All Topics  
Schläfli symbol

 

 
   Email Print
   Bookmark   Link
 

 

Schläfli symbol
 
 
  Topics Schläfli symbol Topic Home

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Schläfli symbol is a notation of the form that defines 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 and tessellations.

The Schläfli symbol is named after the 19th-century mathematician Ludwig Schläfli
Ludwig Schläfli

Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces....
 who made important contributions in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and other areas.

See also list 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....
.

Regular polyhedra (3-space) The Schläfli symbol of a regular 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....
 is if its faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
 are p-gons, and each vertex is surrounded by q faces (the vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
 is a q-gon).

For example is the regular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
.






Discussion
Ask a question about 'Schläfli symbol'
Start a new discussion about 'Schläfli symbol'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Schläfli symbol is a notation of the form that defines 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 and tessellations.

The Schläfli symbol is named after the 19th-century mathematician Ludwig Schläfli
Ludwig Schläfli

Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces....
 who made important contributions in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and other areas.

See also list 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....
.

Regular polygons (plane)


The Schläfli symbol of a regular 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 ....
 with n edges is .

For example, a regular pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
 is represented by .

See the convex regular polygon
Regular polygon

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

For example, is the pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
.

Regular polyhedra (3-space)

The Schläfli symbol of a regular 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....
 is if its faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
 are p-gons, and each vertex is surrounded by q faces (the vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
 is a q-gon).

For example is the regular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
. It has pentagonal faces, and 3 pentagons around each vertex.

See the 5 convex 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, the 4 nonconvex Kepler-Poinsot polyhedra.

Schläfli symbols may also be defined for regular tessellation
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
s of Euclidean
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
 or hyperbolic
Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
 space in a similar way.

For example, the Hexagonal tiling
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
 is represented by .

Regular polychora (4-space)


The Schläfli symbol of a regular 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"....
 is of the form . It has regular polygonal faces, cells, regular polyhedral vertex figures, and regular polygonal edge figures.

See the six convex regular
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 ....
 and 10 nonconvex polychora
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....
.

For example, the 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....
 is represented by . It is made of dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 cells , and has 3 cells around each edge.

There is also one regular tessellation of Euclidean 3-space: the cubic 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....
, with a Schläfli symbol of , made of cubic cells, and 4 cubes around each edge.

There are also 4 regular hyperbolic tessellations including , the Hyperbolic small dodecahedral honeycomb
Hyperbolic small dodecahedral honeycomb

The order-4 dodecahedral honeycomb is one of four regular space-filling tessellation in Hyperbolic space.Four dodecahedron exist on each edge, and 8 dodecahedra around each vertex....
, which fills space with dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 cells.

Higher dimensions


For higher dimensional 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 ....
s, the Schläfli symbol is defined recursively as if the facet
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:...
s have Schläfli symbol and the vertex figure
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 have Schläfli symbol .

Notice that a vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: .

There are only 3 regular polytopes in 5 dimensions and above: the simplex
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 ....
, ; the 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....
, ; and the 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....
, . There are no non-convex regular polytopes above 4 dimensions.

Dual polytopes


For dimension 2 or higher, every polytope has a 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....
.

If a polytope has Schläfli symbol then its dual has Schläfli symbol .

If the sequence is the same forwards and backwards, the polytope is self-dual. Every regular polytope in 2 dimensions (polygon) is self-dual.

Prismatic forms


Prismatic polytopes can be defined and named as a Cartesian product
Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
 of lower dimensional polytopes:
  • A p-gonal prism
    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....
    , with vertex figure p.4.4 as .
  • A uniform -hedral prism as .
  • A uniform p-q 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....
     as .


A prism
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....
 can also be represented as the truncation of a hosohedron
Hosohedron

An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....
 as , and an 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....
 (snub hosohedron) as .

Extended Schläfli symbols for uniform polytopes


Uniform polytope
Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
s, made from 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....
, are represented by an extended truncation notation from a regular form . There are a number of parallel symbolic forms that reference the elements of the Schläfli symbol, discussed by dimension below.

Uniform polyhedra and tilings

For polyhedra, one extended Schläfli symbol is used in the 1954 paper 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....
 enumerating the paper tiled 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....
.

Every regular polyhedron or tiling has 7 forms, including the regular form and its dual, corresponding to positions within the fundamental right triangle. An 8th special form, the snubs, correspond to an alternation of the omnitruncated form.

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....
.

A second, more general 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

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....
 mirrors. (They also correspond to ringed nodes in 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....
.)

For example, the truncated hexahedron can be represented by t0,1 and it can be seen as midway between 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....
, t0, and the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
, t1.

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
Omnitruncation (geometry)

In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets....
 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.
OperationExtended
Schläfli
Symbols		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....
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....
Parent t0
Dynkins 100
 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....

(Quasiregular)
Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
  t1
Dynkins 010
 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
Dynkins 001
 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....
 t0,1
Dynkins 110
 2 q | p
Bitruncated
(or truncated dual)		 t1,2
Dynkins 011
 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 ....
)		  t0,2
Dynkins 101
 p q | 2
Cantitruncated
(or omnitruncated
Omnitruncation (geometry)

In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets....
)		  t0,1,2
Dynkins 111
 2 p q |
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
Dynkins Sss
 | 2 p q


Uniform polychora and honeycombs


There are up to 15 different truncation forms for polychora and honeycombs based on each regular form.

See 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....
 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....
.

The subscripted-t notation is parallel to the graphical 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....
, with each graph node representing the 4 hyperplanes of the reflection mirrors in the fundamental domain.

OperationExtended
Schläfli
symbols		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....
Parent t0
Dynkins 1000
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
Dynkins 0100
Birectified
(or rectified dual)		 t2
Dynkins 0010
Trirectifed
(or dual)		 t3
Dynkins 0001
 
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....
 t0,1
Dynkins 1100
Bitruncated t1,2
Dynkins 0110
Tritruncated
(or truncated dual)		 t2,3
Dynkins 0011
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....
 t0,2
Dynkins 1010
Bicantellated
(or cantellated dual)		 t1,3
Dynkins 0101
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 ....
)		 t0,3
Dynkins 1001
Cantitruncated t0,1,2
Dynkins 1110
Bicantitruncated
(or cantitruncated dual)		 t1,2,3
Dynkins 0111
Runcitruncated t0,1,3
Dynkins 1101
Runcicantellated
(or runcitruncated dual)		 t0,2,3
Dynkins 1011
Runcicantitruncated
(or omnitruncated
Omnitruncation (geometry)

In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets....
)		 t0,1,2,3
Dynkins 1111


External links