Uniform polytope

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Uniform polytope

A uniform polytope is a vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
polytope

Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
made from uniform polytope facets

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
. A uniform polytope must also have only regular polygon

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
faces.

Uniformity is a generalization of the older category semiregular, but also includes the regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , 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....
s. Further, nonconvex regular

Star polygon

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
s (star polygon

Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
s) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes be finite, while a more expansive definition allows uniform tessellation

Uniform tessellation

In mathematics, a uniform tessellation is a tessellation of a d-dimensional space, or a surface, such that all its Vertex-transitive, i.e., there is the same combination and arrangement of faces at each vertex....
s (tilings and honeycombs

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
) of Euclidean and hyperbolic space to be considered polytopes as well.

Nearly all uniform polytopes can be generated by a Wythoff construction

Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
, and represented by a Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
.

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Encyclopedia

A uniform polytope is a vertex-transitive

Vertex-transitive

Polytope

Facet (mathematics)

Regular polygon

Regular polytope

Star polygon

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
s (star polygon

Star polygon

Uniform tessellation

Honeycomb (geometry)

Wythoff construction

Coxeter-Dynkin diagram

Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
, uniform polychoron

Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
, uniform polyteron

Uniform polyteron

In geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron Facet ....
, uniform polypeton, uniform tiling

Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane....
, and convex uniform honeycomb

Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
articles were coined by Norman Johnson.

Rectification operators

Regular n-polytopes have n orders of rectification

Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
. 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

Schläfli symbol

k-th rectification = t_k

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
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
- applied to polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s and higher. A truncation is a form that exists between adjacent rectified forms.
- Schläfli symbol
  Schläfli symbol
  
  In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
  for the nth truncation is t_n-1,n
Cantellation
Cantellation (geometry)

In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex....
- applied to polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
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
Runcination
Runcination

In geometry, a 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....
- applied to polychoron
Polychoron

In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space"....
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
Sterication - applied to 5-polytope
5-polytope

In geometry, a five-dimensional polytope, or 5-polytope, is a polytope in 5-dimensional space. Each polyhedron cell being shared by exactly two polychoron facets....
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

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 of regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s and star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
s (one for each rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
greater than 2).

Uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
:

Convex forms

5 convex regular (Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s);
13 convex semiregular (Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s);
an infinite set of semiregular prisms
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
(one for each convex regular polygon);
an infinite set of semiregular antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
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
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
(one for each regular star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
);
an infinite set of nonconvex uniform antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
s (one for each noninteger rational number greater than 3/2).

Uniform polychoron
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
:

Convex forms

6 convex regular polychora
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
41 convex uniform polychora;
18 convex hyperprisms based on the Platonic and Archimedean solids (including the cube-prism, better known as the regular tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
);
an infinite set of hyperprisms based on the convex antiprisms;
an infinite set of convex duoprism
Duoprism

In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
s;

Nonconvex forms

10 nonconvex regular polychora (Schläfli-Hess polychora
Schläfli-Hess polychoron

In four dimensional geometry, Schl?fli-Hess polychora are the complete set of 10 Regular polytope self-intersecting Star polytope . They are named in honor of their discoverers: Ludwig Schl?fli and Edmund Hess....
)
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 nonconvex duoprism
Duoprism

In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
s based on star polygons.

Higher dimensional uniform polytopes are not fully known. Most may be generated from a Wythoff construction

Wythoff construction

Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
, hypercube

Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
, and cross-polytope

Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
.

The demihypercube family, derived from the hypercubes by removing alternate vertices, includes the tetrahedron

Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
derived from the cube

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
and the 16-cell

16-cell

In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
derived from the tesseract

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
. 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

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
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

List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
defines another family.

The Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
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

A₁: [ ] - line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....

2-polytope

D₂^p: [p]

3-polytope

A₃: [3,3] # C₃: [4,3] # G₃: [5,3] # D₂^pxA₁: [p] x [ ]

4-polytope

A₄: [3,3,3] # C₄: [4,3,3] # F₄: [3,4,3] # G₄: [5,3,3] # B₄: [3^1,1,1] # A₃xA₁: [3,3] x [ ] - # C₃xA₁: [4,3] x [ ] - # G₃xA₁: [5,3] x [ ] - # D₂^pxD₂^q: [p] x [q]

5-polytope
5-polytope

In geometry, a five-dimensional polytope, or 5-polytope, is a polytope in 5-dimensional space. Each polyhedron cell being shared by exactly two polychoron facets....

A₅: [3,3,3,3] # C₅: [4,3,3,3] # B₅: [3^2,1,1] # A₄xA₁: [3,3,3] x [ ] # C₄xA₁: [4,3,3] x [ ] # F₄xA₁: [3,4,3] x [ ] # G₄xA₁: [5,3,3] x [ ] # B₄xA₁: [3^1,1,1] x [ ] # A₃xD₂^p: [3,3] x [p] - # C₃xD₂^p: [4,3] x [p] - # G₃xD₂^p: [5,3] x [p] - # D₂^pxD₂^qxA₁: [p] x [q] x [ ]
6-polytope
6-polytope

In geometry, a six-dimensional polytope, or 6-polytope, is a polytope in 6-dimensional space. Each polychoron Ridge being shared by exactly two 5-polytope Facet ....
A₆:[3,3,3,3,3] # C₆:[4,3,3,3,3] # B₆: [3^3,1,1] # E₆
E6 (mathematics)

In mathematics, E6 is the name of some Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups....
: [3^2,2,1] # A₅xA₁: [3,3,3,3] x [ ] # C₅xA₁:[4,3,3,3] x [ ] # B₅xA₁: [3^2,1,1] x [ ] # A₄xD₂^p: [3,3,3] x [p] # C₄xD₂^p: [4,3,3] x [p] # F₄xD₂^p: [3,4,3] x [p] # G₄xD₂^p: [5,3,3] x [p] # B₄xD₂^p: [3^1,1,1] x [p] # A₃xA₃: [3,3] x [3,3] # A₃xC₃: [3,3] x [4,3] # A₃xG₃: [3,3] x [5,3] # C₃xC₃: [4,3] x [4,3] # C₃xG₃: [4,3] x [5,3]# G₃xA₃: [5,3] x [5,3] # A₃xD₂^pxA₁: [3,3] x [p] x [ ] #C₃xD₂^pxA₁: [4,3] x [p] x [ ] #G₃xD₂^pxA₁: [5,3] x [p] x [ ] #D₂^pxD₂^qxD₂^r: [p] x [q] x [r]

7-polytope
7-polytope

In geometry, a seven-dimensional polytope, or 7-polytope, is a polytope in 7-dimensional space. Each polyteron Ridge being shared by exactly two 6-polytope Facet ....

A₇: [3⁶] # C₇: [4,3⁵] # B₇: [3^4,1,1] # E₇
E7 (mathematics)

In mathematics, E7 is the name of several Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups....
: [3^3,2,1] # A₆xA₁: [3⁵] x [ ] # C₆xA₁: [4,3⁴] x [ ] # B₆xA₁: [3^3,1,1] x [ ] # E₆xA₁: [3^2,2,1] x [ ] # A₅xD₂^p: [3,3,3] x [p] # C₅xD₂^p: [4,3,3] x [p] # B₅xD₂^p: [3^2,1,1] x [p] # A₄xA₃: [3,3,3] x [3,3] # A₄xC₃: [3,3,3] x [4,3] # A₄xG₃: [3,3,3] x [5,3] # C₄xA₃: [4,3,3] x [3,3] # C₄xC₃: [4,3,3] x [4,3] # C₄xG₃: [4,3,3] x [5,3] # G₄xA₃: [5,3,3] x [3,3] # G₄xC₃: [5,3,3] x [4,3] # G₄xG₃: [5,3,3] x [5,3] # F₄xA₃: [3,4,3] x [3,3] # F₄xC₃: [3,4,3] x [4,3] # F₄xG₃: [3,4,3] x [5,3] # B₄xA₃: [3^1,1,1] x [3,3] # B₄xC₃: [3^1,1,1] x [4,3] # B₄xG₃: [3^1,1,1] x [5,3] # A₄xD₂^pxA₁: [3,3,3] x [p] x [ ] # C₄xD₂^pxA₁: [4,3,3] x [p] x [ ] # F₄xD₂^pxA₁: [3,4,3] x [p] x [ ] # G₄xD₂^pxA₁: [5,3,3] x [p] x [ ] # B₄xD₂^pxA₁: [3^1,1,1] x [p] x [ ] #A₃xA₃xA₁: [3,3] x [3,3] x [ ] #A₃xC₃xA₁: [3,3] x [4,3] x [ ] #A₃xG₃xA₁: [3,3] x [5,3] x [ ] #C₃xC₃xA₁: [4,3] x [4,3] x [ ] #C₃xG₃xA₁: [4,3] x [5,3] x [ ] #G₃xA₃xA₁: [5,3] x [5,3] x [ ] # A₃xD₂^pxD₂^q: [3,3] x [p] x [q] # C₃xD₂^pxD₂^q: [4,3] x [p] x [q] # G₃xD₂^pxD₂^q: [5,3] x [p] x [q] # D₂^pxD₂^qxD₂^rA₁: [p] x [q] x [r] x [ ]
8-polytope
8-polytope

In geometry, an eight-dimensional polytope, or 8-polytope, is a polytope in 8-dimensional space. Each 6-polytope Ridge being shared by exactly two 7-polytope Facet ....
(incomplete)
A₈: [3,3,3,3,3,3,3] # C₈: [4,3,3,3,3,3,3] # B₈: [3^1,4,1] # E₈
E8 (mathematics)

In mathematics, E8 is the name given to a family of closely related structures. In particular, it is the name of four exceptional simple Lie algebra Lie algebras as well as that of the six associated simple Lie group Lie groups....
: [3^4,2,1] # A₇xA₁: [3,3,3,3,3,3] x [ ] # C₇xA₁: [4,3,3,3,3,3] x [ ] # B₇xA₁: [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] - tetraprism

Special cases of products become hypercube

Hypercube

[ ] 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

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
for a p-gon. Regular polygons are self-dual, so the rectification produces the same polygon. The uniform truncation operation doubles the sides to . The snub operation, alternatingly truncating the truncation returns it back to the original polygon . Thus all uniform polygons are also regular.

Operation	Extended Schläfli Symbols	Regular result	Position
Operation	Extended Schläfli Symbols	Regular result	(1)	(0)
Parent	t₀			--
Rectified (Dual)	t₁		--
Truncated	t_0,1
Snub	s		--	--

Uniform polyhedra and tilings

Every regular polyhedron or tiling has these five operations that create semiregular polyhedra. The short-hand notation is equivalent to the longer name. For instance, t simply means truncated tetrahedron

Truncated tetrahedron

The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....
.

The vertical notation is used for dual-symmetric operations - those that generate the same polyhedron from as .

A second extended notation, also used by Coxeter

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, Order of Canada is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
applies to all dimensions, and are specified by a t followed by a list of indices corresponding to Wythoff construction

Wythoff construction

Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
.)

In each a Wythoff construction operational name is given first. Second some have alternate terminology (given in parentheses) apply only for a given dimension. Specifically omnitruncation and expansion

Expansion (geometry)

In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements ....
, as well as dual relations apply differently in each dimension.

The final columns offer the elements centered on each position. A single positional index is a node. A double positional index is an edge. A triple positional index is the triangle interior.

The symbol -- implies a vertex at the position. The symbol implies an edge at that position. The symbol x is a square face .

Operation	Extended Schläfli Symbols		Wythoff symbol Wythoff construction File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....	Position
Operation	Extended Schläfli Symbols			(2)	(1)	(0)	(0,1)	(0,2)	(1,2)
Parent		t₀	q \| 2 p			--	--	--
Rectified Rectification (geometry) In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....		t₁	2 \| p q		--		--		--
Birectified (or dual Dual polyhedron In geometry, polyhedron are associated into pairs called *duals*, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron.... )		t₂	p \| 2 q	--				--	--
Truncated Truncation (geometry) In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....		t_0,1	2 q \| p				--
Bitruncated (or truncated dual)		t_1,2	2 p \| q						--
Cantellated Cantellation (geometry) In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex.... (or expanded Expansion (geometry) In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements .... )		t_0,2	p q \| 2		x			--
Cantitruncated (or omnitruncated)		t_0,1,2	2 p q \|		x
Snub Snub (geometry) In geometry, an alternation is an operation on a polyhedron or tessellation that fully truncates alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedron....		s	\| 2 p q				--	--	--

Generating triangles

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-dimensional hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the hypersphere

Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real num...
; thus the mirrors form an irregular tetrahedron

Tetrahedron

List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
is generated by one of four symmetry groups, as follows:

group [3,3,3]: the 5-cell , which is self-dual;
group [3,3,4]: 16-cell
16-cell

In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
and its dual tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
;
group [3,4,3]: the 24-cell
24-cell

In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
, self-dual;
group [3,3,5]: 600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
, its dual 120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
, and their ten regular stellations.

(The groups are named in Coxeter

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, Order of Canada is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
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

Convex uniform honeycomb

Cubic honeycomb

The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....
.

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 : x.
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.

Operation	Extended Schläfli symbols	Position
Operation	Extended Schläfli symbols	(3)	(2)	(1)	(0)
Parent	t₀				--
Rectified Rectification (geometry) In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....	t₁	t₁		--
Birectified (or rectified dual)	t₂		--		t₁
Trirectifed (or dual)	t₃	--			t₂

Truncated Truncation (geometry) In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....	t_0,1	t_0,1
Bitruncated	t_1,2	t_1,2			t_0,1
Tritruncated (or truncated dual)	t_2,3				t_1,2
Cantellated Cantellation (geometry) In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex....	t_0,2	t_0,2		x	t₁
Bicantellated (or cantellated dual)	t_1,3	t₁	x		t_0,2
Runcinated (or expanded Expansion (geometry) In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements .... )	t_0,3		x	x	t₂
Cantitruncated	t_0,1,2	t_0,1,2		x	t_0,1
Bicantitruncated (or cantitruncated dual)	t_1,2,3	t_1,2	x		t_0,1,2
Runcitruncated	t_0,1,3	t_0,1	x	x	t_0,2
Runcicantellated (or runcitruncated dual)	t_0,2,3	t_0,1,2	x	x	t_1,2
Runcicantitruncated (or omnitruncated)	t_0,1,2,3	t_0,1,2	x	x	t_0,1,2