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, equilateral 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) |
In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces (a dihedron).
Chiral snub polyhedra do not always have reflection symmetry and hence sometimes have two enantiomorphous (left- and right-handed) forms which are reflections of each other. Their symmetry groups are all point groups.
For example, the snub cube:
Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3.p.3.q.3.r. Retrosnub polyhedra (a subset of the snub polyhedron, containing the great icosahedron, small retrosnub icosicosidodecahedron, and great retrosnub icosidodecahedron) still have this form of Wythoff symbol, but their vertex configurations are instead
There are 12 uniform snub polyhedra, not including the antiprisms, the icosahedron as a snub tetrahedron, the great icosahedron as a retrosnub tetrahedron and the great disnub dirhombidodecahedron, also known as Skilling's figure.
When the Schwarz triangle of the snub polyhedron is isosceles, the snub polyhedron is not chiral. This is the case for the antiprisms, the icosahedron, the great icosahedron, the small snub icosicosidodecahedron, and the small retrosnub icosicosidodecahedron.
In the pictures of the snub derivation (showing a distorted snub polyhedron, topologically identical to the uniform version, arrived at from geometrically alternating the parent uniform omnitruncated polyhedron) where green is not present, the faces derived from alternation are coloured red and yellow, while the snub triangles are blue. Where green is present (only for the snub icosidodecadodecahedron and great snub dodecicosidodecahedron), the faces derived from alternation are red, yellow, and blue, while the snub triangles are green.
Snub polyhedron | Image | Original omnitruncated polyhedron | Image | Snub derivation | Symmetry group | Wythoff symbol Vertex description |
---|---|---|---|---|---|---|
Icosahedron (snub tetrahedron) | Truncated octahedron | Ih (Th) | | 3 3 2 3.3.3.3.3 | |||
Great icosahedron (retrosnub tetrahedron) | Truncated octahedron | Ih (Th) | | 2 3/23/2 (3.3.3.3.3)/2 | |||
Snub cube or snub cuboctahedron | Truncated cuboctahedron | O | | 4 3 2 3.3.3.3.4 | |||
Snub dodecahedron or snub icosidodecahedron | Truncated icosidodecahedron | I | | 5 3 2 3.3.3.3.5 | |||
Small snub icosicosidodecahedron | Doubly covered truncated icosahedron | Ih | | 3 3 5/2 3.3.3.3.3.5/2 | |||
Snub dodecadodecahedron | Small rhombidodecahedron with extra 12{10/2} faces | I | | 5 5/2 2 3.3.5/2.3.5 | |||
Snub icosidodecadodecahedron | Icositruncated dodecadodecahedron | I | | 5 3 5/3 3.5/3.3.3.3.5 | |||
Great snub icosidodecahedron | Rhombicosahedron with extra 12{10/2} faces | I | | 3 5/2 2 3.3.5/2.3.3 | |||
Inverted snub dodecadodecahedron | Truncated dodecadodecahedron | I | | 5 2 5/3 3.5/3.3.3.3.5 | |||
Great snub dodecicosidodecahedron | Great dodecicosahedron with extra 12{10/2} faces | no image yet | I | | 3 5/25/3 3.5/3.3.5/2.3.3 | ||
Great inverted snub icosidodecahedron | Great truncated icosidodecahedron | I | | 3 2 5/3 3.5/3.3.3.3 | |||
Small retrosnub icosicosidodecahedron | Doubly covered truncated icosahedron | no image yet | Ih | |5/23/23/2 (3.3.3.3.3.5/2)/2 | ||
Great retrosnub icosidodecahedron | Great rhombidodecahedron with extra 20{6/2} faces | no image yet | I | | 2 5/33/2 (3.3.3.5/2.3)/2 | ||
Great dirhombicosidodecahedron | — | — | — | Ih | |3/25/3 3 5/2 (4.3/2.4.5/3.4.3.4.5/2)/2 | |
Great disnub dirhombidodecahedron | — | — | — | Ih | | (3/2) 5/3 (3) 5/2 (3/2.3/2.3/2.4.5/3.4.3.3.3.4.5/2.4)/2 |
Notes:
There is also the infinite set of antiprisms. They are formed from prisms, which are truncated hosohedra, degenerate regular polyhedra. Those up to hexagonal are listed below. In the pictures showing the snub derivation, the faces derived from alternation (of the prism bases) are coloured red, and the snub triangles are coloured yellow. The exception is the tetrahedron, for which all the faces are derived as red snub triangles, as alternating the square bases of the cube results in degenerate digons as faces.
Snub polyhedron | Image | Original omnitruncated polyhedron | Image | Snub derivation | Symmetry group | Wythoff symbol Vertex description |
---|---|---|---|---|---|---|
Tetrahedron | Cube | Td (D2d) | | 2 2 2 3.3.3 | |||
Octahedron | Hexagonal prism | Oh (D3d) | | 3 2 2 3.3.3.3 | |||
Square antiprism | Octagonal prism | D4d | | 4 2 2 3.4.3.3 | |||
Pentagonal antiprism | Decagonal prism | D5d | | 5 2 2 3.5.3.3 | |||
Pentagrammic antiprism | Doubly covered pentagonal prism | D5h | |5/2 2 2 3.5/2.3.3 | |||
Pentagrammic crossed-antiprism | Decagrammic prism | D5d | | 2 2 5/3 3.5/3.3.3 | |||
Hexagonal antiprism | Dodecagonal prism | D6d | | 6 2 2 3.6.3.3 |
Notes:
Two Johnson solids are snub polyhedra: the snub disphenoid and the snub square antiprism. Neither is chiral.
Snub polyhedron | Image | Original polyhedron | Image | Symmetry group |
---|---|---|---|---|
Snub disphenoid | Disphenoid | D2d | ||
Snub square antiprism | Square antiprism | D4d |
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids, excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron. They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, the truncated cuboctahedron 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, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces. Hence, it is a non-regular dodecahedron.
In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß{3,5}.
In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. It is given a Schläfli symbol ß{3⁄2,5}.
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both.
In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.
In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" or "icosahedrons".
Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
---|---|---|---|---|---|---|---|---|---|
t0{p,q} {p,q} | t01{p,q} t{p,q} | t1{p,q} r{p,q} | t12{p,q} 2t{p,q} | t2{p,q} 2r{p,q} | t02{p,q} rr{p,q} | t012{p,q} tr{p,q} | ht0{p,q} h{q,p} | ht12{p,q} s{q,p} | ht012{p,q} sr{p,q} |