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 the ridge or (n−2)-face of the polygon 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
Polychoron (4-polytope) Vertex the (n−4)-face of the polychoron 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) 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 Point Line segment Vertex figure Peak – (n−3)-face Ridge – (n−2)-face Facet – (n−1)-face
Two dimensional (polygons) Triangle Equilateral triangle Isosceles triangle Golden triangle (mathematics) Scalene triangle Right triangle Oblique triangle Acute triangle Obtuse Triangle
Quadrilateral Rectangle Square Unit square Oblong Golden rectangle Silver rectangle Ailles rectangle Rhombus Golden Rhombus Parallelogram Rhomboid Trapezoid Isosceles trapezoid Kite Lute of pythagoras 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 polygons Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram Icositetragram
Families Concave polygon Cyclic polygon Regular polygon Polyform 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 (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 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 Trigonal 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