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Platonic solids - regular polyhedra (all faces of the same type)
 {5,3} |  {3,5} |
Archimedean solids - polyhedra with more than one polygon face type.
 3.10.10 |  4.6.10 |  5.6.6 |  3.4.5.4 |  3.5.3.5 |
Catalan solids - duals of the Archimedean solids.
 V3.10.10 |  V4.6.10 |  V5.6.6 |  V3.4.5.4 |  V3.5.3.5 |
| Name | Picture | Faces | Edges | Vertices | Edges per face | Faces meeting at each vertex |
|---|---|---|---|---|---|---|
| dodecahedron | | 12 | 30 | 20 | 5 | 3 |
| icosahedron | | 20 | 30 | 12 | 3 | 5 |
| Name | picture | Faces | Edges | Vertices | Vertex configuration | |
|---|---|---|---|---|---|---|
| icosidodecahedron (quasi-regular: vertex- and edge-uniform) | (Video) | 32 | 20 triangles 12 pentagons | 60 | 30 | 3,5,3,5 |
| truncated dodecahedron | (Video) | 32 | 20 triangles 12 decagons | 90 | 60 | 3,10,10 |
| truncated icosahedron or commonly football (soccer ball) | (Video) | 32 | 12 pentagons 20 hexagons | 90 | 60 | 5,6,6 |
| rhombicosidodecahedron or small rhombicosidodecahedron | (Video) | 62 | 20 triangles 30 squares 12 pentagons | 120 | 60 | 3,4,5,4 |
| truncated icosidodecahedron or great rhombicosidodecahedron | (Video) | 62 | 30 squares 20 hexagons 12 decagons | 180 | 120 | 4,6,10 |
| Name | picture | Dual Archimedean solid | Faces | Edges | Vertices | Face Polygon |
|---|---|---|---|---|---|---|
| rhombic triacontahedron (quasi-regular dual: face- and edge-uniform) | (Video) | icosidodecahedron | 30 | 60 | 32 | rhombus |
| triakis icosahedron | (Video) | truncated dodecahedron | 60 | 90 | 32 | isosceles triangle |
| pentakis dodecahedron | (Video) | truncated icosahedron | 60 | 90 | 32 | isosceles triangle |
| deltoidal hexecontahedron | (Video) | rhombicosidodecahedron | 60 | 120 | 62 | kite |
| disdyakis triacontahedron or hexakis icosahedron | (Video) | truncated icosidodecahedron | 120 | 180 | 62 | scalene triangle |
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| Name | picture | Faces | Edges | Vertices | Vertex configuration | |
|---|---|---|---|---|---|---|
| snub dodecahedron or snub icosidodecahedron (2 chiral forms) |  (Video)  (Video) | 92 | 80 triangles 12 pentagons | 150 | 60 | 3,3,3,3,5 |
| Name | picture | Dual Archimedean solid | Faces | Edges | Vertices | Face Polygon |
|---|---|---|---|---|---|---|
| pentagonal hexecontahedron |   (Video)(Video) | snub dodecahedron | 60 | 50 | 92 | irregular pentagon |
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In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. The convex polyhedra with regular faces and symmetric vertices include also the five Platonic solids and the two infinite families of prisms and antiprisms; these are not counted as Archimedean solids. The pseudorhombicuboctahedron has regular faces, and vertices that are symmetric in a weaker sense; it is also not generally counted as an Archimedean solid. The Archimedean solids are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ; it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a "Johnson solid".
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
In mathematics, a Catalan solid, or Archimedean dual, is a polyhedron that is dual to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
In geometry, a triakis tetrahedron is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.
In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.
In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.
In geometry, a deltoidal hexecontahedron is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
In geometry, the term semiregular polyhedron is used variously by different authors.
In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.
In geometry, the pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the truncated rhombic triacontahedron.