Polyhedron | |
Class | Number and properties |
---|---|
Platonic solids |
(5, convex, regular) |
Archimedean solids |
(13, convex, uniform) |
Kepler-Poinsot polyhedra |
(4, regular, non-convex) |
Uniform polyhedra |
(75, uniform) |
Prismatoid: prisms, antiprisms etc. |
(4 infinite uniform classes) |
Polyhedra tilings | (11 regular, in the plane) |
Quasi-regular polyhedra |
(8) |
Johnson solids | (92, convex, non-uniform) |
Pyramids and Bipyramids | (infinite) |
Stellations | Stellations |
Polyhedral compounds | (5 regular) |
Deltahedra | (Deltahedra, equalatial triangle faces) |
Snub polyhedra |
(12 uniform, not mirror image) |
Zonohedron | (Zonohedra, faces have 180°symmetry) |
Dual polyhedron | |
Self-dual polyhedron | (infinite) |
Catalan solid | (13, Archimedean dual) |
There are many relations among the uniform polyhedron. Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others share the same vertices and edges as other polyhedron. The grouping below exhibit some of these relations.
Contents |
The relations can be made apparent by examining the vertex figures. obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example the cube has vertex figure 4.4.4 that is three adjacent square faces. The possible faces are
Some faces will appear with reverse orientation which is written here as
Others pass through the origin which we write as
The Wythoff symbol relates the polyhedron to spherical triangles. Wythoff symbols are written p|q r, p q|r, p q r| where the spherical triangle has angles π/p,π/q,π/r, the bar indicates the position of the vertices in relation to the triangle.
Johnson (2000) classified uniform polyhedra according to the following:
The format of each figure follows the same basic pattern
The vertex figures are on the left, followed by the Point groups in three dimensions#The seven remaining point groups, either tetrahedral Td, octahedral Oh or icosahedral Ih.
Column A lists all the regular polyhedra, column B list their truncated forms. Regular polyhedra all have vertex figures pr: p.p.p etc. and Whycroft symbol p|q r. The truncated forms have vertex figure q.q.r (where q=2p and r) and Whycroft p q|r.
vertex figure | group | A: regular: p.p.p | B: truncated regular: p.p.r |
3.3.3 |
Td |
Tetrahedron |
Truncated tetrahedron |
3.3.3.3 4.6.6 |
Oh |
Octahedron |
Truncated octahedron |
4.4.4
3.8.8 |
Oh |
Hexahedron |
Truncated hexahedron |
3.3.3.3.3 |
Ih |
Icosahedron |
Truncated icosahedron |
5.5.5 4.10.10 |
Ih |
Dodecahedron |
Truncated dodecahedron |
5.5.5.5.5 |
Ih |
Great dodecahedron |
Truncated great dodecahedron |
3.3.3.3.3 5/2.6.6. |
Ih |
Great icosahedron |
Great truncated icosahedron |
5/2.5/2.5/2.5/2.5/2 |
Ih |
Small stellated dodecahedron |
|
5/2.5/2.5/2 |
Ih |
Great stellated dodecahedron |
In addition there are three quasi-truncated forms. These also class as truncated-regular polyhedra.
vertex figures | Group Oh | Group Ih | Group Ih |
3.8/3.8/3 |
Stellated truncated hexahedron |
Small stellated truncated dodecahedron |
Great stellated truncated dodecahedron |
Column A lists some quasi-regular polyhedra, column B lists normal truncated forms, column C shows quasi-truncated forms, column D shows a different method of truncation. These truncated forms all have a vertex figure p.q.r and a Wythoff symbol p q r|.
vertex figure | group | A: quasi-regular: p.q.p.q | B: truncated quasi-regular: p.q.r | C: truncated quasi-regular: p.q.r | D: truncated quasi-regular: p.q.r |
3.4.3.4
4.6.8 |
Oh |
Cuboctahedron |
Great rhombicuboctahedron |
Great truncated cuboctahedron |
Cubitruncated cuboctahedron |
3.5.3.5
4.6.10 |
Ih |
Icosidodecahedron |
Great rhombicosidodecahedron |
Great truncated icosidodecahedron |
Icositruncated dodecadodecahedron |
5/2.5.5/2.5 |
Ih |
Dodecadodecahedron |
Truncated dodecadodecahedron |
||
3.5/2.3.5/2 |
Ih |
Great icosidodecahedron |
These are all mentioned elsewhere, but this table shows some relations. They are all regular apart from the tetrahemihexahedron which is versi-regular.
vertex figure | V | E | group | regular | regular/versi-regular |
3.3.3.3 3.4*.-3.4* |
6 | 12 | Oh |
Octahedron |
Tetrahemihexahedron |
3.3.3.3.3 |
12 | 30 | Ih |
Icosahedron |
Great dodecahedron |
5/2.5/2.5/2.5/2.5/2 |
12 | 30 | Ih |
Small stellated dodecahedron |
Great icosahedron |
Rectangular vertex figures, or crossed rectangles first column are quasi-regular second and third columns are hemihedra with faces passing through the origin, called versi-regular by some authors.
vertex figure | V | E | group | quasi-regular: p.q.p.q | versi-regular: p.s*.-p.s* | versi-regular: q.s*.-q.s* |
3.4.3.4 |
12 | 24 | Oh |
Cuboctahedron |
Octahemioctahedron |
Cubohemioctahedron |
3.5.3.5 |
30 | 60 | Ih |
Icosidodecahedron |
Small icosihemidodecahedron |
Small dodecahemidodecahedron |
3.5/2.3.5/2 |
30 | 60 | Ih |
Great icosidodecahedron |
Great icosihemidodecahedron |
Great dodecahemidodecahedron |
5.5/2.5.5/2 |
30 | 60 | Ih |
Dodecadodecahedron |
Great dodecahemicosahedron |
Small dodecahemicosahedron |
Ditrigonal (that is di(2) -tri(3)-ogonal) vertex figures are the 3-fold analog of a rectangle. These are all quasi-regular as all edges are isomorphic. The compound of 5-cubes shares the same set of edges and vertices. The cross forms have a non-orientable vertex figure so the "-" notation has not been used and the "*" faces pass near rather than through the origin.
vertex figure | V | E | group | ditrogonal | crossed-ditrogonal | crossed-ditrogonal |
5/2.3.5/2.3.5/2.3 |
20 | 60 | Ih |
Small ditrigonal icosidodecahedron |
Ditrigonal dodecadodecahedron |
Great ditrigonal icosidodecahedron |
Group III: trapezoid or crossed trapezoid vertex figures. The first column include the convex rhombic polyhedra, created by inserting two squares into the vertex figures of the Cuboctahedron and Icosidodecahedron.
vertex figure | V | E | group | trapezoid: p.q.r.q | crossed-trapezoid: p.s*.-r.s* | crossed-trapezoid: q.s*.-q.s* |
3.4.4.4 |
24 | 48 | Oh |
Small rhombicuboctahedron |
Small cubicuboctahedron |
Small rhombihexahedron |
3.8/3.4.8/3 |
24 | 48 | Oh |
Great cubicuboctahedron |
Nonconvex great rhombicuboctahedron |
Great rhombihexahedron |
3.4.5.4 |
60 | 120 | Ih |
Small rhombicosidodecahedron |
Small dodecicosidodecahedron |
Small rhombidodecahedron |
5/2.4.5.4 |
60 | 120 | Ih |
Rhombidodecadodecahedron |
Icosidodecadodecahedron |
Rhombicosahedron |
3.10/3.5/2.10/3 |
60 | 120 | Ih |
Great dodecicosidodecahedron |
Nonconvex great rhombicosidodecahedron |
Great rhombidodecahedron |
3.6.5/2.6 |
60 | 120 | Ih |
Small icosicosidodecahedron |
Small ditrigonal dodecicosidodecahedron |
Small dodecicosahedron |
3.10/3.5.10/3 |
60 | 120 | Ih |
Great ditrigonal dodecicosidodecahedron |
Great icosicosidodecahedron |
Great dodecicosahedron |