Semiregular polyhedron

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 in an isometry mapping one into the other (the elongated square gyrobicupola shows that this condition makes a difference).

Naming by vertex configuration

Semiregular polyhedra can be named by their vertex configuration.

For example the notation 4.4.4 represents the cube because it has 3 squares on each vertex, and 3.4.3.4 represents the cuboctohedron which alternates triangles and squares of each vertex.

NOTE: The notation for semiregular vertex figures is a 2d generalization of the Schläfli symbols for regular polyhedra.

For example, the cube's 4.4.4 can be "reduced to a shorthand" 4³, or Schläfli's notation {4,3}, meaning three squares per vertex.

Vertex count

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.

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

Semiregular polyhedron example: A truncated cube 3.8.8 has an angle defect of 30 degrees. Therefore it has 720/30=24 vertices.

Semiregular plane tiling example: The tiling 4.8.8 has an angle defect of zero, and so it doesn't have a finite number of faces.

Existence

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 alternatingly a q-gon and an r-gon, so 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, 4.4.4, 4.4.5, 4.4.6, 4.4.7,..., and 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 (in parentheses the number of vertices, as can be computed from the configuration, see above):

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), 4.4.5 (10), 4.4.6 (12), 4.4.7 (14),...
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 tessellation 6.6.6
semiregular tessellations 3.12.12, 4.6.12, 4.8.8

Similarly configurations with four faces meeting at each vertex, p.q.r.s, require that if one number is odd, two of the other three are the same:

Platonic solid 3.3.3.3 (6)
antiprisms 3.3.3.3 (6; also listed above), 3.3.3.4 (8), 3.3.3.5 (10), 3.3.3.6 (12), 3.3.3.7 (14),...
Archimedean solids 3.4.3.4 (12), 3.5.3.5 (30), 3.4.4.4 (24), 3.4.5.4 (60)
regular tessellation 4.4.4.4
semiregular tessellations 3.6.3.6, 4.4.4.4

Finally configurations with five and six faces meeting at each vertex:

Platonic solid 3.3.3.3.3 (12)
Archimedean solids 3.3.3.3.4 (24), 3.3.3.3.5 (60) (both chiral)
regular tessellation 3.3.3.3.3.3
semiregular tessellations 3.6.3.6, 3.3.3.3.6, 3.3.3.4.4, 3.3.4.3.4 (note that the two different orders of the same numbers give two different patterns)

Subclassifications

Platonic_solid - Five regular polyhedra. (All faces of same type)

{3,3}

{4,3}

{3,4}

{5,3}

{3,5}

Kepler-Poinsot solid - 4 Stellated/Intersecting "Regular" forms.

File:Kepler poinsot solids.gif

Archimedean solid - 13 polyhedra with more than one polygon face type.

3.6.6	3.4.3.4	3.8.8	4.6.6	3.4.4.4	4.6.8	3.3.3.3.4
3.5.3.5	3.10.10	5.6.6	3.4.5.4	4.6.4.10	3.3.3.3.5

There are two infinite subgroups with symmetry across two parallel planes:
- Prism - 2 N-gons and N-squares.

3.4.4 | 4.4.5 | 4.4.6

4.4.4

- Antiprism - 2 N-gons and 2N triangles.

3.3.3.3

3.3.3.5

3.3.3.17

Nonclosed extensions

If closed (finite) requirement is removed, there are two extended classes that can be considered. Both have an infinite number of faces.

Planar semiregular tessellations - angle defect as zero.

{3,6}	{4,4}	{6,3}
3.3.3.3.6	3.6.3.6	File:Tile33344bc.gif 3.3.3.4.4	3.3.4.3.3
3.4.6.4	4.8.8	3.12.12	4.6.12

Hyperbolic semiregular tessellations - angle defect is negative. (Won't fit on a plane, but some forms can fit periodically within 3d space.)

4.4.4.6 hyperbolic tessellation embedded in 3-space
4.4.4.6

There are an infinite number of hyperbolic semiregular tessellations and their topology is not determined fully by the vertex configuration. i.e. Two or more different geometries can be formed by the same vertex configuration.

Duals

The dual of a semiregular polyhedron has all identical irregular polygons.

Dual subclassifications

The duals of the Platonic solids are Platonic solids!
The duals of the Kepler-Poinsot solids are other Kepler-Poinsot solids!
The duals of the Archimedean solids are called the Catalan_solids.

Duals can be named by their source semiregular polyhedron vertex configuration name, prefixed by a V.

V3.6.6	V3.4.3.4	V3.8.8	V4.6.6	V3.4.4.4	V4.6.8	V3.3.3.3.4
V3.5.3.5	V3.10.10	V5.6.6	V3.4.5.4	V4.6.10	V3.3.3.3.3.5

The duals of the Prisms are called Bipyramids

V3.4.4

V4.4.4

V4.4.5

The duals of antiprisms have kite-shaped faces.

V3.3.3.5

This fact means the duals make for "good dice" of varied number of sides beyond the platonic solids with equal area. For example, the dual of the icosidodecahedron has 30 rhombic faces and is called a rhombic_triacontahedron.