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 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

Quadrilateral

Star polygons

Families

Tilings

List of uniform tilings

Uniform tilings in hyperbolic plane

Archimedean tiling

Three dimensional (polyhedra)

Three-dimensional space

Bipyramids and Trapezohedron

Uniform star polyhedra

Uniform star polyhedron

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

Dual uniform star polyhedra

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

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

Prismatic uniform 4-polytope

Uniform antiprismatic prism

Honeycombs

Five dimensions

Five-dimensional space, 5-polytope and uniform 5-polytope

Prismatic uniform 5-polytope

Honeycombs

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope

Honeycombs

Seven dimensions

Seven-dimensional space, uniform 7-polytope

Honeycombs

Eight dimension

Eight-dimensional space, uniform 8-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

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 polychoron	Pentachoron	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

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

See also