- Uniform polytope
A

**uniform polytope**is avertex-transitive polytope made from uniform polytope facets. A uniform polytope must also have onlyregular polygon faces.Uniformity is a generalization of the older category

**semiregular**, but also includes theregular polytope s. Further, nonconvex regular faces andvertex figure s (star polygon s) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes be finite, while a more expansive definition allowsuniform tessellation s (tilings and honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.Nearly all uniform polytopes can be generated by a

Wythoff construction , and represented by aCoxeter-Dynkin diagram . The terminology for the convex uniform polytopes used inuniform polyhedron ,uniform polychoron ,uniform tiling , andconvex uniform honeycomb articles were coined by Norman Johnson.**Rectification operators**Regular n-polytopes have "n" orders of rectification. The zeroth rectification is the original form. The nth 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 = "t_{k}{p_{1}, p_{2}, ..., p_{n-1}}"**Truncation operators**Regular n-polytopes have "n" orders of truncations that can be applied in any combination, and which can create new uniform polytopes.

# Truncation - applied to

polygon s and higher. A truncation is a form that exists between adjacent rectified forms.

#*Schläfli symbol for the nth truncation is t_{n-1,n}{p,q,...}

# Cantellation - applied topolyhedron s and higher and creates uniform polytopes that exists between alternate rectified forms.

#* "Schläfli symbol" for the n-th cantellation is t_{n-1,n+1}{p,q,...}

#Runcination - applied topolychoron s and higher and creates uniform polytopes that exists between third alternate rectified forms.

#* "Schläfli symbol" for the n-th runcination is t_{n-1,n+2}{p,q,...}

#Sterication - applied to5-polytope s and higher and creates uniform polytopes that exists between fourth alternate rectified forms.

#* "Schläfli symbol" for the n-th sterication is t_{n-1,n+3}{p,q,...}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 .**Alternation**One special operation, called alternation removes alternate vertices on polytope with all even-sided faces. An "alternation" applied to an 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.

**Classes of polytopes by dimension***

**Uniform polygons**: an infinite set ofregular polygon s andstar polygon s (one for eachrational number greater than 2).*

**Uniform polyhedra**:

** Convex forms

*** 5 convex regular (Platonic solid s);

*** 13 convex semiregular (Archimedean solid s);

*** an infinite set of semiregular prisms (one for each convex regular polygon);

*** an infinite set of semiregularantiprism s (one for each convex regular polygon);

** Nonconvex forms

*** 4 nonconvex regular (Kepler-Poinsot polyhedra );

*** 53 other nonconvex forms;

*** an infinite set of nonconvex uniform prisms (one for each regularstar polygon );

*** an infinite set of nonconvex uniformantiprism s (one for each noninteger rational number greater than 3/2).*

:Uniform polychoron

** Convex forms

*** 6 convex regular polychora

*** 41 "convex uniform" polychora;

*** 18 "convex hyperprisms" based on the Platonic and Archimedean solids (including the cube-prism, better known as the regulartesseract );

*** an infinite set of hyperprisms based on the convex antiprisms;

*** an infinite set of convexduoprism s;

** Nonconvex forms

*** 10 nonconvex regular polychora (Schläfli-Hess polychora)

*** 57 "nonconvex hyperprisms" based on the nonconvex uniform polyhedra;

*** an unknown number of nonconvex nonprismatic uniform polychora (over a thousand have been found);

*** an infinite set of hyperprisms based on the nonconvex antiprisms;

*** an infinite set of nonconvexduoprism s based on star polygons.Higher dimensional uniform polytopes are not fully known. Most may be generated from a

Wythoff construction applied to the regular forms.Regular "n"-polytope families include the

simplex ,hypercube , andcross-polytope .The

demihypercube family, derived from the hypercubes by removing alternate vertices, includes thetetrahedron derived from thecube and the16-cell derived from thetesseract . Higher members of the family are uniform but not regular.**Families of convex uniform polytopes**Families of convex uniform polytopes are defined by

Coxeter group s. In addition prismatic families exist as products of this groups.Categorical regular and prismatic family groups, up to 8-polytopes, are given below. Each permutation of indices of regular polytopes defines another family.

The

Coxeter-Dynkin diagram is given for the first form in each family. Every combination of rings, with each prismatic group having at least one ring, produces another uniform prismatic polytope.; Convex uniform polytope families by dimension

1-polytope # A

_{1}: [ ]2-polytope # D

_{2}^{p}: [p]3-polytope # A

_{3}: [3,3]

# C_{3}: [4,3]

# G_{3}: [5,3]

# D_{2}^{p}xA_{1}: [p] x [ ]4-polytope # A

_{4}: [3,3,3]

# C_{4}: [4,3,3]

# F_{4}: [3,4,3]

# G_{4}: [5,3,3]

# B_{4}: [3^{1,1,1}]

# A_{3}xA_{1}: [3,3] x [ ] -

# C_{3}xA_{1}: [4,3] x [ ] -

# G_{3}xA_{1}: [5,3] x [ ] -

# D_{2}^{p}xD_{2}^{q}: [p] x [q]5-polytope # A

_{5}: [3,3,3,3]

# C_{5}: [4,3,3,3]

# B_{5}: [3^{2,1,1}]

# A_{4}xA_{1}: [3,3,3] x [ ]

# C_{4}xA_{1}: [4,3,3] x [ ]

# F_{4}xA_{1}: [3,4,3] x [ ]

# G_{4}xA_{1}: [5,3,3] x [ ]

# B_{4}xA_{1}: [3^{1,1,1}] x [ ]

# A_{3}xD_{2}^{p}: [3,3] x [p] -

# C_{3}xD_{2}^{p}: [4,3] x [p] -

# G_{3}xD_{2}^{p}: [5,3] x [p] -

# D_{2}^{p}xD_{2}^{q}xA_{1}: [p] x [q] x [ ]6-polytope # A

_{6}: [3,3,3,3,3]

# C_{6}: [4,3,3,3,3]

# B_{6}: [3^{3,1,1}]

#

# A_{5}xA_{1}: [3,3,3,3] x [ ]

# C_{5}xA_{1}: [4,3,3,3] x [ ]

# B_{5}xA_{1}: [3^{2,1,1}] x [ ]

# A_{4}xD_{2}^{p}: [3,3,3] x [p]

# C_{4}xD_{2}^{p}: [4,3,3] x [p]

# F_{4}xD_{2}^{p}: [3,4,3] x [p]

# G_{4}xD_{2}^{p}: [5,3,3] x [p]

# B_{4}xD_{2}^{p}: [3^{1,1,1}] x [p]

# A_{3}xA_{3}: [3,3] x [3,3]

# A_{3}xC_{3}: [3,3] x [4,3]

# A_{3}xG_{3}: [3,3] x [5,3]

# C_{3}xC_{3}: [4,3] x [4,3]

# C_{3}xG_{3}: [4,3] x [5,3]

# G_{3}xA_{3}: [5,3] x [5,3]

# A_{3}xD_{2}^{p}xA_{1}: [3,3] x [p] x [ ]

#C_{3}xD_{2}^{p}xA_{1}: [4,3] x [p] x [ ]

#G_{3}xD_{2}^{p}xA_{1}: [5,3] x [p] x [ ]

#D_{2}^{p}xD_{2}^{q}xD_{2}^{r}: [p] x [q] x [r]7-polytope # A

_{7}: [3^{6}]

# C_{7}: [4,3^{5}]

# B_{7}: [3^{4,1,1}]

#

# A_{6}xA_{1}: [3^{5}] x [ ]

# C_{6}xA_{1}: [4,3^{4}] x [ ]

# B_{6}xA_{1}: [3^{3,1,1}] x [ ]

# E_{6}xA_{1}: [3^{2,2,1}] x [ ]

# A_{5}xD_{2}^{p}: [3,3,3] x [p]

# C_{5}xD_{2}^{p}: [4,3,3] x [p]

# B_{5}xD_{2}^{p}: [3^{2,1,1}] x [p]

# A_{4}xA_{3}: [3,3,3] x [3,3]

# A_{4}xC_{3}: [3,3,3] x [4,3]

# A_{4}xG_{3}: [3,3,3] x [5,3]

# C_{4}xA_{3}: [4,3,3] x [3,3]

# C_{4}xC_{3}: [4,3,3] x [4,3]

# C_{4}xG_{3}: [4,3,3] x [5,3]

# G_{4}xA_{3}: [5,3,3] x [3,3]

# G_{4}xC_{3}: [5,3,3] x [4,3]

# G_{4}xG_{3}: [5,3,3] x [5,3]

# F_{4}xA_{3}: [3,4,3] x [3,3]

# F_{4}xC_{3}: [3,4,3] x [4,3]

# F_{4}xG_{3}: [3,4,3] x [5,3]

# B_{4}xA_{3}: [3^{1,1,1}] x [3,3]

# B_{4}xC_{3}: [3^{1,1,1}] x [4,3]

# B_{4}xG_{3}: [3^{1,1,1}] x [5,3]

# A_{4}xD_{2}^{p}xA_{1}: [3,3,3] x [p] x [ ]

# C_{4}xD_{2}^{p}xA_{1}: [4,3,3] x [p] x [ ]

# F_{4}xD_{2}^{p}xA_{1}: [3,4,3] x [p] x [ ]

# G_{4}xD_{2}^{p}xA_{1}: [5,3,3] x [p] x [ ]

# B_{4}xD_{2}^{p}xA_{1}: [3^{1,1,1}] x [p] x [ ]

#A_{3}xA_{3}xA_{1}: [3,3] x [3,3] x [ ]

#A_{3}xC_{3}xA_{1}: [3,3] x [4,3] x [ ]

#A_{3}xG_{3}xA_{1}: [3,3] x [5,3] x [ ]

#C_{3}xC_{3}xA_{1}: [4,3] x [4,3] x [ ]

#C_{3}xG_{3}xA_{1}: [4,3] x [5,3] x [ ]

#G_{3}xA_{3}xA_{1}: [5,3] x [5,3] x [ ]

# A_{3}xD_{2}^{p}xD_{2}^{q}: [3,3] x [p] x [q]

# C_{3}xD_{2}^{p}xD_{2}^{q}: [4,3] x [p] x [q]

# G_{3}xD_{2}^{p}xD_{2}^{q}: [5,3] x [p] x [q]

# D_{2}^{p}xD_{2}^{q}xD_{2}^{r}A_{1}: [p] x [q] x [r] x [ ]8-polytope (incomplete)# A

_{8}: [3,3,3,3,3,3,3]

# C_{8}: [4,3,3,3,3,3,3]

# B_{8}: [3^{1,4,1}]

#

# A_{7}xA_{1}: [3,3,3,3,3,3] x [ ]

# C_{7}xA_{1}: [4,3,3,3,3,3] x [ ]

# B_{7}xA_{1}: [3^{1,3,1}] x [ ]

# [p,q,r,s,t] x [u]

# [p,q,r,s] x [t,u]

# [p,q,r] x [s,t,u]

# [p,q,r,s] x [t] x [ ]

# [p,q,r] x [s,t] x [ ]

# [p,q,r] x [s] x [t]

# [p,q] x [r,s] x [t]

# [p,q] x [r] x [s] x [ ]

# [p] x [q] x [r] x [s] - tetraprismSpecial cases of products become

hypercube s:

* [ ] x [ ] = [4]

* [ ] x [ ] x [ ] = [4,3]

* [ ] x [ ] x [ ] = [4,3,3]

* [ ] x [ ] x [ ] x [ ] = [4,3,3,3]

* ....**Uniform polygons**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, alternatingly truncating the truncation returns it back to the original polygon {p}. Thus all uniform polygons are also regular.**Uniform polychora and 3-space honeycombs**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-dimensionalhyperplane , but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of thehypersphere ; thus the mirrors form an irregulartetrahedron .Each of the sixteen regular polychora is generated by one of four symmetry groups, as follows:

* group [3,3,3] : the5-cell {3,3,3}, which is self-dual;

* group [3,3,4] :16-cell {3,3,4} and its dualtesseract {4,3,3};

* group [3,4,3] : the24-cell {3,4,3}, self-dual;

* group [3,3,5] :600-cell {3,3,5}, its dual120-cell {5,3,3}, and their ten regular stellations.(The groups are named in Coxeter notation.)

A set of up to 13 (nonregular) uniform polychora can be generated from each regular polychoron and its dual. Eight of the

convex uniform honeycomb s in Euclidean 3-space are analogously generated from thecubic honeycomb {4,3,4}.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.

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}x{2}.

* The green background is shown on forms that are equivalent from either the parent or dual.

* The red background shows truncations of the parent, and blue as truncations of the dual.**ee also**

*List of regular polytopes

*Schläfli symbol

*List of uniform polyhedra

*regular polygon

*uniform polyhedron

*uniform polychoron

*5-polytope

*6-polytope

*7-polytope

*8-polytope **External links***

**References*** 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**, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [*http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html*]

** (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)

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