List of polygons, polyhedra and polytopes

2025-02-09T15:23:53Z

2601:247:C803:1CD0:49AE:19A6:7058:4E38: /* Other */ Add Goldberg Polyhedron

{{Short description|none}}
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 (geometry)|Vertex]] the ''ridge'' or ''(n−2)-face'' of the polygon
*[[Edge (geometry)|Edge]] the ''facet'' or ''(n−1)-face'' of the polygon

===Polyhedron (3-polytope)===
*[[Vertex (geometry)|Vertex]] the ''peak'' or ''(n−3)-face'' of the polyhedron
*[[Edge (geometry)|Edge]] the ''ridge'' or ''(n−2)-face'' of the polyhedron
*[[Face (geometry)|Face]] the ''facet'' or ''(n−1)-face'' of the polyhedron

===Polychoron (4-polytope)===
*[[Vertex (geometry)|Vertex]] the ''(n−4)-face'' of the polychoron
*[[Edge (geometry)|Edge]] the ''peak'' or ''(n−3)-face'' of the polychoron
*[[Face (geometry)|Face]] the ''ridge'' or ''(n−2)-face'' of the polychoron
*[[Cell (geometry)|Cell]] the ''facet'' or ''(n−1)-face'' of the polychoron

===5-polytope===
*[[Vertex (geometry)|Vertex]] the ''(n−5)-face'' of the 5-polytope
*[[Edge (geometry)|Edge]] the ''(n−4)-face'' of the 5-polytope
*[[Face (geometry)|Face]] the ''peak'' or ''(n−3)-face'' of the 5-polytope
*[[Cell (geometry)|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===
*[[Point (geometry)|Point]]
*[[Line segment]]
*[[Vertex figure]]
*[[Peak (geometry)|Peak]] – ''(n−3)-face''
*[[Ridge (geometry)|Ridge]] – ''(n−2)-face''
*[[Facet (geometry)|Facet]] – ''(n−1)-face''

==Two dimensional ([[polygon]]s)==
* [[Triangle]]
**[[Equilateral triangle]]
**[[Isosceles triangle]]
***[[Golden triangle (mathematics)]]
**[[Scalene triangle]]
**[[Right triangle]]
**[[Oblique triangle]]
***[[Acute triangle]]
***[[Oblique triangle|Obtuse Triangle]]

* [[Quadrilateral]]
**[[Rectangle]]
***[[Square (geometry)|Square]]
****[[Unit square]]
***[[Oblong (description)|Oblong]]
****[[Golden rectangle]]
****[[Silver rectangle]]
****[[Ailles rectangle]]
**[[Rhombus]]
***[[Golden rhombus|Golden Rhombus]]
**[[Parallelogram]]
***[[Rhomboid]]
**[[Trapezoid]]
***[[Isosceles trapezoid]]
**[[Kite (geometry)|Kite]]
***[[Lute of Pythagoras|Lute of pythagoras]]
***[[Right kite|Right Kite]]
**[[Antiparallelogram]]

*[[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]]
*[[Pentacontagon]]
*[[Hexacontagon]]
*[[Heptacontagon]]
*[[Octacontagon]]
*[[Enneacontagon]]
*[[Hectogon]]
*[[257-gon]]
*[[Chiliagon]]
*[[Myriagon]]
*[[65537-gon]]
*[[Megagon]]
*[[Gigagon]]
*[[Teragon]]
*[[Apeirogon]]

===[[Star polygon]]s===
*[[Pentagram]]
*[[Hexagram]]
*[[Heptagram]]
*[[Octagram]]
*[[Enneagram (geometry)|Enneagram]]
*[[Decagram (geometry)|Decagram]]
*[[Hendecagram]]
*[[Dodecagram]]
*[[Icositetragram]]

===Families===
*[[Concave polygon]]
*[[Cyclic polygon]]
*[[Regular polygon]]
*[[Polyform]]
*[[Gnomon (figure)|Gnomon]]
*[[Golygon]]

===Tilings===
[[List of uniform tilings]]

[[Uniform tilings in hyperbolic plane]]

;[[Archimedean tiling]]
*[[Square tiling]]
*[[Triangular tiling]]
*[[Hexagonal tiling]]
*[[Truncated square tiling]]
*[[Snub square tiling]]
*[[Trihexagonal tiling]]
*[[Truncated hexagonal tiling]]
*[[Rhombitrihexagonal tiling]]
*[[Truncated trihexagonal tiling]]
*[[Snub hexagonal tiling]]
*[[Elongated triangular tiling]]

==Three dimensional ([[polyhedron|polyhedra]])==
;[[Three-dimensional space]]

===Regular===
[[Regular polyhedron]]
*[[Platonic solid]]:
**[[Tetrahedron]], [[Cube]], [[Octahedron]], [[Dodecahedron]], [[Icosahedron]]
*Regular [[spherical polyhedron]]
**[[Dihedron]], [[Hosohedron]]
*[[Kepler–Poinsot polyhedron]] (Regular star polyhedra)
**[[Small stellated dodecahedron]], [[Great stellated dodecahedron]], [[Great icosahedron]], [[Great dodecahedron]]
*Abstract regular polyhedra ([[Projective polyhedron]])
**[[Hemicube (geometry)]], [[hemi-octahedron]], [[hemi-dodecahedron]], [[hemi-icosahedron]]

;[[Tetrahedron]]
*[[Disphenoid]]
;[[Pentahedron]]
*[[Square pyramid]], [[Triangular prism]]
;[[Hexahedron]]
*[[Parallelepiped]], [[Cuboid]], [[Rhombohedron]], [[Trigonal trapezohedron]], [[Cube]], [[Pentagonal pyramid]], [[Triangular bipyramid]], quadrilateral [[frustum]]
;[[Heptahedron]]
*[[hexagonal pyramid]], [[pentagonal prism]], [[tetrahemihexahedron]]
;[[Octahedron]]
*[[Hexagonal prism]], [[Truncated tetrahedron]], [[Tetragonal trapezohedron]]
;[[Enneahedron]]
*[[Octagonal pyramid]], [[Heptagonal prism]]
;[[Decahedron]]
*[[Octagonal prism]], [[Square antiprism]], [[Square cupola]], [[Pentagonal bipyramid]], [[Augmented pentagonal prism]]
;[[Dodecahedron]]
*[[Pentagonal antiprism]], [[Decagonal prism]], [[Pentagonal cupola]], [[Snub disphenoid]], [[Elongated square bipyramid]], [[Metabidiminished icosahedron]], [[Hexagonal bipyramid]], [[Hexagonal trapezohedron]], [[Triakis tetrahedron]], [[Rhombic dodecahedron]], [[Hendecagonal pyramid]], [[Trapezo-rhombic dodecahedron]], [[Rhombo-hexagonal dodecahedron]]

===Archimedean solids===
;[[Archimedean solid]]
*[[Truncated tetrahedron]], [[Cuboctahedron]], [[Truncated cube]], [[Truncated octahedron]], [[Rhombicuboctahedron]], [[Truncated cuboctahedron]], [[Snub cube]], [[Icosidodecahedron]], [[Truncated dodecahedron]], [[Truncated icosahedron]], [[Rhombicosidodecahedron]], [[Truncated icosidodecahedron]], [[Snub dodecahedron]]

===Prisms and antiprisms===
;[[Prism (geometry)|Prism]]
*[[Triangular prism]], [[Pentagonal prism]], [[Hexagonal prism]], [[Heptagonal prism]], [[Octagonal prism]], [[Enneagonal prism]], [[Decagonal prism]], [[Hendecagonal prism]], [[Dodecagonal prism]]
;[[Antiprism]]
*[[Square antiprism]], [[Pentagonal antiprism]], [[Hexagonal antiprism]], [[Heptagonal antiprism]], [[Octagonal antiprism]], [[Enneagonal antiprism]], [[Decagonal antiprism]], Dodecagonal antiprism

===Catalan solids===
;[[Catalan solid]]
*[[Triakis tetrahedron]], [[Rhombic dodecahedron]], [[Triakis octahedron]], [[Tetrakis hexahedron]], [[Deltoidal icositetrahedron]], [[Disdyakis dodecahedron]], [[Pentagonal icositetrahedron]], [[Rhombic triacontahedron]], [[Triakis icosahedron]], [[Pentakis dodecahedron]], [[Deltoidal hexecontahedron]], [[Disdyakis triacontahedron]], [[Pentagonal hexecontahedron]]

===Bipyramids and Trapezohedron===
*[[Bipyramid]]
**[[Triangular bipyramid]], [[Pentagonal bipyramid]], [[Hexagonal bipyramid]], [[Heptagonal bipyramid]], [[Octagonal bipyramid]], [[Decagonal bipyramid]]
*[[Trapezohedron]]

===Uniform star polyhedra===
;[[Uniform star polyhedron]]
*[[Cubitruncated cuboctahedron]]
*[[Cubohemioctahedron]]
*[[Ditrigonal dodecadodecahedron]]
*[[Dodecadodecahedron]]
*[[Great cubicuboctahedron]]
*[[Great dirhombicosidodecahedron]]
*[[Great disnub dirhombidodecahedron]]
*[[Great ditrigonal dodecicosidodecahedron]]
*[[Great ditrigonal icosidodecahedron]]
*[[Great dodecahemicosahedron]]
*[[Great dodecahemidodecahedron]]
*[[Great dodecicosahedron]]
*[[Great dodecicosidodecahedron]]
*[[Great icosicosidodecahedron]]
*[[Great icosidodecahedron]]
*[[Great icosihemidodecahedron]]
*[[Great inverted snub icosidodecahedron]]
*[[Great retrosnub icosidodecahedron]]
*[[Great rhombidodecahedron]]
*[[Great rhombihexahedron]]
*[[Great snub dodecicosidodecahedron]]
*[[Great snub icosidodecahedron]]
*[[Great stellated truncated dodecahedron]]
*[[Great truncated cuboctahedron]]
*[[Great truncated icosidodecahedron]]
*[[Icosidodecadodecahedron]]
*[[Icositruncated dodecadodecahedron]]
*[[Inverted snub dodecadodecahedron]]
*[[Nonconvex great rhombicosidodecahedron]]
*[[Nonconvex great rhombicuboctahedron]]
*[[Octahemioctahedron]]
*[[Rhombicosahedron]]
*[[Rhombidodecadodecahedron]]
*[[Small cubicuboctahedron]]
*[[Small ditrigonal dodecicosidodecahedron]]
*[[Small ditrigonal icosidodecahedron]]
*[[Small dodecahemicosahedron]]
*[[Small dodecahemidodecahedron]]
*[[Small dodecicosahedron]]
*[[Small dodecicosidodecahedron]]
*[[Small icosicosidodecahedron]]
*[[Small icosihemidodecahedron]]
*[[Small retrosnub icosicosidodecahedron]]
*[[Small rhombidodecahedron]]
*[[Small rhombihexahedron]]
*[[Small snub icosicosidodecahedron]]
*[[Small stellated truncated dodecahedron]]
*[[Snub dodecadodecahedron]]
*[[Snub icosidodecadodecahedron]]
*[[Stellated truncated hexahedron]]
*[[Tetrahemihexahedron]]
*[[Truncated dodecadodecahedron]]
*[[Truncated great dodecahedron]]
*[[Truncated great icosahedron]]

===Uniform prismatic star polyhedra===
;[[Prismatic uniform polyhedron]]
*[[Pentagrammic prism]], [[Pentagrammic antiprism]], [[Pentagrammic crossed-antiprism]]
*[[Heptagrammic antiprism (7/2)]], [[Heptagrammic antiprism (7/3)]]
*[[Enneagrammic antiprism (9/2)]]. [[Enneagrammic antiprism (9/4)]]
*[[Enneagrammic crossed-antiprism]], [[Enneagrammic prism (9/2)]], [[Enneagrammic prism (9/4)]]
*[[Decagrammic prism]], [[Decagrammic antiprism]]

===Johnson solids===
;[[Johnson solid]]
#[[Augmented dodecahedron]]
#[[Augmented hexagonal prism]]
#[[Augmented pentagonal prism]]
#[[Augmented sphenocorona]]
#[[Augmented triangular prism]]
#[[Augmented tridiminished icosahedron]]
#[[Augmented truncated cube]]
#[[Augmented truncated dodecahedron]]
#[[Augmented truncated tetrahedron]]
#[[Biaugmented pentagonal prism]]
#[[Biaugmented triangular prism]]
#[[Biaugmented truncated cube]]
#[[Bigyrate diminished rhombicosidodecahedron]]
#[[Bilunabirotunda]]
#[[Diminished rhombicosidodecahedron]]
#[[Disphenocingulum]]
#[[Elongated pentagonal bipyramid]]
#[[Elongated pentagonal cupola]]
#[[Elongated pentagonal gyrobicupola]]
#[[Elongated pentagonal gyrobirotunda]]
#[[Elongated pentagonal gyrocupolarotunda]]
#[[Elongated pentagonal orthobicupola]]
#[[Elongated pentagonal orthobirotunda]]
#[[Elongated pentagonal orthocupolarotunda]]
#[[Elongated pentagonal pyramid]]
#[[Elongated pentagonal rotunda]]
#[[Elongated square bipyramid]]
#[[Elongated square cupola]]
#[[Elongated square gyrobicupola]]
#[[Elongated square pyramid]]
#[[Elongated triangular bipyramid]]
#[[Elongated triangular cupola]]
#[[Elongated triangular gyrobicupola]]
#[[Elongated triangular orthobicupola]]
#[[Elongated triangular pyramid]]
#[[Gyrate bidiminished rhombicosidodecahedron]]
#[[Gyrate rhombicosidodecahedron]]
#[[Gyrobifastigium]]
#[[Gyroelongated pentagonal bicupola]]
#[[Gyroelongated pentagonal birotunda]]
#[[Gyroelongated pentagonal cupola]]
#[[Gyroelongated pentagonal cupolarotunda]]
#[[Gyroelongated pentagonal pyramid]]
#[[Gyroelongated pentagonal rotunda]]
#[[Gyroelongated square bicupola]]
#[[Gyroelongated square bipyramid]]
#[[Gyroelongated square cupola]]
#[[Gyroelongated square pyramid]]
#[[Gyroelongated triangular bicupola]]
#[[Gyroelongated triangular cupola]]
#[[Hebesphenomegacorona]]
#[[Metabiaugmented dodecahedron]]
#[[Metabiaugmented hexagonal prism]]
#[[Metabiaugmented truncated dodecahedron]]
#[[Metabidiminished icosahedron]]
#[[Metabidiminished rhombicosidodecahedron]]
#[[Metabigyrate rhombicosidodecahedron]]
#[[Metagyrate diminished rhombicosidodecahedron]]
#[[Parabiaugmented dodecahedron]]
#[[Parabiaugmented hexagonal prism]]
#[[Parabiaugmented truncated dodecahedron]]
#[[Parabidiminished rhombicosidodecahedron]]
#[[Parabigyrate rhombicosidodecahedron]]
#[[Paragyrate diminished rhombicosidodecahedron]]
#[[Pentagonal bipyramid]]
#[[Pentagonal cupola]]
#[[Pentagonal gyrobicupola]]
#[[Pentagonal gyrocupolarotunda]]
#[[Pentagonal orthobicupola]]
#[[Pentagonal orthobirotunda]]
#[[Pentagonal orthocupolarotunda]]
#[[Pentagonal pyramid]]
#[[Pentagonal rotunda]]
#[[Snub disphenoid]]
#[[Snub square antiprism]]
#[[Sphenocorona]]
#[[Sphenomegacorona]]
#[[Square cupola]]
#[[Square gyrobicupola]]
#[[Square orthobicupola]]
#[[Square pyramid]]
#[[Triangular bipyramid]]
#[[Triangular cupola]]
#[[Triangular hebesphenorotunda]]
#[[Triangular orthobicupola]]
#[[Triaugmented dodecahedron]]
#[[Triaugmented hexagonal prism]]
#[[Triaugmented triangular prism]]
#[[Triaugmented truncated dodecahedron]]
#[[Tridiminished icosahedron]]
#[[Tridiminished rhombicosidodecahedron]]
#[[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]]
*[[Cubic honeycomb]]
*[[Truncated cubic honeycomb]]
*[[Bitruncated cubic honeycomb]]
*[[Cantellated cubic honeycomb]]
*[[Cantitruncated cubic honeycomb]]
*[[Rectified cubic honeycomb]]
*[[Runcitruncated cubic honeycomb]]
*[[Omnitruncated cubic honeycomb]]
*[[Tetrahedral-octahedral honeycomb]]
*[[Truncated alternated cubic honeycomb]]
*[[Cantitruncated alternated cubic honeycomb]]
*[[Runcinated alternated cubic honeycomb]]
*[[Quarter cubic honeycomb]]
*[[Gyrated tetrahedral-octahedral honeycomb]]
*[[Gyrated triangular prismatic honeycomb]]
*[[Gyroelongated alternated cubic honeycomb]]
*[[Gyroelongated triangular prismatic honeycomb]]
*[[Elongated triangular prismatic honeycomb]]
*[[Elongated alternated cubic honeycomb]]
*[[Hexagonal prismatic honeycomb]]
*[[Triangular prismatic honeycomb]]
*[[Triangular-hexagonal prismatic honeycomb]]
*[[Truncated hexagonal prismatic honeycomb]]
*[[Truncated square prismatic honeycomb]]
*[[Rhombitriangular-hexagonal prismatic honeycomb]]
*[[Omnitruncated triangular-hexagonal prismatic honeycomb]]
*[[Snub triangular-hexagonal prismatic honeycomb]]
*[[Snub square prismatic honeycomb]]

;Dual uniform honeycomb
*[[Disphenoid tetrahedral honeycomb]]
*[[Rhombic dodecahedral honeycomb]]

;Others
*[[Trapezo-rhombic dodecahedral honeycomb]]
*[[Weaire–Phelan structure]]

;[[Convex uniform honeycombs in hyperbolic space]]
*[[Order-4 dodecahedral honeycomb]]
*[[Order-5 cubic honeycomb]]
*[[Order-5 dodecahedral honeycomb]]
*[[Icosahedral honeycomb]]

===Other===
*[[Apeirogonal prism]]
*[[Apeirohedron]]
*[[Bicupola (geometry)|Bicupola]]
*[[Cupola (geometry)|Cupola]]
*[[Bifrustum]]
*[[Boerdijk–Coxeter helix]]
*[[Császár polyhedron]]
*[[Flexible polyhedron]]
*[[Goldberg polyhedron]]
*[[Gyroelongated square bipyramid]]
*[[Heronian tetrahedron]]
*[[Hexagonal bifrustum]]
*[[Hexagonal truncated trapezohedron]]
*[[Hill tetrahedron]]
*[[Holyhedron]]
*[[Infinite skew polyhedron]]
*[[Jessen's icosahedron]]
*[[Near-miss Johnson solid]]
*[[Parallelepiped]]
*[[Pentagonal bifrustum]]
*[[Polytetrahedron]]
*[[Pyritohedron]]
*[[Rhombic enneacontahedron]]
*[[Rhombic icosahedron]]
*[[Rhombo-hexagonal dodecahedron]]
*[[Rhombohedron]]
*[[Scalenohedron]]
*[[Schönhardt polyhedron]]
*[[Square bifrustum]]
*[[Square truncated trapezohedron]]
*[[Szilassi polyhedron]]
*[[Tetradecahedron]]
*[[Tetradyakis hexahedron]]
*[[Tetrated dodecahedron]]
*[[Triangular bifrustum]]
*[[Triaugmented triangular prism]]
*[[Truncated rhombic dodecahedron]]
*[[Truncated trapezohedron]]
*[[Truncated triakis tetrahedron]]
*[[Tridyakis icosahedron]]
*[[Trigonal trapezohedron]]
*[[Regular skew polyhedron]]
*[[Waterman polyhedron]]
*[[Wedge (geometry)|Wedge]]

===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 prisms]]
*Compound of six pentagrammic crossed antiprisms
*[[Compound of six pentagrammic prisms]]
*[[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 tetrahedra]]
*[[Compound of twelve pentagonal antiprisms with rotational freedom]]
*[[Compound of twelve pentagonal prisms]]
*[[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]]
*[[11-cell]], [[57-cell]]

;[[Regular star 4-polytope]]
*[[Icosahedral 120-cell]], [[Small stellated 120-cell]], [[Great 120-cell]], [[Grand 120-cell]], [[Great stellated 120-cell]], [[Grand stellated 120-cell]], [[Great grand 120-cell]], [[Great icosahedral 120-cell]], [[Grand 600-cell]], [[Great grand stellated 120-cell]]

;[[Uniform 4-polytope]]
*[[Rectified 5-cell]], [[Truncated 5-cell]], [[Cantellated 5-cell]], [[Runcinated 5-cell]]
*[[Rectified tesseract]], [[Truncated tesseract]], [[Cantellated tesseract]], [[Runcinated tesseract]]
*[[Rectified 16-cell]], [[Truncated 16-cell]]
*[[Rectified 24-cell]], [[Truncated 24-cell]], [[Cantellated 24-cell]], [[Runcinated 24-cell]], [[Snub 24-cell]]
*[[Rectified 120-cell]], [[Truncated 120-cell]], [[Cantellated 120-cell]], [[Runcinated 120-cell]]
*[[Rectified 600-cell]], [[Truncated 600-cell]], [[Cantellated 600-cell]]

;[[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]], [[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]], [[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]]
*[[1 22 polytope|122 polytope]], [[2 21 polytope|221 polytope]]

===Honeycombs===
*[[6-cubic honeycomb]]
*[[6-simplex honeycomb]]
*[[6-demicubic honeycomb]]
*[[2 22 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]], [[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]], [[Hexicated 7-orthoplexes|Hexicated 7-orthoplex]]
*[[1 32 polytope|132 polytope]], [[2 31 polytope|231 polytope]], [[3 21 polytope|321 polytope]]

===Honeycombs===
*[[7-cubic honeycomb]]
*[[7-demicubic honeycomb]]
*[[3 31 honeycomb|331 honeycomb]], [[1 33 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-demicube]], [[Truncated 8-demicube]], [[Cantellated 8-demicube]], [[Runcinated 8-demicube]], [[Stericated 8-demicube]], [[Pentellated 8-demicube]], [[Hexicated 8-demicube]]
*[[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-orthoplex]], [[Rectified 8-orthoplex]], [[Truncated 8-orthoplex]], [[Cantellated 8-orthoplex]], [[Runcinated 8-orthoplex]], [[Stericated 8-orthoplex]], [[Pentellated 8-orthoplex]], [[Hexicated 8-orthoplex]],
*[[1 42 polytope|142 polytope]], [[2 41 polytope|241 polytope]], [[4 21 polytope|421 polytope]], [[Truncated 4 21 polytope|Truncated 421 polytope]], [[Truncated 2 41 polytope|Truncated 241 polytope]], [[Truncated 1 42 polytope|Truncated 142 polytope]], [[Cantellated 4 21 polytope|Cantellated 421 polytope]], [[Cantellated 2 41 polytope|Cantellated 241 polytope]], [[Runcinated 4 21 polytope|Runcinated 421 polytope]]

===Honeycombs===
*[[8-cubic honeycomb]]
*[[8-demicubic honeycomb]]
*[[5 21 honeycomb|521 honeycomb]], [[2 51 honeycomb|251 honeycomb]], [[1 52 honeycomb|152 honeycomb]]

==Nine dimensions==
;[[9-polytope]]
*[[9-simplex]]
*[[9-demicube]]
*[[9-cube]]
*[[9-orthoplex]]

===Hyperbolic honeycombs===
*[[E9 honeycomb|E9 honeycomb]]

==Ten dimensions==
;[[10-polytope]]
*[[10-cube|10-simplex]]
*[[10-demicube]]
*[[10-cube]]
*[[10-simplex|10-orthoplex]]

==Dimensional families==
;[[Regular polytope]] and [[List of regular polytopes]]
*[[Simplex]]
*[[Hypercube]]
*[[Cross-polytope]]
;[[Uniform polytope]]
*[[Demihypercube]]
*[[Uniform 1 k2 polytope|Uniform 1''k''2 polytope]]
*[[Uniform 2 k1 polytope|Uniform 2''k''1 polytope]]
*[[Uniform k 21 polytope|Uniform ''k''21 polytope]]

;Honeycombs
*[[Hypercubic honeycomb]]
*[[Alternated hypercubic honeycomb]]

==Geometric operators==
*[[Rectification (geometry)]]
*[[Truncation (geometry)]]
*[[Bitruncation]]
*[[Cantellation]]
*[[Runcination]]
*[[Sterication]]
*[[Omnitruncation]]
*[[Expansion (geometry)]]
*[[Snub (geometry)]]
*[[Alternation (geometry)]]
*[[Dual polyhedron]]
*[[Gyration (geometry)]]
*[[Elongation (geometry)]]
*[[Augmentation (geometry)]]
*[[Diminishment (geometry)]]
*[[Greatening (geometry)]]
*[[Aggrandizement (geometry)]]
*[[Stellation]]
*[[Kleetope]]
*[[Conway polyhedron notation]]

==See also==
*[[List of geometry topics]]

{{Polygons}}
{{Polyhedron navigator}}
{{Nonconvex polyhedron navigator}}
{{Polytopes}}

[[Category:Polyhedra| ]]
[[Category:Polygons| ]]
[[Category:Polytopes| ]]
[[Category:Lists of shapes|polygons]]

Vero - Wikipedia - User contributions [en]

List of polygons, polyhedra and polytopes