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2D | 3D |
---|---|

Truncated triangle or uniform hexagon, with Coxeter diagram . | Truncated octahedron, |

4D | 5D |

Truncated 16-cell, | Truncated 5-orthoplex, |

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 (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges).

- Operations
- Rectification operators
- Truncation operators
- Alternation
- Vertex figure
- Circumradius
- Uniform polytopes by dimension
- One dimension
- Two dimensions
- Three dimensions
- Four dimensions
- Five and higher dimensions
- Uniform honeycombs
- See also
- References
- External links

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.

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. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.^{[ citation needed ]}

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.

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**{p_{k}_{1}, p_{2}, ..., p_{n-1}} =*k***r**.

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.

**t**or_{0,1}**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**or_{1,2}**2t**,**tritruncation****t**or_{2,3}**3t**,**quadritruncation****t**or_{3,4}**4t**,**quintitruncation****t**or_{4,5}**5t**, etc.

- There are higher truncations also:
**t**or_{0,2}**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**or_{1,3}**r2r**,**tricantellation****t**or_{2,4}**r3r**,**quadricantellation****t**or_{3,5}**r4r**, etc. **t**or_{0,1,2}**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**or_{1,2,3}**t2r**,**tricantitruncation****t**or_{2,3,4}**t3r**,**quadricantitruncation****t**or_{3,4,5}**t4r**, etc.

- There are higher cantellations also:

- There are higher cantellations also:
**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**, etc._{2,5}

- There are higher runcinations also:
**t**or_{0,4}**2r2r**:**Sterication**- applied to Uniform 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****t**or_{1,5}**2r3r**,**tristerication****t**or_{2,6}**2r4r**, etc. **t**or_{0,2,4}**2t2r**:**Stericantellation**- applied to Uniform 5-polytopes and higher. It can be seen as bitruncating its birectification.- There are higher sterications also:
**bistericantellation****t**or_{1,3,5}**2t3r**,**tristericantellation****t**or_{2,4,6}**2t4r**, etc.

- There are higher sterications also:

- There are higher sterications also:
**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**, etc._{2,7}

- There are also higher pentellations:
**t**or_{0,6}**3r3r**:**Hexication**- applied to Uniform 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**:**t**or_{1,7}**3r4r**,**trihexication**:**t**or_{2,8}**3r5r**, etc. **t**or_{0,3,6}**3t3r**:**Hexiruncinated**- applied to Uniform 7-polytopes and higher. It can be seen as tritruncating its trirectification.- There are also higher hexiruncinations:
**bihexiruncinated**:**t**or_{1,4,7}**3t4r**,**trihexiruncinated**:**t**or_{2,5,8}**3t5r**, etc.

- There are also higher hexiruncinations:

- There are higher hexications also:
**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**, etc._{2,9}

- There are higher heptellations also:
**t**or_{0,8}**4r4r**:**Octellation**- applied to Uniform 9-polytopes and higher.**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.

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*.

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.

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.

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.

The only one-dimensional polytope is the line segment. It corresponds to the Coxeter family A_{1}.

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-agrams | ||
---|---|---|---|---|---|---|---|---|

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} | t_{0}{p} | {p} | {} | -- | [p] (order 2p) | |

Rectified(Dual) | r{p} | t_{1}{p} | {p} | -- | {} | [p] (order 2p) | |

Truncated | t{p} | t_{0,1}{p} | {2p} | {} | {} | [[p]]=[2p] (order 4p) | |

Half | h{2p} | {p} | -- | -- | [1^{+},2p]=[p](order 2p) | ||

Snub | s{p} | {p} | -- | -- | [[p]]^{+}=[p](order 2p) |

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} | T_{d} | (self) | ||||

Cube (3-cube) (Hexahedron) | {4,3} | 6 {4} | 12 | 8 {3} | O_{h} | Octahedron | ||||

Octahedron (3-orthoplex) | {3,4} | 8 {3} | 12 | 6 {4} | O_{h} | Cube | ||||

Dodecahedron | {5,3} | 12 {5} | 30 | 20 {3}2 | I_{h} | Icosahedron | ||||

Icosahedron | {3,5} | 20 {3} | 30 | 12 {5} | I_{h} | 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 |
---|---|---|---|---|---|

P_{2p} | Prism | tr{2,p} | |||

A_{p} | 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).

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 | {p,q} | t_{0}{p,q} | 2 p | {p} | { } | -- | -- | -- | { } | ||||

Birectified (or dual ) | {q,p} | t_{2}{p,q} | p | 2 q | -- | { } | {q} | { } | -- | -- | ||||

Truncated | t{p,q} | t_{0,1}{p,q} | p | {2p} | { } | {q} | -- | { } | { } | ||||

Bitruncated (or truncated dual) | t{q,p} | t_{1,2}{p,q} | q | {p} | { } | {2q} | { } | { } | -- | ||||

Rectified | r{p,q} | t_{1}{p,q} | p q | {p} | -- | {q} | -- | { } | -- | ||||

Cantellated (or expanded) | rr{p,q} | t_{0,2}{p,q} | p q | 2 | {p} | { }×{ } | {q} | { } | -- | { } | ||||

Cantitruncated (or Omnitruncated) | tr{p,q} | t_{0,1,2}{p,q} | 2 p q | | {2p} | { }×{} | {2q} | { } | { } | { } |

Operation | Schläfli Symbol | Coxeter diagram | Wythoff symbol | Position: | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Snub rectified | sr{p,q} | | 2 p q | {p} | {3} {3} | {q} | -- | -- | -- | |||||

Snub | s{p,2q} | ht_{0,1}{p,q} | s{2p} | {3} | {q} | -- | {3} |

Generating triangles |

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-polytope, including the non-Wythoffian grand antiprism and the snub 24-cell. Both of these special 4-polytope are composed of subgroups of the vertices of the 600-cell.

The four-dimensional uniform star polytopes have not all been enumerated. The ones that have include the 10 regular star (Schläfli-Hess) 4-polytopes and 57 prisms on the uniform star 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 4-polytope 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 4-polytopes 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 [3
^{1,1,1}]: 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 4-polytopes.

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 4-polytope.

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 4-polytope (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 4-polytopes, "dual" means a similar 4-polytope in dual position.) Two or more subscripts mean that the generating point is between the corners indicated.

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 diagrams. The final 10th row lists the snub 24-cell constructions. This includes all nonprismatic uniform 4-polytopes, except for the non-Wythoffian grand antiprism, which has no Coxeter family.

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}×{ }. - 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 | Schläfli symbol | Coxeter diagram | Cells by position: | ||||
---|---|---|---|---|---|---|---|

(3) | (2) | (1) | (0) | ||||

Parent | {p,q,r} | t_{0}{p,q,r} | {p,q} | -- | -- | -- | |

Rectified | r{p,q,r} | t_{1}{p,q,r} | r{p,q} | -- | -- | {q,r} | |

Birectified (or rectified dual) | 2r{p,q,r} = r{r,q,p} | t_{2}{p,q,r} | {q,p} | -- | -- | r{q,r} | |

Trirectifed (or dual) | 3r{p,q,r} = {r,q,p} | t_{3}{p,q,r} | -- | -- | -- | {r,q} | |

Truncated | t{p,q,r} | t_{0,1}{p,q,r} | t{p,q} | -- | -- | {q,r} | |

Bitruncated | 2t{p,q,r} | 2t{p,q,r} | t{q,p} | -- | -- | t{q,r} | |

Tritruncated (or truncated dual) | 3t{p,q,r} = t{r,q,p} | t_{2,3}{p,q,r} | {q,p} | -- | -- | t{r,q} | |

Cantellated | rr{p,q,r} | t_{0,2}{p,q,r} | rr{p,q} | -- | { }×{r} | r{q,r} | |

Bicantellated (or cantellated dual) | r2r{p,q,r} = rr{r,q,p} | t_{1,3}{p,q,r} | r{p,q} | {p}×{ } | -- | rr{q,r} | |

Runcinated (or expanded) | e{p,q,r} | t_{0,3}{p,q,r} | {p,q} | {p}×{ } | { }×{r} | {r,q} | |

Cantitruncated | tr{p,q,r} | tr{p,q,r} | tr{p,q} | -- | { }×{r} | t{q,r} | |

Bicantitruncated (or cantitruncated dual) | t2r{p,q,r} = tr{r,q,p} | t_{1,2,3}{p,q,r} | t{q,p} | {p}×{ } | -- | tr{q,r} | |

Runcitruncated | e_{t}{p,q,r} | t_{0,1,3}{p,q,r} | t{p,q} | {2p}×{ } | { }×{r} | rr{q,r} | |

Runcicantellated (or runcitruncated dual) | e_{3t}{p,q,r}= e _{t}{r,q,p} | t_{0,2,3}{p,q,r} | tr{p,q} | {p}×{ } | { }×{2r} | t{r,q} | |

Runcicantitruncated (or omnitruncated) | o{p,q,r} | t_{0,1,2,3}{p,q,r} | tr{p,q} | {2p}×{ } | { }×{2r} | tr{q,r} |

Half constructions exist with *holes* rather than ringed nodes. Branches neighboring *holes* and inactive nodes must be even-order. Half construction have the vertices of an identically ringed construction.

Operation | Schläfli symbol | Coxeter diagram | Cells by position: | ||||
---|---|---|---|---|---|---|---|

(3) | (2) | (1) | (0) | ||||

Half Alternated | h{p,2q,r} | ht_{0}{p,2q,r} | h{p,2q} | -- | -- | -- | |

Alternated rectified | hr{2p,2q,r} | ht_{1}{2p,2q,r} | hr{2p,2q} | -- | -- | h{2q,r} | |

Snub Alternated truncation | s{p,2q,r} | ht_{0,1}{p,2q,r} | s{p,2q} | -- | -- | h{2q,r} | |

Bisnub Alternated bitruncation | 2s{2p,q,2r} | ht_{1,2}{2p,q,2r} | s{q,2p} | -- | -- | s{q,2r} | |

Snub rectified Alternated truncated rectified | sr{p,q,2r} | ht_{0,1,2}{p,q,2r} | sr{p,q} | -- | s{2,2r} | s{q,2r} | |

Omnisnub Alternated omnitruncation | os{p,q,r} | ht_{0,1,2,3}{p,q,r} | sr{p,q} | {p}×{ } | { }×{r} | sr{q,r} |

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, E_{6}, E_{7} and E_{8} come into play. By placing rings on a nonzero number of nodes of the Coxeter diagrams, one can obtain 63 new 6-polytopes, 127 new 7-polytopes and 255 new 8-polytopes. A notable example is the 4_{21} polytope.

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 paracompact. Uniform honeycombs generated by compact groups have finite facets and vertex figures, and exist in 2 through 4 dimensions. Paracompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and exist in 2 through 10 dimensions.

In elementary geometry, a **polytope** is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. Flat sides mean that the sides of a (

In geometry, a **4-polytope** is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.

In geometry, the **Schläfli symbol** is a notation of the form {*p*,*q*,*r*,...} that defines regular polytopes and tessellations.

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or *j*-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

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 4-polytope** is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In Euclidean geometry, **rectification**, also known as **critical truncation** or **complete-truncation** is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

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 **cantellation** is a 2nd order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tilings and honeycombs. Cantellating is also rectifying its rectification.

In geometry, **expansion** is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.

In geometry, **runcination** is an operation that cuts a regular polytope simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

In geometry, a **star polyhedron** is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

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 **snub** is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

In mathematics, a **regular 4-polytope** is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In six-dimensional geometry, a **six-dimensional polytope** or **6-polytope** is a polytope, bounded by 5-polytope facets.

The **order-6 cubic honeycomb** is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is *paracompact* because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

- 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

- N.W. Johnson:
- 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

- H.S.M. Coxeter, M.S. Longuet-Higgins and J.C.P. Miller:
*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- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

- 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) (in German)

- Olshevsky, George. "Uniform polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - uniform, convex polytopes in four dimensions:, Marco Möller (in German)

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