A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other. This article is about the geometric shape. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ...

There are 3 classes of semiregular polyhedra:

• Archimedean solids - 13 polyhedra with more than one polygon face type.
• Prisms - infinite set: 2 N-gons and N-squares.
• Antiprisms - infinite set: 2 N-gons and 2N triangles.

The Platonic solids are a set of 5 regular polyhedra. In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with: All its faces being congruent regular polygons The same number of faces meeting at each of its vertices These are in contrast to: The Kepler-Poinsot solids, which are not convex The Archimedean and...

These 4 sets compose the convex polyhedra, along with a set of 53 nonconvex forms compose the larger set of uniform polyhedra. A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...

There's one example of a polyhedron with every vertex containing the same sequence of faces, that fails to have, for every pair of vertices, an isometry mapping the first vertex into the second, is the elongated square gyrobicupola. In geometry, the elongated square gyrobicupola is one of the Johnson solids (J37). ...

The regular and semiregular polyhedra are related to the regular and semiregular tilings of the plane. See List of uniform planar tilings. This table shows the 11 uniform tilings of the plane, and their dual tilings. ...

Existence of the semiregular polyhedra

The existence of these polyehedra can be enumerated by looking at their vertex configuration and the angle defect: A set of regular faces must have internal angles less than 360 degees. In polyhedral geometry a vertex configuration is a short-hand notation for representing a vertex as the sequence of faces around a vertex. ...

NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if equal to 360. It can represent a tiling of the hyperbolic plane if greater than 360 degrees.

For uniform polyhedra, the angle defect can be used to compute the number of vertices. (The angle defect is defined as 360 degrees minus the sum of all the internal angles of the polygons that meet at the vertex.) Descartes' theorem states that the sum of all the angle defects in a topological sphere must add to 4*π radians or 720 degrees. In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ...

Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices: Vertices = 720/(angle-defect).

Example: A truncated cube 3.8.8 has an angle defect of 30 degrees. Therefore it has 720/30=24 vertices. The truncated cube, or truncated hexahedron, is an Archimedean solid. ...

In particular it follows that {a,b} has 4/(2-b(1-2/a)) vertices.

Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However not all configurations are possible.

Topological requirements limit existence. Specifically p.q.r implies that a p-gon is surrounded by alternating a q-gons and r-gons, so either p is even or q=r. Similarly q is even or p=r. Therefore potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.n (for any n>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6. In fact, all these configurations with three faces meeting at each vertex turn out to exist.

Similarly when four faces meet at each vertex, p.q.r.s, if one number is odd its neighbors must be equal.

A complete enumeration of convex uniform polyhedra

The number in parentheses is the number of vertices, determined by the angle defect.

Triples

• Platonic solids 3.3.3 (4), 4.4.4 (8), 5.5.5 (20)
• prisms 3.4.4 (6), 4.4.4 (8; also listed above), 4.4.n (2n)
• Archimedean solids 3.6.6 (12), 3.8.8 (24), 3.10.10 (60), 4.6.6 (24), 4.6.8 (48), 4.6.10 (120), 5.6.6 (60).
• regular tiling 6.6.6
• semiregular tilings 3.12.12, 4.6.12, 4.8.8

Quadruples A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A cube [1] (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each vertex. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... The truncated tetrahedron is an Archimedean solid. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... The truncated dodecahedron is an Archimedean solid. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... The truncated icosahedron is an Archimedean solid. ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ...

• Platonic solid 3.3.3.3 (6)
• antiprisms 3.3.3.3 (6; also listed above), 3.3.3.n (2n)
• Archimedean solids 3.4.3.4 (12), 3.5.3.5 (30), 3.4.4.4 (24), 3.4.5.4 (60)
• regular tiling 4.4.4.4
• semiregular tilings 3.6.3.6, 3.4.6.4

Quintuples Finally configurations with five and six faces meeting at each vertex: An octahedron (plural: octahedra) is a polyhedron with eight faces. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... A colored model The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...

• Platonic solid 3.3.3.3.3 (12)
• Archimedean solids 3.3.3.3.4 (24), 3.3.3.3.5 (60) (both chiral)
• semiregular tilings 3.3.3.3.6 (chiral), 3.3.3.4.4, 3.3.4.3.4 (note that the two different orders of the same numbers give two different patterns)

Sextuples An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant, which has faces which are equilateral triangles. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. ...

• regular tiling 3.3.3.3.3.3