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

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

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
Scalene triangle
Right triangle
Oblique triangle
- Acute triangle
- Obtuse Triangle

Quadrilateral

Star polygons

Families

Concave polygon
Cyclic polygon
Regular polygon
Polyform
Gnomon
Golygon

Tilings

List of uniform tilings

Uniform tilings in hyperbolic plane

Archimedean 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 dipyramid, 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 dipyramid, Augmented pentagonal prism

Dodecahedron

Pentagonal antiprism, Decagonal prism, Pentagonal cupola, Snub disphenoid, Elongated square dipyramid, Metabidiminished icosahedron, Hexagonal bipyramid, Hexagonal trapezohedron, Triakis tetrahedron, Rhombic dodecahedron, Hendecagonal pyramid, Trapezo-rhombic dodecahedron, Rhombo-hexagonal dodecahedron

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

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

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 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

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, 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

Nine dimensions

9-polytope

Hyperbolic honeycombs

E₉ honeycomb

Ten dimensions

10-polytope

Dimensional families

Regular polytope and List of regular polytopes

Uniform polytope

Honeycombs

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

v t e Polygons (List)
Triangles	Acute Equilateral Ideal Isosceles Obtuse Right
Quadrilaterals	Antiparallelogram Bicentric Cyclic Equidiagonal Ex-tangential Harmonic Isosceles trapezoid Kite Lambert Orthodiagonal Parallelogram Rectangle Right kite Rhombus Saccheri Square Tangential Tangential trapezoid Trapezoid
By number of sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon (Enneagon, 9) Decagon (10) Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Enneadecagon (19) Icosagon (20) Icosidigon (22) Icositetragon (24) Icosihexagon (26) Icosioctagon (28) Triacontagon (30) Triacontadigon (32) Triacontatetragon (34) Tetracontagon (40) Tetracontadigon (42) Tetracontaoctagon (48) Pentacontagon (50) Hexacontagon (60) Hexacontatetragon (64) Heptacontagon (70) Octacontagon (80) Enneacontagon (90) Enneacontahexagon (96) Hectogon (100) 120-gon 257-gon 360-gon Chiliagon (1000) Myriagon (10000) 65537-gon Megagon (1000000) Apeirogon (∞)
Star polygons	Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram
Classes	Concave Convex Cyclic Equiangular Equilateral Isogonal Isotoxal Pseudotriangle Regular Simple Skew Star-shaped Tangential

v t e Star-polyhedra navigator
Kepler-Poinsot polyhedra (nonconvex regular polyhedra)	small stellated dodecahedron great dodecahedron great stellated dodecahedron great icosahedron
Uniform truncations of Kepler-Poinsot polyhedra	dodecadodecahedron truncated great dodecahedron rhombidodecadodecahedron truncated dodecadodecahedron snub dodecadodecahedron great icosidodecahedron truncated great icosahedron nonconvex great rhombicosidodecahedron great truncated icosidodecahedron
Nonconvex uniform hemipolyhedra	tetrahemihexahedron cubohemioctahedron octahemioctahedron small dodecahemidodecahedron small icosihemidodecahedron great dodecahemidodecahedron great icosihemidodecahedron great dodecahemicosahedron small dodecahemicosahedron
Duals of nonconvex uniform polyhedra	medial rhombic triacontahedron small stellapentakis dodecahedron medial deltoidal hexecontahedron small rhombidodecacron medial pentagonal hexecontahedron medial disdyakis triacontahedron great rhombic triacontahedron great stellapentakis dodecahedron great deltoidal hexecontahedron great disdyakis triacontahedron great pentagonal hexecontahedron
Duals of nonconvex uniform polyhedra with infinite stellations	tetrahemihexacron hexahemioctacron octahemioctacron small dodecahemidodecacron small icosihemidodecacron great dodecahemidodecacron great icosihemidodecacron great dodecahemicosacron small dodecahemicosacron

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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List of polygons, polyhedra and polytopes

Polytope elements

Polygon (2-polytope)

Polyhedron (3-polytope)

Polychoron (4-polytope)

5-polytope

Other

Two dimensional (polygons)

Star polygons

Families

Tilings

Three dimensional (polyhedra)

Regular

Archimedean solids

Prisms and antiprisms

Catalan solids

Bipyramids and Trapezohedron

Uniform star polyhedra

Uniform prismatic star polyhedra

Johnson solids

Dual uniform star polyhedra

Honeycombs

Other

Regular and uniform compound polyhedra

Four dimensions

Honeycombs

Five dimensions

Honeycombs

Six dimensions

Honeycombs

Seven dimensions

Honeycombs

Eight dimension

Honeycombs

Nine dimensions

Hyperbolic honeycombs

Ten dimensions

Dimensional families

Geometric operators

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