
  • Compound of five uniform great rhombicuboctahedra
  • Compound of four hexagonal prisms
  • Compound of four octahedra
  • Compound of four octahedra with rotational freedom
  • Compound of four tetrahedra
  • Compound of four triangular prisms
  • Compound of great icosahedron and great stellated dodecahedron
  • Compound of six cubes with rotational freedom
  • Compound of six decagonal prisms
  • Compound of six decagrammic prisms
  • Compound of six pentagonal antiprisms
  • Compound of six pentagonal prisms
  • Compound of six pentagrammic antiprisms
  • Compound of six pentagrammic crossed antiprisms
  • Compound of six pentagrammic prisms
  • Compound of six square antiprisms
  • Compound of six tetrahedra
  • Compound of six tetrahedra with rotational freedom
  • Compound of small stellated dodecahedron and great dodecahedron
  • Compound of ten hexagonal prisms
  • Compound of ten octahedra
  • Compound of ten tetrahedra
  • Compound of ten triangular prisms
  • Compound of ten truncated tetrahedra
  • Compound of three cubes
  • Compound of three square antiprisms
  • Compound of three tetrahedra
  • Compound of twelve pentagonal antiprisms with rotational freedom
  • Compound of twelve pentagonal prisms
  • Compound of twelve pentagrammic antiprisms
  • Compound of twelve pentagrammic crossed antiprisms with rotational freedom
  • Compound of twelve pentagrammic prisms
  • Compound of twelve tetrahedra with rotational freedom
  • Compound of twenty octahedra
  • Compound of twenty octahedra with rotational freedom
  • Compound of twenty tetrahemihexahedra
  • Compound of twenty triangular prisms
  • Compound of two great dodecahedra
  • Compound of two great icosahedra
  • Compound of two great inverted snub icosidodecahedra
  • Compound of two great retrosnub icosidodecahedra
  • Compound of two great snub icosidodecahedra
  • Compound of two icosahedra
  • Compound of two inverted snub dodecadodecahedra
  • Compound of two small stellated dodecahedra
  • Compound of two snub cubes
  • Compound of two snub dodecadodecahedra
  • Compound of two snub dodecahedra
  • Compound of two snub icosidodecadodecahedra
  • Compound of two truncated tetrahedra
  • Prismatic compound of antiprisms
  • Prismatic compound of antiprisms with rotational freedom
  • Prismatic compound of prisms
  • Prismatic compound of prisms with rotational freedom

Four dimensions

Four-dimensional space

4-polytope – general term for a four dimensional polytope

Regular 4-polytope
  • 5-cell, Tesseract, 16-cell, 24-cell, 120-cell, 600-cell
Abstract regular polytope
Regular star 4-polytope
Uniform 4-polytope
Prismatic uniform 4-polytope
  • Grand antiprism
  • Duoprism
  • Tetrahedral prism, Truncated tetrahedral prism
  • Truncated cubic prism, Truncated octahedral prism, Cuboctahedral prism, Rhombicuboctahedral prism, Truncated cuboctahedral prism, Snub cubic prism
  • Truncated dodecahedral prism, Truncated icosahedral prism, Icosidodecahedral prism, Rhombicosidodecahedral prism, Truncated icosidodecahedral prism, Snub dodecahedral prism
Uniform antiprismatic prism
  • Triangular antiprismatic prism, Square antiprismatic prism, Pentagonal antiprismatic prism, Hexagonal antiprismatic prism, Heptagonal antiprismatic prism, Octagonal antiprismatic prism, Enneagonal antiprismatic prism, Decagonal antiprismatic prism
  • Pentagrammic antiprismatic prism, Hexagrammic antiprismatic prism, Heptagrammic antiprismatic prism, Octagrammic antiprismatic prism, Enneagrammic antiprismatic prism, Decagrammic antiprismatic prism
  • Pentagrammic crossed antiprismatic prism, Hexagrammic crossed antiprismatic prism, Heptagrammic crossed antiprismatic prism, Octagrammic crossed antiprismatic prism, Enneagrammic crossed antiprismatic prism, Decagrammic crossed antiprismatic prism

Honeycombs

  • Tesseractic honeycomb
  • 24-cell honeycomb
  • Snub 24-cell honeycomb
  • Rectified 24-cell honeycomb
  • Truncated 24-cell honeycomb
  • 16-cell honeycomb
  • 5-cell honeycomb
  • Omnitruncated 5-cell honeycomb
  • Truncated 5-cell honeycomb
  • Omnitruncated 5-simplex honeycomb

Five dimensions

Five-dimensional space, 5-polytope and uniform 5-polytope
  • 5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex
  • 5-demicube, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube
  • 5-cube, Rectified 5-cube, 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube
  • 5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplex, Cantellated 5-orthoplex, Runcinated 5-orthoplex
Prismatic uniform 5-polytope
  • 5-cell prism, Rectified 5-cell prism, Truncated 5-cell prism, Cantellated 5-cell prism, Runcinated 5-cell prism, Bitruncated 5-cell prism, Cantitruncated 5-cell prism, Runcitruncated 5-cell prism, Omnitruncated 5-cell prism
  • Tesseractic prism, Rectified tesseractic prism, Truncated tesseractic prism, Cantellated tesseractic prism, Runcinated tesseractic prism, Bitruncated tesseractic prism, Cantitruncated tesseractic prism, Runcitruncated tesseractic prism, Omnitruncated tesseractic prism
  • 16-cell prism, Truncated 16-cell prism, Runcitruncated 16-cell prism
  • 24-cell prism, rectified 24-cell prism, truncated 24-cell prism, cantellated 24-cell prism, runcinated 24-cell prism, bitruncated 24-cell prism, cantitruncated 24-cell prism, runcitruncated 24-cell prism, omnitruncated 24-cell prism, snub 24-cell prism
  • 120-cell prism, Rectified 120-cell prism, Truncated 120-cell prism, Cantellated 120-cell prism, Runcinated 120-cell prism, Bitruncated 120-cell prism, Cantitruncated 120-cell prism, Runcitruncated 120-cell prism, Omnitruncated 120-cell prism
  • 600-cell prism, Rectified 600-cell prism, Truncated 600-cell prism, Cantellated 600-cell prism, Cantitruncated 600-cell prism, Runcitruncated 600-cell prism
  • Grand antiprism prism

Honeycombs

  • 5-cubic honeycomb
  • 5-simplex honeycomb
  • Truncated 5-simplex honeycomb
  • 5-demicubic honeycomb

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope
  • 6-simplex, Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex
  • 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube
  • 6-cube, Rectified 6-cube, 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube
  • 6-orthoplex, Rectified 6-orthoplex, Truncated 6-orthoplex, Cantellated 6-orthoplex, Runcinated 6-orthoplex, Stericated 6-orthoplex
  • 122 polytope, 221 polytope

Honeycombs

  • 6-cubic honeycomb
  • 6-simplex honeycomb
  • 6-demicubic honeycomb
  • 222 honeycomb

Seven dimensions

Seven-dimensional space, uniform 7-polytope
  • 7-simplex, Rectified 7-simplex, Truncated 7-simplex, Cantellated 7-simplex, Runcinated 7-simplex, Stericated 7-simplex, Pentellated 7-simplex, Hexicated 7-simplex
  • 7-demicube, Truncated 7-demicube, Cantellated 7-demicube, Runcinated 7-demicube, Stericated 7-demicube, Pentellated 7-demicube
  • 7-cube, Rectified 7-cube, 7-cube, Truncated 7-cube, Cantellated 7-cube, Runcinated 7-cube, Stericated 7-cube, Pentellated 7-cube, Hexicated 7-cube
  • 7-orthoplex, Rectified 7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex
  • 132 polytope, 231 polytope, 321 polytope

Honeycombs

  • 7-cubic honeycomb
  • 7-demicubic honeycomb
  • 331 honeycomb, 133 honeycomb

Eight dimension

Eight-dimensional space, uniform 8-polytope
  • 8-simplex, Rectified 8-simplex, Truncated 8-simplex, Cantellated 8-simplex, Runcinated 8-simplex, Stericated 8-simplex, Pentellated 8-simplex, Hexicated 8-simplex, Heptellated 8-simplex
  • 8-orthoplex, Rectified 8-orthoplex, Truncated 8-orthoplex, Cantellated 8-orthoplex, Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex,
  • 8-cube, Rectified 8-cube, Truncated 8-cube, Cantellated 8-cube, Runcinated 8-cube, Stericated 8-cube, Pentellated 8-cube, Hexicated 8-cube, Heptellated 8-cube
  • 8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube
  • 142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated 241 polytope, Runcinated 421 polytope

Honeycombs

  • 8-cubic honeycomb
  • 8-demicubic honeycomb
  • 521 honeycomb, 251 honeycomb, 152 honeycomb

Nine dimensions

9-polytope
  • 9-cube
  • 9-demicube
  • 9-orthoplex
  • 9-simplex

Hyperbolic honeycombs

  • E9 honeycomb

Ten dimensions

10-polytope
  • 10-cube
  • 10-demicube
  • 10-orthoplex
  • 10-simplex

Dimensional families

Regular polytope and List of regular polytopes
Uniform polytope
  • Demihypercube
  • Uniform 1k2 polytope
  • Uniform 2k1 polytope
  • Uniform k21 polytope
Honeycombs

Geometric operators

See also

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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