Uniform polytope

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Convex uniform polytopes
2D 3D
Truncated triangle.png
Truncated triangle is a uniform hexagon, with Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.png.
Truncated octahedron.png
Truncated octahedron, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
4D 5D
Schlegel half-solid truncated 16-cell.png
Truncated 16-cell, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t34 B4.svg
Truncated 5-orthoplex, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

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, although even-sided polygons can be seen as uniform by alternating two colors of edges, represented by a 2-ring Coxeter diagram CDel node 1.pngCDel p.pngCDel node 1.png.

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.

Operations[edit]

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[edit]

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:

  • k-th rectification = tk{p1, p2, ..., pn-1} = kr.

Truncation operators[edit]

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 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'.)
    Cube truncation sequence.svg
    • There are higher truncations also: bitruncation t1,2 or 2t, tritruncation t2,3 or 3t, quadritruncation t3,4 or 4t, quintitruncation t4,5 or 5t, etc.
  2. t0,2 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.)
    Cube cantellation sequence.svg
    • There are higher cantellations also: bicantellation t1,3 or r2r, tricantellation t2,4 or r3r, quadricantellation t3,5 or r4r, etc.
    • t0,1,2 or tr: Cantitruncation - applied to polyhedra and higher. It can be seen as a truncation of 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 t1,2,3 or t2r, tricantitruncation t2,3,4 or t3r, quadricantitruncation t3,4,5 or t4r, etc.
  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'.)
    • There are higher runcinations also: biruncination t1,4, triruncination t2,5, etc.
  4. t0,4 or 2r2r: Sterication - applied to 5-polytopes and higher. It can be seen as birectifying its birectification. 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 t1,5 or 2r3r, tristerication t2,6 or 2r4r, etc.
    • t0,2,4 or 2t2r: Stericantellation - applied to 5-polytopes and higher. It can be seen as bitruncation its birectification.
      • There are higher sterications also: bistericantellation t1,3,5 or 2t3r, tristericantellation t2,4,6 or 2t4r, etc.
  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'.)
    • There are higher pentellations also: bipentellation t1,6, tripentellation t2,7, etc.
  6. t0,6 or 3r3r: Hexication - applied to 7-polytopes and higher. It can be seen as trirectifying its trirectification. 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: t1,7 or 3r4r, trihexication: t2,8 or 3r5r, etc.
    • t0,3,6 or 3t3r: Hexiruncinated - applied to 7-polytopes and higher. It can be seen as tritruncation its trirectification.
      • There are higher hexiruncinated also: bihexiruncinated: t1,4,7 or 3t4r, trihexiruncinated: t2,5,8 or 3t5r, etc.
  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. 8-faces are replaced by topologically expanded copies of themselves. (Heptellation is derived from Greek hepta 'seven'.)
    • There are higher heptellations also: biheptellation t1,8, triheptellation t2,9, etc.

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[edit]

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.

Snubcubes in grCO.svg
An alternation of a truncated cuboctahedron 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[edit]

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[edit]

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[edit]

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[edit]

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

Two dimensions[edit]

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 regular polygons of twice as many sides. The first few regular polygons are displayed below:

Name Triangle
(2-simplex)
Square
(2-orthoplex)
(2-cube)
Pentagon Hexagon Heptagon Octagon Enneagon Decagon Hendecagon Dodecagon
Schläfli {3} {4} {5} {6} {7} {8} {9} {10} {11} {12}
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 7.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 9.pngCDel node.png CDel node 1.pngCDel 10.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.png
CDel node 1.pngCDel 11.pngCDel node.png CDel node 1.pngCDel 12.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.png
Image Regular triangle.svg Regular quadrilateral.svg
Truncated digon.png
Regular pentagon.svg Regular hexagon.svg
Truncated triangle.png
Regular heptagon.svg Regular octagon.svg
Truncated square.png
Regular nonagon.svg Regular decagon.svg
Truncated pentagon.png
Regular hendecagon.svg Regular dodecagon.svg
Truncated hexagon.png

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}
Coxeter
diagram
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 7.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 7.pngCDel rat.pngCDel d3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel rat.pngCDel d3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel rat.pngCDel d3.pngCDel node 1.png
CDel node 1.pngCDel 9.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 9.pngCDel rat.pngCDel d4.pngCDel node.png CDel node 1.pngCDel 10.pngCDel rat.pngCDel d3.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node 1.png
CDel node 1.pngCDel p.pngCDel rat.pngCDel dq.pngCDel node.png
Image Star polygon 5-2.svg Star polygon 7-2.svg Star polygon 7-3.svg Star polygon 8-3.svg Star polygon 9-2.svg Star polygon 9-4.svg Star polygon 10-3.svg  

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} CDel node 1.pngCDel p.pngCDel node.png {} -- [p]
(order 2p)
Rectified
(Dual)
r{p} t1{p} {p} CDel node.pngCDel p.pngCDel node 1.png -- {} [p]
(order 2p)
Truncated t{p} t0,1{p} {2p} CDel node 1.pngCDel p.pngCDel node 1.png {} {} [[p]]=[2p]
(order 4p)
Half h{2p} {p} CDel node h.pngCDel 2x.pngCDel p.pngCDel node.png -- -- [1+,2p]=[p]
(order 2p)
Snub s{p} {p} CDel node h.pngCDel p.pngCDel node h.png -- -- [[p]]+=[p]
(order 2p)

Three dimensions[edit]

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
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Image
(transparent)
Image
(solid)
Image
(sphere)
Faces
{p}
Edges Vertices
{q}
Symmetry Dual
Tetrahedron
(3-simplex)
(Pyramid)
{3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Tetrahedron.svg Tetrahedron.png Uniform tiling 332-t0-1-.png 4
{3}
6 4
{3}
Td (self)
Cube
(3-cube)
(Hexahedron)
{4,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png Hexahedron.svg Hexahedron.png Uniform tiling 432-t0.png 6
{4}
12 8
{3}
Oh Octahedron
Octahedron
(3-orthoplex)
{3,4} CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png Octahedron.svg Octahedron.png Uniform tiling 432-t2.png 8
{3}
12 6
{4}
Oh Cube
Dodecahedron {5,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png Dodecahedron.svg Dodecahedron.png Uniform tiling 532-t0.png 12
{5}
30 20
{3}2
Ih Icosahedron
Icosahedron {3,5} CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png Icosahedron.svg Icosahedron.png Uniform tiling 532-t2.png 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
Uniform polyhedron-33-t0.png
{3,3}
Uniform polyhedron-33-t01.png
(3.6.6)
Uniform polyhedron-33-t1.png
(3.3.3.3)
Uniform polyhedron-33-t12.png
(3.6.6)
Uniform polyhedron-33-t2.png
{3,3}
Uniform polyhedron-33-t02.png
(3.4.3.4)
Uniform polyhedron-33-t012.png
(4.6.6)
Uniform polyhedron-33-s012.png
(3.3.3.3.3)
Octahedral
4-3-2
Uniform polyhedron-43-t0.png
{4,3}
Uniform polyhedron-43-t01.png
(3.8.8)
Uniform polyhedron-43-t1.png
(3.4.3.4)
Uniform polyhedron-43-t12.png
(4.6.6)
Uniform polyhedron-43-t2.png
{3,4}
Uniform polyhedron-43-t02.png
(3.4.4.4)
Uniform polyhedron-43-t012.png
(4.6.8)
Uniform polyhedron-43-s012.png
(3.3.3.3.4)
Icosahedral
5-3-2
Uniform polyhedron-53-t0.png
{5,3}
Uniform polyhedron-53-t01.png
(3.10.10)
Uniform polyhedron-53-t1.png
(3.5.3.5)
Uniform polyhedron-53-t12.png
(5.6.6)
Uniform polyhedron-53-t2.png
{3,5}
Uniform polyhedron-53-t02.png
(3.4.5.4)
Uniform polyhedron-53-t012.png
(4.6.10)
Uniform polyhedron-53-s012.png
(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 Dodecagonal prism.png Spherical truncated hexagonal prism.png Dodecagonal prism vf.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{2,p}
Ap Antiprism Hexagonal antiprism.png Spherical hexagonal antiprism.png Hexagonal antiprism vertfig.png CDel node h.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node h.png
sr{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[edit]

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} {p,q} t0{p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png q | 2 p {p} {} -- -- -- {}
Birectified
(or dual)
\begin{Bmatrix} q , p \end{Bmatrix} {q,p} t2{p,q} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png p | 2 q -- {} {q} {} -- --
Truncated t\begin{Bmatrix} p , q \end{Bmatrix} t{p,q} t0,1{p,q} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png 2 q | p {2p} {} {q} -- {} {}
Bitruncated
(or truncated dual)
t\begin{Bmatrix} q , p \end{Bmatrix} t{q,p} t1,2{p,q} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png 2 p | q {p} {} {2q} {} {} --
Rectified \begin{Bmatrix} p \\ q \end{Bmatrix} r{p,q} t1{p,q} CDel node 1.pngCDel split1-pq.pngCDel nodes.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png 2 | p q {p} -- {q} -- {} --
Cantellated
(or expanded)
r\begin{Bmatrix} p \\ q \end{Bmatrix} rr{p,q} t0,2{p,q} CDel node.pngCDel split1-pq.pngCDel nodes 11.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png p q | 2 {p} {}x{} {q} {} -- {}
Cantitruncated
(or Omnitruncated)
t\begin{Bmatrix} p \\ q \end{Bmatrix} tr{p,q} t0,1,2{p,q} CDel node 1.pngCDel split1-pq.pngCDel nodes 11.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png 2 p q | {2p} {}x{} {2q} {} {} {}
Snub rectified s\begin{Bmatrix} p \\ q \end{Bmatrix} sr{p,q} CDel node h.pngCDel split1-pq.pngCDel nodes hh.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png | 2 p q {p} {3}
{3}
{q} -- -- --
Snub s\begin{Bmatrix} p , q \end{Bmatrix} s{p,2q} ht0,1{p,q} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node.png s{2p} {3} {q} -- {3}
Polyhedron truncation example3.png Wythoffian construction diagram.png
Generating triangles

Four dimensions[edit]

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.

Example tetrahedron in cubic honeycomb cell.
There are 3 right dihedral angles (2 intersecting perpendicular mirrors):
Edges 1 to 2, 0 to 2, and 1 to 3.
Summary chart of truncation operations

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:

  • group [3,3,3]: the 5-cell {3,3,3}, which is self-dual;
  • group [3,3,4]: 16-cell {3,3,4} and its dual tesseract {4,3,3};
  • group [3,4,3]: the 24-cell {3,4,3}, self-dual;
  • group [3,3,5]: 600-cell {3,3,5}, its dual 120-cell {5,3,3}, and their ten regular stellations.
  • group [31,1,1]: contains only repeated members of the [3,3,4] family.

(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

  • 0: vertex of the parent polychoron (center of the dual's cell)
  • 1: center of the parent's edge (center of the dual's face)
  • 2: center of the parent's face (center of the dual's edge)
  • 3: center of the parent's cell (vertex of the dual)

(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[edit]

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 H4
[3,3,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,31,1]
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
[3,4,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[5,3,3]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cell
Schlegel wireframe 5-cell.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3,3}
16-cell
Schlegel wireframe 16-cell.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,3,4}
tesseract
Schlegel wireframe 8-cell.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{4,3,3}
demitesseract
Schlegel wireframe 16-cell.png
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
{3,31,1}
24-cell
Schlegel wireframe 24-cell.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{3,4,3}
600-cell
Schlegel wireframe 600-cell vertex-centered.png
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,3,5}
120-cell
Schlegel wireframe 120-cell.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5,3,3}
rectified 5-cell
Schlegel half-solid rectified 5-cell.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{3,3,3}
rectified 16-cell
Schlegel half-solid rectified 16-cell.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
r{3,3,4}
rectified tesseract
Schlegel half-solid rectified 8-cell.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{4,3,3}
rectified demitesseract
Schlegel wireframe 24-cell.png
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.png
r{3,31,1}
rectified 24-cell
Schlegel half-solid cantellated 16-cell.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
r{3,4,3}
rectified 600-cell
Rectified 600-cell schlegel halfsolid.png
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
r{3,3,5}
rectified 120-cell
Rectified 120-cell schlegel halfsolid.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{5,3,3}
truncated 5-cell
Schlegel half-solid truncated pentachoron.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{3,3,3}
truncated 16-cell
Schlegel half-solid truncated 16-cell.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{3,3,4}
truncated tesseract
Schlegel half-solid truncated tesseract.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{4,3,3}
truncated demitesseract
Schlegel half-solid truncated 16-cell.png
CDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.png
t{3,31,1}
truncated 24-cell
Schlegel half-solid truncated 24-cell.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
t{3,4,3}
truncated 600-cell
Schlegel half-solid truncated 600-cell.png
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{3,3,5}
truncated 120-cell
Schlegel half-solid truncated 120-cell.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{5,3,3}
cantellated demitesseract
Schlegel half-solid rectified 8-cell.png
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png
2r{3,31,1}
cantellated 16-cell
Schlegel half-solid cantellated 16-cell.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3,4}
cantellated tesseract
Schlegel half-solid cantellated 8-cell.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{4,3,3}
cantellated 5-cell
Schlegel half-solid cantellated 5-cell.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{3,3,3}
cantellated 24-cell
Cantel 24cell1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{3,4,3}
cantellated 600-cell
Cantellated 600 cell center.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3,5}
cantellated 120-cell
Cantellated 120 cell center.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{5,3,3}
runcinated 5-cell
Schlegel half-solid runcinated 5-cell.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,3}
runcinated 16-cell
Schlegel half-solid runcinated 16-cell.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,4}
runcinated tesseract
Schlegel half-solid runcinated 8-cell.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{4,3,3}
runcinated 24-cell
Runcinated 24-cell Schlegel halfsolid.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,4,3}
runcinated 600-cell
runcinated 120-cell
Runcinated 120-cell.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,5}
bitruncated 5-cell
Schlegel half-solid bitruncated 5-cell.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{3,3,3}
bitruncated 16-cell
Schlegel half-solid bitruncated 16-cell.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,3,4}
bitruncated tesseract
Schlegel half-solid bitruncated 8-cell.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{4,3,3}
cantitruncated demitesseract
Schlegel half-solid bitruncated 16-cell.png
CDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea 1.png
2t{3,31,1}
bitruncated 24-cell
Bitruncated 24-cell Schlegel halfsolid.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,4,3}
bitruncated 600-cell
bitruncated 120-cell
Bitruncated 120-cell schlegel halfsolid.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,3,5}
cantitruncated 5-cell
Schlegel half-solid cantitruncated 5-cell.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{3,3,3}
cantitruncated 16-cell
Schlegel half-solid cantitruncated 16-cell.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3,4}
cantitruncated tesseract
Schlegel half-solid cantitruncated 8-cell.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{4,3,3}
omnitruncated demitesseract
Schlegel half-solid truncated 24-cell.png
CDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.png
tr{3,31,1}
cantitruncated 24-cell
Cantitruncated 24-cell schlegel halfsolid.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{3,4,3}
cantitruncated 600-cell
Cantitruncated 600-cell.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3,5}
cantitruncated 120-cell
Cantitruncated 120-cell.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{5,3,3}
runcitruncated 5-cell
Schlegel half-solid runcitruncated 5-cell.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,3}
runcitruncated 16-cell
Schlegel half-solid runcitruncated 16-cell.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,4}
runcitruncated tesseract
Schlegel half-solid runcitruncated 8-cell.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{4,3,3}
runcicantellated demitesseract
Schlegel half-solid cantellated 16-cell.png
CDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.png
rr{3,31,1}
runcitruncated 24-cell
Runcitruncated 24-cell.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,4,3}
runcitruncated 600-cell
Runcitruncated 600-cell.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,5}
runcitruncated 120-cell
Runcitruncated 120-cell.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{5,3,3}
omnitruncated 5-cell
Schlegel half-solid omnitruncated 5-cell.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,3}
omnitruncated 16-cell
Schlegel half-solid omnitruncated 16-cell.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,4}
omnitruncated tesseract
Schlegel half-solid omnitruncated 8-cell.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,4}
omnitruncated 24-cell
Omnitruncated 24-cell.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,4,3}
omnitruncated 120-cell
omnitruncated 600-cell
Omnitruncated 120-cell wireframe.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{5,3,3}
alternated cantitruncated 16-cell
Schlegel half-solid alternated cantitruncated 16-cell.png
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3,4}
snub demitesseract
Ortho solid 969-uniform polychoron 343-snub.png
CDel nodea h.pngCDel 3a.pngCDel branch hh.pngCDel 3a.pngCDel nodea h.png
sr{3,31,1}
Alternated truncated 24-cell
Ortho solid 969-uniform polychoron 343-snub.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
s{3,4,3}

Truncated forms[edit]

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.

  • An n-gonal prism is represented as : {n}×{2}.
  • The green background is shown on forms that are equivalent to either the parent or the dual.
  • The red background shows the truncations of the parent, and blue the truncations of the dual.
Operation Extended
Schläfli
symbols
Coxeter-
Dynkin
Diagram
Position
(3) (2) (1) (0)
Parent {p,q,r} t0{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
{p}
CDel node 1.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png
{}
CDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
--
Rectified r{p,q,r} t1{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
t1{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png
{p}
CDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png
--
CDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
{q,r}
Birectified
(or rectified dual)
2r{p,q,r}
= r{r,q,p}
t2{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
{q,p}
CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
--
CDel node.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node.png
{r}
CDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
t1{q,r}
Trirectifed
(or dual)
3r{p,q,r}
= {r,q,p}
t3{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
--
CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png
{}
CDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node 1.png
{r}
CDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
t2{q,r}
Truncated t{p,q,r} t0,1{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
t0,1{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png
{2p}
CDel node 1.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png
{}
CDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
{q,r}
Bitruncated 2t{p,q,r} t1,2{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
t1,2{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png
{p}
CDel node.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node.png
{r}
CDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
t0,1{q,r}
Tritruncated
(or truncated dual)
3t{p,q,r}
= t{r,q,p}
t2,3{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
{q,p}
CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png
{}
CDel node.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node 1.png
{2r}
CDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png
t1,2{q,r}
Cantellated rr{p,q,r} t0,2{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
t0,2{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
{p}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node.png
{}×{r}
CDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
t1{q,r}
Bicantellated
(or cantellated dual)
r2r{p,q,r}
= rr{r,q,p}
t1,3{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
t1{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{p}×{}
CDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node 1.png
{r}
CDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
t0,2{q,r}
Runcinated
(or expanded)
e{p,q,r} t0,3{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png
{p}×{}
CDel node 1.pngCDel 2.pngCDel node.pngCDel r.pngCDel node 1.png
{}×{r}
CDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
t2{q,r}
Cantitruncated tr{p,q,r} t0,1,2{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
t0,1,2{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png
{2p}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node.png
{}×{r}
CDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
t0,1{q,r}
Bicantitruncated
(or cantitruncated dual)
t2r{p,q,r}
= tr{r,q,p}
t1,2,3{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
t1,2{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{p}×{}
CDel node.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node 1.png
{2r}
CDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png
t0,1,2{q,r}
Runcitruncated et{p,q,r} t0,1,3{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
t0,1{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{2p}×{}
CDel node 1.pngCDel 2.pngCDel node.pngCDel r.pngCDel node 1.png
{}×{r}
CDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
t0,2{q,r}
Runcicantellated
(or runcitruncated dual)
e3t{p,q,r}
= et{r,q,p}
t0,2,3{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
t0,1,2{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png
{p}×{}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node 1.png
{}×{2r}
CDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png
t1,2{q,r}
Runcicantitruncated
(or omnitruncated)
o{p,q,r} t0,1,2,3{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
t0,1,2{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{2p}×{}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel r.pngCDel node 1.png
{}×{2r}
CDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png
t0,1,2{q,r}
Half
Alternated
h{p,q,r} ht0{p,q,r} CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
s{p,q}
CDel node h.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
{p}
CDel node h.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png
--
CDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
--
Alternated rectified hr{p,q,r} ht1{p,q,r} CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.png
ht1{p,q}
CDel node.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node.png
{p/2}
CDel node.pngCDel 2.pngCDel node h.pngCDel r.pngCDel node.png
{r/2}
CDel node h.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
h{q,r}
Snub
Alternated truncation
s{p,q,r} ht0,1{p,q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.png
s{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node.png
{p}
CDel node h.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png
--
CDel node h.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
h{q,r}
Bisnub
Alternated bitruncation
2s{p,q,r} ht1,2{p,q,r} CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
s{q,p}
CDel node.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node.png
{p}
CDel node.pngCDel 2.pngCDel node h.pngCDel r.pngCDel node.png
s{2,r}
CDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node.png
s{q,r}
Snub rectified
Alternated truncated rectified
sr{p,q,r} ht0,1,2{p,q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
sr{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node.png
{p}
CDel node h.pngCDel 2.pngCDel node h.pngCDel r.pngCDel node.png
s{2,r}
CDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node.png
s{q,r}
Omnisnub
Alternated omnitruncation
os{p,q,r} ht0,1,2,3{p,q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
sr{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node h.png
{p}×{}
CDel node h.pngCDel 2.pngCDel node h.pngCDel r.pngCDel node h.png
{}×{r}
CDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node h.png
sr{q,r}

Five and higher dimensions[edit]

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[edit]

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 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 in 2 through 4 dimensions. Noncompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and exist in 2 through 10 dimensions.

Scaliform polytope[edit]

A scaliform polytope or honeycomb is vertex-transitive, like a uniform polytope, but only requires regular polygon faces while cells and higher elements are only required to be orbiforms, equilateral, with their vertices lying on hyperspheres. For 4-polytopes, this allows a subset of Johnson solids along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps.[1] Individual orbiform cells contain multiple vertex figures, but the whole polytope contains a single vertex figure with each orbiform facet's vertex orientation.[2]

All four known convex scaliform polychora
s3{2,4,3}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png s3{3,4,3}, CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png Non-Wythoffian Non-Wythoffian
Truncated tetrahedral cupoliprism net.png
Net
(truncated tetrahedral cupoliprism)
(runcic snub cubic hosochoron)
(parabidiminished rectified tesseract)
Prismatorhombisnub icositetrachoron net.png
Net
(prismatorhombisnub 24-cell)
(runcic snub 24-cell)
Swirlprismatodiminished rectified hexacosichoron net.png
Net
(swirlprismatodiminished rectified 600-cell)[3]
Biicositetradiminished hexacosichoron net.png
Net
(biicositetradiminished 600-cell)[4]
Runcic snub 243 verf.png Runcic snub 24-cell verf.png
Euclidean honeycomb scaliforms
s3{2,6,3}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png s3{2,4,4}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png s{2,4,4}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png 3s4{4,4,2,∞}, CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
Runcic snub 263 honeycomb.png
Triangular cupola, trihexagonal tiling and octahedra
Runcic snub 244 honeycomb.png
Square cupola, truncated square tiling, and tetrahedra
Alternated cubic slab honeycomb.png
square pyramids, square tiling, and tetrahedra
Elongated square antiprismatic celluation.png
square pyramids, cubes, and tetrahedra

See also[edit]

References[edit]

Sources[edit]

  • Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins and J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
  • Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50. (Extended Schläfli notation used)
  • Marco Möller, Vierdimensionale Archimedische Polytope, Dissertation, Universität Hamburg, Hamburg (2004) (German)

External links[edit]