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) |
There are many relations among the uniform polyhedra. [1] [2] [3] 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.
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, which is to say, 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 Wythoff symbol p|q r. The truncated forms have vertex figure q.q.r (where q=2p and r) and Wythoff p q|r.
vertex figure | group | A: regular: p.p.p | B: truncated regular: p.p.r |
Td | | | |
3.3.3.3 | Oh | | |
4.4.4 | Oh | | |
Ih | | | |
5.5.5 | Ih | | |
Ih | | | |
3.3.3.3.3 | Ih | | |
Ih | | ||
Ih | |
In addition there are three quasi-truncated forms. These also class as truncated-regular polyhedra.
vertex figures | Group Oh | Group Ih | Group Ih |
| | |
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 | Oh | | | | |
3.5.3.5 | Ih | | | | |
Ih | | | |||
3.5/2.3.5/2 | Ih | |
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 | | |
12 | 30 | Ih | | | |
12 | 30 | Ih | | |
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 | | | |
3.5.3.5 | 30 | 60 | Ih | | | |
3.5/2.3.5/2 | 30 | 60 | Ih | | | |
5.5/2.5.5/2 | 30 | 60 | Ih | | | |
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 | ditrigonal | crossed-ditrigonal | crossed-ditrigonal |
5/2.3.5/2.3.5/2.3 | 20 | 60 | Ih | | | |
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 | | | |
3.8/3.4.8/3 | 24 | 48 | Oh | | | |
3.4.5.4 | 60 | 120 | Ih | | | |
5/2.4.5.4 | 60 | 120 | Ih | | | |
3.10/3.5/2.10/3 | 60 | 120 | Ih | | | |
3.6.5/2.6 | 60 | 120 | Ih | | | |
3.10/3.5.10/3 | 60 | 120 | Ih | | | |
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
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, 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.
In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol a{5,3}, as an altered dodecahedron, and Coxeter diagram or .
In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U63. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
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, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other polyhedron – hence the "hemi" prefix.
In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra.
In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal.
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle,, defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles. Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror. Up to 3 face types exist centered on the fundamental triangle corners. Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling.