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
Vertex the ridge or (n−2)-face of the polygonEdge the facet or (n−1)-face of the polygon
Vertex the peak or (n−3)-face of the polyhedronEdge the ridge or (n−2)-face of the polyhedronFace the facet or (n−1)-face of the polyhedron
Vertex the (n−4)-face of the polychoronEdge the peak or (n−3)-face of the polychoronFace the ridge or (n−2)-face of the polychoronCell the facet or (n−1)-face of the polychoron
5-polytope
Vertex the (n−5)-face of the 5-polytopeEdge the (n−4)-face of the 5-polytopeFace the peak or (n−3)-face of the 5-polytopeCell the ridge or (n−2)-face of the 5-polytopeHypercell or Teron the facet or (n−1)-face of the 5-polytope
Other
Families
Tilings
List of uniform tilings
Uniform tilings in hyperbolic plane
Archimedean tiling Three-dimensional space
Regular
Regular polyhedron
Tetrahedron Pentahedron Hexahedron Heptahedron Octahedron Enneahedron Decahedron 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 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
Uniform star polyhedron
Prismatic uniform polyhedron
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
Honeycombs
Convex uniform honeycomb Dual uniform honeycomb Others Convex uniform honeycombs in hyperbolic space
Other
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 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
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
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 122 polytope , 221 polytope
Honeycombs
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-orthoplex 132 polytope , 231 polytope , 321 polytope
Honeycombs
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 ,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
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
Kepler-Poinsot (nonconvex Uniform truncations Nonconvex uniform hemipolyhedra Duals of nonconvex Duals of nonconvex
