Uniform polytope

Convex uniform polytopes

2D	3D
Truncated triangle or uniform hexagon, with Coxeter diagram .	Truncated octahedron,
4D	5D
Truncated 16-cell,	Truncated 5-orthoplex,

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions (and for two-dimensional faces of higher-dimensional polytopes) a stronger definition is used: only the regular polygons are considered as uniform, disallowing polygons that alternate between two different lengths of edges.

This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.

Operations

Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombicosidodecahedron in three dimensions and the grand antiprism in four dimensions.

Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation.

Rectification operators

Regular n-polytopes have n orders of rectification. The zeroth rectification is the original form. The (n−1)-th rectification is the dual. A rectification reduces edges to vertices, a birectification reduces faces to vertices, a trirectification reduces cells to vertices, a quadirectification reduces 4-faces to vertices, a quintirectification reduced 5-faces to vertices, and so on.

An extended Schläfli symbol can be used for representing rectified forms, with a single subscript:

• k-th rectification = t_k{p₁, p₂, ..., p_n−1} = kr.

Truncation operators

Truncation operations that can be applied to regular n-polytopes in any combination. The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, sterication cut cells. Each higher operation also cuts lower ones too, so a cantellation also truncates vertices.

1. t_0,1 or t: Truncation - applied to polygons and higher. A truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges. (The term, coined by Kepler, comes from Latin truncare 'to cut off'.)

   

   • There are higher truncations also: bitruncation t_1,2 or 2t, tritruncation t_2,3 or 3t, quadritruncation t_3,4 or 4t, quintitruncation t_4,5 or 5t, etc.
2. t_0,2 or rr: Cantellation - applied to polyhedra and higher. It can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves. (The term, coined by Johnson, is derived from the verb cant, like bevel, meaning to cut with a slanted face.)

   

   • There are higher cantellations also: bicantellation t_1,3 or r2r, tricantellation t_2,4 or r3r, quadricantellation t_3,5 or r4r, etc.
   • t_0,1,2 or tr: Cantitruncation - applied to polyhedra and higher. It can be seen as truncating its rectification. A cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves. (The composite term combines cantellation and truncation)
     • There are higher cantellations also: bicantitruncation t_1,2,3 or t2r, tricantitruncation t_2,3,4 or t3r, quadricantitruncation t_3,4,5 or t4r, etc.
3. t_0,3: Runcination - applied to Uniform 4-polytope and higher. Runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves. (The term, coined by Johnson, is derived from Latin runcina 'carpenter's plane'.)
   • There are higher runcinations also: biruncination t_1,4, triruncination t_2,5, etc.
4. t_0,4 or 2r2r: Sterication - applied to Uniform 5-polytopes and higher. Sterication truncates vertices, edges, faces, and cells, replacing each with new facets. 5-faces are replaced by topologically expanded copies of themselves. (The term, coined by Johnson, is derived from Greek stereos 'solid'.)
   • There are higher sterications also: bisterication t_1,5 or 2r3r, tristerication t_2,6 or 2r4r, etc.
   • t_0,2,4 or 2t2r: Stericantellation - applied to Uniform 5-polytopes and higher.
     • There are higher sterications also: bistericantellation t_1,3,5 or 2t3r, tristericantellation t_2,4,6 or 2t4r, etc.
5. t_0,5: Pentellation - applied to Uniform 6-polytopes and higher. Pentellation truncates vertices, edges, faces, cells, and 4-faces, replacing each with new facets. 6-faces are replaced by topologically expanded copies of themselves. (Pentellation is derived from Greek pente 'five'.)
   • There are also higher pentellations: bipentellation t_1,6, tripentellation t_2,7, etc.
6. t_0,6 or 3r3r: Hexication - applied to Uniform 7-polytopes and higher. Hexication truncates vertices, edges, faces, cells, 4-faces, and 5-faces, replacing each with new facets. 7-faces are replaced by topologically expanded copies of themselves. (Hexication is derived from Greek hex 'six'.)
   • There are higher hexications also: bihexication: t_1,7 or 3r4r, trihexication: t_2,8 or 3r5r, etc.
   • t_0,3,6 or 3t3r: Hexiruncinated - applied to Uniform 7-polytopes and higher.
     • There are also higher hexiruncinations: bihexiruncinated: t_1,4,7 or 3t4r, trihexiruncinated: t_2,5,8 or 3t5r, etc.
7. t_0,7: Heptellation - applied to Uniform 8-polytopes and higher. Heptellation truncates vertices, edges, faces, cells, 4-faces, 5-faces, and 6-faces, replacing each with new facets. 8-faces are replaced by topologically expanded copies of themselves. (Heptellation is derived from Greek hepta 'seven'.)
   • There are higher heptellations also: biheptellation t_1,8, triheptellation t_2,9, etc.
8. t_0,8 or 4r4r: Octellation - applied to Uniform 9-polytopes and higher.
9. t_0,9: Ennecation - applied to Uniform 10-polytopes and higher.

In addition combinations of truncations can be performed which also generate new uniform polytopes. For example, a runcitruncation is a runcination and truncation applied together.

If all truncations are applied at once, the operation can be more generally called an omnitruncation.

Alternation

An alternation of a truncated cuboctahedron produces a snub cube.

One special operation, called alternation, removes alternate vertices from a polytope with only even-sided faces. An alternated omnitruncated polytope is called a snub.

The resulting polytopes always can be constructed, and are not generally reflective, and also do not in general have uniform polytope solutions.

The set of polytopes formed by alternating the hypercubes are known as demicubes. In three dimensions, this produces a tetrahedron; in four dimensions, this produces a 16-cell, or demitesseract.

Vertex figure

Uniform polytopes can be constructed from their vertex figure, the arrangement of edges, faces, cells, etc. around each vertex. Uniform polytopes represented by a Coxeter diagram, marking active mirrors by rings, have reflectional symmetry, and can be simply constructed by recursive reflections of the vertex figure.

A smaller number of nonreflectional uniform polytopes have a single vertex figure but are not repeated by simple reflections. Most of these can be represented with operations like alternation of other uniform polytopes.

Vertex figures for single-ringed Coxeter diagrams can be constructed from the diagram by removing the ringed node, and ringing neighboring nodes. Such vertex figures are themselves vertex-transitive.

Multiringed polytopes can be constructed by a slightly more complicated construction process, and their topology is not a uniform polytope. For example, the vertex figure of a truncated regular polytope (with 2 rings) is a pyramid. An omnitruncated polytope (all nodes ringed) will always have an irregular simplex as its vertex figure.

Circumradius

Uniform polytopes have equal edge-lengths, and all vertices are an equal distance from the center, called the circumradius.

Uniform polytopes whose circumradius is equal to the edge length can be used as vertex figures for uniform honeycombs. For example, the regular hexagon divides into 6 equilateral triangles and is the vertex figure for the regular triangular tiling. Also the cuboctahedron divides into 8 regular tetrahedra and 6 square pyramids (half octahedron), and it is the vertex figure for the alternated cubic honeycomb.

Uniform polytopes by dimension

It is useful to classify the uniform polytopes by dimension. This is equivalent to the number of nodes on the Coxeter diagram, or the number of hyperplanes in the Wythoffian construction. Because (n+1)-dimensional polytopes are tilings of n-dimensional spherical space, tilings of n-dimensional Euclidean and hyperbolic space are also considered to be (n+1)-dimensional. Hence, the tilings of two-dimensional space are grouped with the three-dimensional solids.

One dimension

The only one-dimensional polytope is the line segment. It corresponds to the Coxeter family A₁.

Two dimensions

In two dimensions, there is an infinite family of convex uniform polytopes, the regular polygons, the simplest being the equilateral triangle. Truncated regular polygons become bicolored geometrically quasiregular polygons of twice as many sides, t{p}={2p}. The first few regular polygons (and quasiregular forms) are displayed below:

Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon	Hexagon	Heptagon	Octagon	Enneagon	Decagon	Hendecagon
Schläfli	{3}	{4} t{2}	{5}	{6} t{3}	{7}	{8} t{4}	{9}	{10} t{5}	{11}
Coxeter diagram
Image
Name	Dodecagon	Tridecagon	Tetradecagon	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon
Schläfli	{12} t{6}	{13}	{14} t{7}	{15}	{16} t{8}	{17}	{18} t{9}	{19}	{20} t{10}
Coxeter diagram
Image

There is also an infinite set of star polygons (one for each rational number greater than 2), but these are non-convex. The simplest example is the pentagram, which corresponds to the rational number 5/2. Regular star polygons, {p/q}, can be truncated into semiregular star polygons, t{p/q}=t{2p/q}, but become double-coverings if q is even. A truncation can also be made with a reverse orientation polygon t{p/(p−q)}={2p/(p−q)}, for example t{5/3}={10/3}.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-grams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3} t{4/3}	{9/2}	{9/4}	{10/3} t{5/3}	{p/q}
Coxeter diagram
Image

Regular polygons, represented by Schläfli symbol {p} for a p-gon. Regular polygons are self-dual, so the rectification produces the same polygon. The uniform truncation operation doubles the sides to {2p}. The snub operation, alternating the truncation, restores the original polygon {p}. Thus all uniform polygons are also regular. The following operations can be performed on regular polygons to derive the uniform polygons, which are also regular polygons:

Operation	Extended Schläfli Symbols		Regular result	Coxeter diagram	Position		Symmetry
					(1)	(0)
Parent	{p}	t0{p}	{p}		{}	--	[p] (order 2p)
Rectified (Dual)	r{p}	t1{p}	{p}		--	{}	[p] (order 2p)
Truncated	t{p}	t0,1{p}	{2p}		{}	{}	[[p]]=[2p] (order 4p)
Half	h{2p}		{p}		--	--	[1+,2p]=[p] (order 2p)
Snub	s{p}		{p}		--	--	[[p]]+=[p] (order 2p)

Three dimensions

Main article: Uniform polyhedron

In three dimensions, the situation gets more interesting. There are five convex regular polyhedra, known as the Platonic solids:

Name	Schläfli {p,q}	Diagram	Image (transparent)	Image (solid)	Image (sphere)	Faces {p}	Edges	Vertices {q}	Symmetry	Dual
Tetrahedron (3-simplex) (Pyramid)	{3,3}					4 {3}	6	4 {3}	Td	(self)
Cube (3-cube) (Hexahedron)	{4,3}					6 {4}	12	8 {3}	Oh	Octahedron
Octahedron (3-orthoplex)	{3,4}					8 {3}	12	6 {4}	Oh	Cube
Dodecahedron	{5,3}					12 {5}	30	20 {3}2	Ih	Icosahedron
Icosahedron	{3,5}					20 {3}	30	12 {5}	Ih	Dodecahedron

In addition to these, there are also 13 semiregular polyhedra, or Archimedean solids, which can be obtained via Wythoff constructions, or by performing operations such as truncation on the Platonic solids, as demonstrated in the following table:

	Parent	Truncated	Rectified	Bitruncated (tr. dual)	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Tetrahedral 3-3-2	{3,3}	(3.6.6)	(3.3.3.3)	(3.6.6)	{3,3}	(3.4.3.4)	(4.6.6)	(3.3.3.3.3)
Octahedral 4-3-2	{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4.4)	(4.6.8)	(3.3.3.3.4)
Icosahedral 5-3-2	{5,3}	(3.10.10)	(3.5.3.5)	(5.6.6)	{3,5}	(3.4.5.4)	(4.6.10)	(3.3.3.3.5)

There is also the infinite set of prisms, one for each regular polygon, and a corresponding set of antiprisms.

#	Name	Picture	Tiling	Vertex figure	Diagram and Schläfli symbols
P2p	Prism				tr{2,p}
Ap	Antiprism				sr{2,p}

The uniform star polyhedra include a further 4 regular star polyhedra, the Kepler-Poinsot polyhedra, and 53 semiregular star polyhedra. There are also two infinite sets, the star prisms (one for each star polygon) and star antiprisms (one for each rational number greater than 3/2).

Constructions

The Wythoffian uniform polyhedra and tilings can be defined by their Wythoff symbol, which specifies the fundamental region of the object. An extension of Schläfli notation, also used by Coxeter, applies to all dimensions; it consists of the letter 't', followed by a series of subscripted numbers corresponding to the ringed nodes of the Coxeter diagram, and followed by the Schläfli symbol of the regular seed polytope. For example, the truncated octahedron is represented by the notation: t_0,1{3,4}.

Operation	Schläfli Symbol			Coxeter diagram		Wythoff symbol	Position:
Parent	${\begin{Bmatrix}p,q\end{Bmatrix}}$	{p,q}	t0{p,q}			q | 2 p	{p}	{ }	--	--	--	{ }
Birectified (or dual)	${\begin{Bmatrix}q,p\end{Bmatrix}}$	{q,p}	t2{p,q}			p | 2 q	--	{ }	{q}	{ }	--	--
Truncated	$t{\begin{Bmatrix}p,q\end{Bmatrix}}$	t{p,q}	t0,1{p,q}			2 q | p	{2p}	{ }	{q}	--	{ }	{ }
Bitruncated (or truncated dual)	$t{\begin{Bmatrix}q,p\end{Bmatrix}}$	t{q,p}	t1,2{p,q}			2 p | q	{p}	{ }	{2q}	{ }	{ }	--
Rectified	${\begin{Bmatrix}p\\q\end{Bmatrix}}$	r{p,q}	t1{p,q}			2 | p q	{p}	--	{q}	--	{ }	--
Cantellated (or expanded)	$r{\begin{Bmatrix}p\\q\end{Bmatrix}}$	rr{p,q}	t0,2{p,q}			p q | 2	{p}	{ }×{ }	{q}	{ }	--	{ }
Cantitruncated (or Omnitruncated)	$t{\begin{Bmatrix}p\\q\end{Bmatrix}}$	tr{p,q}	t0,1,2{p,q}			2 p q |	{2p}	{ }×{}	{2q}	{ }	{ }	{ }

Operation	Schläfli Symbol			Coxeter diagram		Wythoff symbol	Position:
Snub rectified	$s{\begin{Bmatrix}p\\q\end{Bmatrix}}$	sr{p,q}				| 2 p q	{p}	{3} {3}	{q}	--	--	--
Snub	$s{\begin{Bmatrix}p,q\end{Bmatrix}}$	s{p,2q}	ht0,1{p,q}				s{2p}	{3}	{q}	--	{3}

Generating triangles

Four dimensions

Main article: Uniform 4-polytope

In four dimensions, there are 6 convex regular 4-polytopes, 17 prisms on the Platonic and Archimedean solids (excluding the cube-prism, which has already been counted as the tesseract), and two infinite sets: the prisms on the convex antiprisms, and the duoprisms. There are also 41 convex semiregular 4-polytopes, including the