List of small polyhedra by vertex count

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In geometry, a polyhedron is a solid in three dimensions with flat faces and straight edges. Every edge has exactly two faces, and every vertex is surrounded by alternating faces and edges. The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices. Named polyhedra primarily come from the families of platonic solids, Archimedean solids, Catalan solids, and Johnson solids, as well as dihedral symmetry families including the pyramids, bipyramids, prisms, antiprisms, and trapezohedrons.

Polyhedra by vertex count[edit]

Notes: Polyhedra with different names that are topologically identical are listed together. Except in the cases of four and five vertices, the lists below are by no means exhaustive of all possible polyhedra with the given number of vertices, but rather just include particularly simple/common/well-known/named examples. The "Counting Polyhedra" link below gives the exact number of distinct polyhedra with n vertices for small values of n.

4 Tetrahedron.svg
Tetrahedron
triangular pyramid
5 Square pyramid.png
Square pyramid
Triangular dipyramid.png
Triangular bipyramid
6 Pentagonal pyramid.png
Pentagonal pyramid
Octahedron.svg
Octahedron (regular)
triangular antiprism
square bipyramid
Triangular prism.png
Triangular prism
wedge
7 Hexagonal pyramid.png
Hexagonal pyramid
Pentagonal dipyramid.png
Pentagonal bipyramid
Augmented triangular prism.png
Augmented triangular prism
Elongated triangular pyramid.png
Elongated triangular pyramid
8 Heptagonal pyramid Hexagonale bipiramide.png
Hexagonal bipyramid
Hexahedron.png
Hexahedron
Cube
Square prism
Cuboid
Rhombohedron
Trigonal trapezohedron
Square antiprism.png
Square antiprism
Triakistetrahedron.jpg
Triakis tetrahedron
Elongated triangular dipyramid.png
Elongated triangular bipyramid
Gyrobifastigium.png
Gyrobifastigium
Snub disphenoid.png
Snub disphenoid
Biaugmented triangular prism.png
Biaugmented triangular prism
9 Triangular cupola.png
Triangular cupola
Triaugmented triangular prism.png
Triaugmented triangular prism
Elongated square pyramid.png
Elongated square pyramid
Gyroelongated square pyramid.png
Gyroelongated square pyramid
Tridiminished icosahedron.png
Tridiminished icosahedron
Octagonal pyramid Heptagonal bipyramid
10 Enneagonal pyramid Octagonal bipyramid.png
Octagonal bipyramid
Pentagonal prism.png
Pentagonal prism
Pentagonal antiprism.png
Pentagonal antiprism
Gyroelongated square dipyramid.png
Gyroelongated square bipyramid
Elongated square dipyramid.png
Elongated square bipyramid
Metabidiminished icosahedron.png
Metabidiminished icosahedron
Augmented tridiminished icosahedron.png
Augmented tridiminished icosahedron
Sphenocorona.png
Sphenocorona
11 Decagonal pyramid Enneagonal bipyramid Augmented pentagonal prism.png
Augmented pentagonal prism
Elongated pentagonal pyramid.png
Elongated pentagonal pyramid
Gyroelongated pentagonal pyramid.png
Gyroelongated pentagonal pyramid
Augmented sphenocorona.png
Augmented sphenocorona
Octadecahedron.png
Octadecahedron
12 Elongated pentagonal dipyramid.png
Elongated pentagonal bipyramid
Truncated tetrahedron.png
Truncated tetrahedron
Cuboctahedron.png
Cuboctahedron
Square cupola.png
Square cupola
Triangular orthobicupola.png
Triangular orthobicupola
Biaugmented pentagonal prism.png
Biaugmented pentagonal prism
Hendecagonal pyramid Icosahedron.jpg
Icosahedron
Decagonal bipyramid.png
Decagonal bipyramid
Hexagonal prism.png
Hexagonal prism
Hexagonal antiprism.png
Hexagonal antiprism
Sphenomegacorona.png
Sphenomegacorona
13 Dodecagonal pyramid Hendecagonal bipyramid Augmented hexagonal prism.png
Augmented hexagonal prism
14 Tridecagonal pyramid Dodecagonal bipyramid Elongated hexagonal dipyramid.png
Elongated hexagonal dipyramid
Prism 7.png
Heptagonal prism
Antiprism 7.png
Heptagonal antiprism
Rhombic dodecahedron.png
Rhombic dodecahedron
Parabiaugmented hexagonal prism.png
Parabiaugmented hexagonal prism
Metabiaugmented hexagonal prism.png
Metabiaugmented hexagonal prism
Hebesphenomegacorona.png
]Hebesphenomegacorona
Bilunabirotunda.png
Bilunabirotunda
15 Tetradecagonal pyramid Tridecagonal bipyramid Pentagonal cupola Elongated triangular cupola
Gyroelongated triangular cupola
16 Octagonal prism.png
Octagonal prism
Octagonal antiprism.png
Octagonal antiprism
Triaugmented hexagonal prism Augmented truncated tetrahedron Square orthobicupola Square gyrobicupola
Disphenocingulum Snub square antiprism
17
18 Prism 9.png
Enneagonal prism
Enneagonal antiprism.png
Enneagonal antiprism
Gyroelongated triangular bicupola Elongated triangular orthobicupola Elongated triangular gyrobicupola Triangular hebesphenorotunda
19
20 Dodecahedron Pentagonal rotunda Elongated square cupola Gyroelongated square cupola Pentagonal orthobicupola Pentagonal gyrobicupola
Decagonal prism.png
Decagonal prism
Decagonal antiprism.png
Decagonal antiprism
21 Augmented dodecahedron
22 Parabiaugmented dodecahedron Metabiaugmented dodecahedron
23 Triaugmented dodecahedron
24 Truncated hexahedron.png
Truncated cube
Truncated octahedron.png
Truncated octahedron
Small rhombicuboctahedron.png
Rhombicuboctahedron
Pseudorhombicuboctahedron.png
Elongated square gyrobicupola
Gyroelongated square bicupola.png
Gyroelongated square bicupola

References[edit]

  • Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN.  The first proof that there are only 92 Johnson solids.

External links[edit]