Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons.

This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, nonconvex 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 tessellations (tilings and honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.

Contents

Operations

Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter-Dynkin diagram. Notable exceptions include the grand antiprism in four dimensions. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform polychoron, uniform polyteron, uniform polypeton, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.

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. The first rectification reduces edges to vertices. The second rectification reduces faces to vertices. The third rectification reduces cells to vertices, etc.

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

Truncation operators

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

  1. t0,1: 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'.)
  2. t0,2: Cantellation - applied to polyhedra and higher. 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.)
  3. t0,3: Runcination - applied to polychora 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'.)
  4. t0,4: Sterication - applied to 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'.)
  5. t0,5: Pentellation - applied to 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'.)
  6. t0,6: Hexication - applied to 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'.)
  7. t0,7: Heptellation - applied to 8-polytopes and higher. Heptellation truncates vertices, edges, faces, cells, 4-faces, 5-faces, and 6-faces, replacing each with new facets. 7-faces are replaced by topologically expanded copies of themselves. (Heptellation is derived from Greek hepta 'seven'.)

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

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

Names are given relative to the first ringed node, and Latin prefixes (used in the numeric powers) is given to the position of the first ring. So for example, t1,2 is bitruncation, t2,3 is tritruncation, t3,4 is quadritruncation, etc.

Alternation

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.

An alternation of a great rhombicuboctahedron produces a snub cube.

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-Dynkin 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-Dynkin 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 tessellations. 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-Dynkin 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. The first few regular polygons are displayed below:

Name Triangle
(2-simplex)		 Square
(2-orthoplex)
(2-cube)		 Pentagon Hexagon Heptagon Octagon
Schläfli {3} {4} {5} {6} {7} {8}
Dynkin
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.

Name Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agrams
Schläfli {5/2} {7/2} {7/3} {8/3} {9/2} {9/4} {10/3} {p/q}
Dynkin
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:

Operation Extended
Schläfli
Symbols		 Regular
result		 Coxeter-
Dynkin
Diagram		 Position
(1) (0)
Parent t0{p} {p} {} --
Rectified
(Dual)		 t1{p} {p} -- {}
Truncated t0,1{p} {2p} {} {}
Snub s{p} {p} -- --

Three dimensions

Main article: Uniform polyhedron

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

Name Schläfli
{p,q}		 Dynkin
 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		 Coxeter-Dynkin
and Schläfli
symbols
P2p Prism
t{2,p}
Ap Antiprism
t{2,p}

The nonconvex uniform polyhedra include a further 4 regular polyhedra, the Kepler-Poinsot polyhedra, and 53 semiregular nonconvex 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-Dynkin diagram, and followed by the Schläfli symbol of the regular seed polytope. For example, the truncated octahedron is represented by the notation: t0,1{3,4}.

Operation Extended
Schläfli
Symbols		 Coxeter-
Dynkin
Diagram		 Wythoff
symbol		 Position
(2) (1) (0) (0,1) (0,2) (1,2)
Parent \begin{Bmatrix} p , q \end{Bmatrix} t0{p,q} q | 2 p {p} {} -- -- -- {}
Rectified \begin{Bmatrix} p \\ q \end{Bmatrix} t1{p,q} 2 | p q {p} -- {q} -- {} --
Birectified
(or dual)		 \begin{Bmatrix} q , p \end{Bmatrix} t2{p,q} p | 2 q -- {} {q} {} -- --
Truncated t\begin{Bmatrix} p , q \end{Bmatrix} t0,1{p,q} 2 q | p {2p} {} {q} -- {} {}
Bitruncated
(or truncated dual)		 t\begin{Bmatrix} q , p \end{Bmatrix} t1,2{p,q} 2 p | q {p} {} {2q} {} {} --
Cantellated
(or expanded)		 r\begin{Bmatrix} p \\ q \end{Bmatrix} t0,2{p,q} p q | 2 {p} {}x{} {q} {} -- {}
Cantitruncated
(or omnitruncated)		 t\begin{Bmatrix} p \\ q \end{Bmatrix} t0,1,2{p,q} 2 p q | {2p} {}x{} {2q} {} {} {}
Snub s\begin{Bmatrix} p \\ q \end{Bmatrix} s{p,q} | 2 p q {p} {3}
{3}		 {q} -- -- --

Generating triangles

Four dimensions

Main article: Uniform polychoron

In four dimensions, there are 6 convex regular polychora, 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 polychora, including the non-Wythoffian grand antiprism and the snub 24-cell. Both of these special polychora are composed of subgroups of the vertices of the 600-cell.

The four-dimensional nonconvex uniform polytopes have not all been enumerated. The ones that have include the 10 regular nonconvex polychora (Schläfli-Hess polychora) and 57 prisms on the nonconvex uniform polyhedra, as well as three infinite families: the prisms on the star antiprisms, the duoprisms formed by multiplying two star polygons, and the duoprisms formed by multiplying an ordinary polygon with a star polygon. There is an unknown number of polychora that do not fit into the above categories; over one thousand have been discovered so far.

Every regular polytope can be seen as the images of a fundamental region in a small number of mirrors. In a 4-dimensional polytope (or 3-dimensional cubic honeycomb) the fundamental region is bounded by four mirrors. A mirror in 4-space is a three-dimensional hyperplane, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the hypersphere; thus the mirrors form an irregular tetrahedron.

Each of the sixteen regular polychora is generated by one of four symmetry groups, as follows:

(The groups are named in Coxeter notation.)

Eight of the convex uniform honeycombs in Euclidean 3-space are analogously generated from the cubic honeycomb {4,3,4}, by applying the same operations used to generate the Wythoffian uniform polychora.

For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume. On each of these 15 elements there is a point whose images, reflected in the four mirrors, are the vertices of a uniform polychoron.

The extended Schläfli symbols are made by a t followed by inclusion of one to four subscripts 0,1,2,3. If there's one subscript, the generating point is on a corner of the fundamental region, i.e. a point where three mirrors meet. These corners are notated as

(For the two self-dual polychora, "dual" means a similar polychoron in dual position.) Two or more subscripts mean that the generating point is between the corners indicated.

Constructive summary

The 15 constructive forms by family are summarized below. The self-dual families are listed in one column, and others as two columns with shared entries on the symmetric Coxeter-Dynkin diagrams. The final 10th row lists the snub 24-cell constructions. This includes all nonprismatic uniform polychora, except for the non-Wythoffian grand antiprism, which has no Coxeter family.

A4 BC4 D4 F4 GH4
[3,3,3]
 [4,3,3]
 [31,1,1]
 [3,4,3]
 [5,3,3]
5-cell


{3,3,3}		 16-cell


{3,3,4}		 tesseract


{4,3,3}		 demitesseract

{31,1,1}

 24-cell


{3,4,3}		 600-cell


{3,3,5}		 120-cell


{5,3,3}
rectified 5-cell


t1{3,3,3}		 rectified 16-cell


t1{3,3,4}		 rectified tesseract


t1{4,3,3}		 rectified demitesseract

t1{31,1,1}

 rectified 24-cell


t1{3,4,3}		 rectified 600-cell


t1{3,3,5}		 rectified 120-cell


t1{5,3,3}
truncated 5-cell


t0,1{3,3,3}		 truncated 16-cell


t0,1{3,3,4}		 truncated tesseract


t0,1{4,3,3}		 truncated demitesseract

t0,1{31,1,1}

 truncated 24-cell


t0,1{3,4,3}		 truncated 600-cell


t0,1{3,3,5}		 truncated 120-cell


t0,1{5,3,3}
cantellated 5-cell


t0,2{3,3,3}		 cantellated 16-cell


t0,2{3,3,4}		 cantellated tesseract


t0,2{4,3,3}		 cantellated demitesseract

t0,2{31,1,1}

 cantellated 24-cell


t0,2{3,4,3}		 cantellated 600-cell


t0,2{3,3,5}		 cantellated 120-cell


t0,2{5,3,3}
runcinated 5-cell


t0,3{3,3,3}		 runcinated 16-cell


t0,3{3,3,4}		 runcinated tesseract


t0,3{4,3,3}		 runcinated 24-cell


t0,3{3,4,3}		 runcinated 600-cell
runcinated 120-cell


t0,3{3,3,5}
bitruncated 5-cell


t1,2{3,3,3}		 bitruncated 16-cell


t1,2{3,3,4}		 bitruncated tesseract


t1,2{4,3,3}		 bitruncated 24-cell


t1,2{3,4,3}		 bitruncated 600-cell
bitruncated 120-cell


t1,2{3,3,5}
cantitruncated 5-cell


t0,1,2{3,3,3}		 cantitruncated 16-cell


t0,1,2{3,3,4}		 cantitruncated tesseract


t0,1,2{4,3,3}		 cantitruncated demitesseract

t0,1,2{31,1,1}

 cantitruncated 24-cell


t0,1,2{3,4,3}		 cantitruncated 600-cell


t0,1,2{3,3,5}		 cantitruncated 120-cell


t0,1,2{5,3,3}
runcitruncated 5-cell


t0,1,3{3,3,3}		 runcitruncated 16-cell


t0,1,3{3,3,4}		 runcitruncated tesseract


t0,1,3{4,3,3}		 runcicantellated demitesseract

t0,2,3{31,1,1}

 runcitruncated 24-cell


t0,1,3{3,4,3}		 runcitruncated 600-cell


t0,1,3{3,3,5}		 runcitruncated 120-cell


t0,1,3{5,3,3}
omnitruncated 5-cell


t0,1,2,3{3,3,3}		 omnitruncated 16-cell


t0,1,2,3{3,3,4}		 omnitruncated tesseract


t0,1,2,3{3,3,4}		 omnitruncated demitesseract

t0,1,2,3{31,1,1}

 omnitruncated 24-cell


t0,1,2,3{3,4,3}		 omnitruncated 120-cell
omnitruncated 600-cell


t0,1,2,3{5,3,3}
alternated cantitruncated 16-cell


h0,1,2{3,3,4}		 snub demitesseract

s{31,1,1}

 Alternated truncated 24-cell


h0,1{3,4,3}

Truncated forms

The following table defines all 15 forms. Each trunction form can have from one to four cell types, located in positions 0,1,2,3 as defined above. The cells are labeled by polyhedral truncation notation.

Operation Extended
Schläfli
symbols		 Coxeter-
Dynkin
Diagram		 Position
(3) (2) (1) (0)
Parent t0{p,q,r}
{p,q}
{p}
{}
--
Rectified t1{p,q,r}
t1{p,q}
{p}
--
{q,r}
Birectified
(or rectified dual)		 t2{p,q,r}
{q,p}
--
{r}
t1{q,r}
Trirectifed
(or dual)		 t3{p,q,r}
--
{}
{r}
t2{q,r}
Truncated t0,1{p,q,r}
t0,1{p,q}
{2p}
{}
{q,r}
Bitruncated t1,2{p,q,r}
t1,2{p,q}
{p}
{r}
t0,1{q,r}
Tritruncated
(or truncated dual)		 t2,3{p,q,r}
{q,p}
{}
{2r}
t1,2{q,r}
Cantellated t0,2{p,q,r}
t0,2{p,q}
{p}
{}x{r}
t1{q,r}
Bicantellated
(or cantellated dual)		 t1,3{p,q,r}
t1{p,q}
{p}x{}
{r}
t0,2{q,r}
Runcinated
(or expanded)		 t0,3{p,q,r}
{p,q}
{p}x{}
{}x{r}
t2{q,r}
Cantitruncated t0,1,2{p,q,r}
t0,1,2{p,q}
{2p}
{}x{r}
t0,1{q,r}
Bicantitruncated
(or cantitruncated dual)		 t1,2,3{p,q,r}
t1,2{p,q}
{p}x{}
{2r}
t0,1,2{q,r}
Runcitruncated t0,1,3{p,q,r}
t0,1{p,q}
{2p}x{}
{}x{r}
t0,2{q,r}
Runcicantellated
(or runcitruncated dual)		 t0,2,3{p,q,r}
t0,1,2{p,q}
{p}x{}
{}x{2r}
t1,2{q,r}
Runcicantitruncated
(or omnitruncated)		 t0,1,2,3{p,q,r}
t0,1,2{p,q}
{2p}x{}
{}x{2r}
t0,1,2{q,r}

Five and higher dimensions

In five and higher dimensions, there are 3 regular polytopes, the hypercube, simplex and cross-polytope. They are generalisations of the three-dimensional cube, tetrahedron and octahedron, respectively. There are no regular star polytopes in these dimensions. Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.

In six, seven and eight dimensions, the exceptional simple Lie groups, E6, E7 and E8 come into play. By placing rings on a nonzero number of nodes of the Coxeter-Dynkin diagrams, one can obtain 63 new 6-polytopes, 127 new 7-polytopes and 255 new 8-polytopes. A notable example is the Gosset 421 polytope.

Uniform honeycombs

Related to the subject of finite uniform polytopes are uniform honeycombs in Euclidean and hyperbolic spaces. Euclidean uniform honeycombs are generated by Affine Coxeter groups and hyperbolic honeycombs are generated by the Coxeter-Dynkin_diagram#Hyperbolic Coxeter groups. Two affine Coxeter groups can be multiplied together

There are two classes of hyperbolic Coxeter groups, compact and noncompact. Uniform honeycombs generated by compact groups have finite facets and vertex figures, and exist from 2 to 4 dimensions. Noncompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and exist from 3 to 10 dimensions.

See also

External links

References