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{{short description|Notation that defines regular polytopes and tessellations}}
[[File:Dodecahedron.png|thumb|The [[dodecahedron]] is a regular polyhedron with Schläfli symbol {5,3}, having 3 [[pentagon]]s around each [[Vertex_(geometry)#Of_a_polytope|vertex]].]]
In [[geometry]], the '''Schläfli symbol''' is a notation of the form <math>\{p,q,r, ...\}</math> that defines [[List of regular polytopes and compounds|regular polytopes and tessellations]].

The Schläfli symbol is named after the 19th-century Swiss mathematician [[Ludwig Schläfli]],<ref name=":0">{{Cite book|title=Regular Polytopes|last=Coxeter|first=H.S.M.|publisher=Dover|year=1973|edition=3rd|location=New York}}</ref>{{Rp|143}} who generalized [[Euclidean space|Euclidean geometry]] to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.
{{toclimit|3}}

== Definition ==
The Schläfli symbol is a [[Recursive definition|recursive]] description,<ref name=":0" />{{Rp|129}} starting with <math>\{p\}</math> for a <math>p</math> for-sided [[regular polygon]] that is [[Convex set|convex]]. For example, {3} is an [[equilateral triangle]], {4} is a [[Square (geometry)|square]], {5} a convex [[regular pentagon]], etc.

Regular [[star polygon]]s are not convex, and their Schläfli symbols <math>\{p/q\}</math> for contain [[irreducible fraction]]s <math>p/q</math>, where <math>p</math> is the number of vertices, and <math>q</math> is their [[turning number]]. Equivalently, <math>\{p/q\}</math> is created from the vertices of {''p''}, connected every ''q''. For example, <math>\{5/2\}</math> is a [[pentagram]]; <math>\{5/1\}</math> is a [[pentagon]].

A [[regular polyhedron]] that has <math>q</math> regular <math>p</math>-sided [[Face (geometry)|polygon faces]] around each [[Vertex_(geometry)#Of_a_polytope|vertex]] is represented by <math>\{p,q\}</math>. For example, the [[cube]] has 3 squares around each vertex and is represented by {4,3}.

A regular [[4-polytope|4-dimensional polytope]], with <math>r\{p,q\}</math> regular [[Face (geometry)|polyhedral cells]] around each edge is represented by <math>\{p,q,r\}</math>. For example, a [[tesseract]], {4,3,3}, has 3 [[cube]]s, {4,3}, around an edge.

In general, a [[regular polytope]] <math>\{p,q,r,\dots,y,z\}</math> has <math>z\{p,q,r,\dots,y\}</math> [[Facet (mathematics)|facet]]s around every [[Peak (geometry)|peak]], where a peak is a [[Vertex_(geometry)#Of_a_polytope|vertex]] in a polyhedron, an edge in a 4-polytope, a [[face (geometry)#Polygonal face|face]] in a 5-polytope, and an [[k-face|(''n''-3)-face]] in an ''n''-polytope.

==Properties==
A regular polytope has a regular [[vertex figure]]. The vertex figure of a regular polytope {''p'',''q'',''r'',...,''y'',''z''} is {''q'',''r'',...,''y'',''z''}.

Regular polytopes can have [[star polygon]] elements, like the [[pentagram]], with symbol {{mset|{{frac|5|2}}}}, represented by the vertices of a [[pentagon]] but connected alternately.

The Schläfli symbol can represent a finite [[convex polyhedron]], an infinite [[tessellation]] of [[Euclidean space]], or an infinite tessellation of [[hyperbolic space]], depending on the [[angle defect]] of the construction. A positive angle defect allows the vertex figure to ''fold'' into a higher dimension and loops back into itself as a polytope. A zero angle defect tessellates space of the same dimension as the facets. A negative angle defect cannot exist in ordinary space, but can be constructed in hyperbolic space.

Usually, a facet or a vertex figure is assumed to be a finite polytope, but can sometimes itself be considered a tessellation.

A regular polytope also has a [[#Dual polytopes|dual polytope]], represented by the ''Schläfli symbol'' elements in reverse order. A self-dual regular polytope will have a symmetric Schläfli symbol.

In addition to describing Euclidean polytopes, Schläfli symbols can be used to describe spherical polytopes or spherical honeycombs.<ref name=":0" />{{Rp|138}}

== History and variations ==
Schläfli's work was almost unknown in his lifetime, and his notation for describing polytopes was rediscovered independently by several others. In particular, [[Thorold Gosset]] rediscovered the Schläfli symbol which he wrote as |{{thin space}}''p''{{thin space}}|{{thin space}}''q''{{thin space}}|{{thin space}}''r''{{thin space}}|{{thin space}}...{{thin space}}|{{thin space}}''z''{{thin space}}| rather than with brackets and commas as Schläfli did.<ref name=":0" />{{Rp|144}}

Gosset's form has greater symmetry, so the number of dimensions is the number of vertical bars, and the symbol exactly includes the sub-symbols for facet and vertex figure. Gosset regarded |{{thin space}}''p'' as an operator, which can be applied to |{{thin space}}''q''{{thin space}}|{{thin space}}...{{thin space}}|{{thin space}}''z''{{thin space}}| to produce a polytope with ''p''-gonal faces whose vertex figure is |{{thin space}}''q''{{thin space}}|{{thin space}}...{{thin space}}|{{thin space}}''z''{{thin space}}|.

== Cases ==

=== Symmetry groups ===

Schläfli symbols are closely related to (finite) [[Reflection symmetry|reflection]] [[symmetry group]]s, which correspond precisely to the finite [[Coxeter group]]s and are specified with the same indices, but square brackets instead [''p'',''q'',''r'',...]. Such groups are often named by the regular polytopes they generate. For example, [3,3] is the Coxeter group for reflective [[tetrahedral symmetry]], [3,4] is reflective [[octahedral symmetry]], and [3,5] is reflective [[icosahedral symmetry]].

=== Regular polygons (plane) ===

[[File:regular star polygons.svg|thumb|400px|Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols]]
The Schläfli symbol of a convex [[regular polygon]] with ''p'' edges is {''p''}. For example, a regular [[pentagon]] is represented by {5}.

For nonconvex [[star polygon]]s, the constructive notation {{mset|{{frac|''p''|''q''}}}} is used, where ''p'' is the number of vertices and {{nowrap|''q''−1}} is the number of vertices skipped when drawing each edge of the star. For example, {{mset|{{frac|5|2}}}} represents the [[pentagram]].

=== Regular polyhedra (3 dimensions) ===
The Schläfli symbol of a regular [[polyhedron]] is {''p'',''q''} if its [[face (geometry)|faces]] are ''p''-gons, and each vertex is surrounded by ''q'' faces (the [[vertex figure]] is a ''q''-gon).

For example, {5,3} is the regular [[dodecahedron]]. It has pentagonal (5 edges) faces, and 3 pentagons around each vertex.

See the 5 convex [[Platonic solid]]s, the 4 nonconvex [[Kepler-Poinsot polyhedra]].

Topologically, a regular 2-dimensional [[tessellation]] may be regarded as similar to a (3-dimensional) polyhedron, but such that the [[angular defect]] is zero. Thus, Schläfli symbols may also be defined for regular [[tessellation]]s of [[Euclidean geometry|Euclidean]] or [[hyperbolic geometry|hyperbolic]] space in a similar way as for polyhedra. The analogy holds for higher dimensions.

For example, the [[hexagonal tiling]] is represented by {6,3}.

=== Regular 4-polytopes (4 dimensions) ===

The Schläfli symbol of a regular [[4-polytope]] is of the form {''p'',''q'',''r''}. Its (two-dimensional) faces are regular ''p''-gons ({''p''}), the cells are regular polyhedra of type {''p'',''q''}, the vertex figures are  regular polyhedra of type {''q'',''r''}, and the edge figures are regular ''r''-gons (type {''r''}).

See the six [[convex regular 4-polytope|convex regular]] and 10 [[List of regular polytopes#Nonconvex forms (4D)|regular star 4-polytopes]].

For example, the [[120-cell]] is represented by {5,3,3}. It is made of [[dodecahedron]] cells {5,3}, and has 3 cells around each edge.

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

There are also 4 regular compact hyperbolic tessellations including {5,3,4}, the [[hyperbolic small dodecahedral honeycomb]], which fills space with [[dodecahedron]] cells.

If a 4-polytope's symbol is palindromic (e.g. {3,3,3} or {3,4,3}), its bitruncation will only have truncated forms of the vertex figure as cells.

=== Regular ''n''-polytopes (higher dimensions) ===

For higher-dimensional [[regular polytope]]s, the Schläfli symbol is defined recursively as {{math|{{mset|''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n'' − 1</sub>}}}} if the [[facet (mathematics)|facet]]s have Schläfli symbol {{math|{{mset|''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n'' − 2</sub>}}}} and the [[vertex figure]]s have Schläfli symbol {{math|{{mset|''p''<sub>2</sub>, ''p''<sub>3</sub>, ..., ''p''<sub>''n'' − 1</sub>}}}}.

A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: {{math|{{mset|''p''<sub>2</sub>, ''p''<sub>3</sub>, ..., ''p''<sub>''n'' − 2</sub>}}}}.

There are only 3 regular polytopes in 5 dimensions and above: the [[simplex]], {3, 3, 3, ..., 3}; the [[cross-polytope]], {3, 3, ..., 3, 4}; and the [[hypercube]], {4, 3, 3, ..., 3}. There are no non-convex regular polytopes above 4 dimensions.
[[File:Schläfli symbol of regular polytopes.gif|thumb]]

=== Dual polytopes ===
If a polytope of dimension n{{thin space}}≥{{thin space}}2 has Schläfli symbol {{{thin space}}''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''{{thin space}}−{{thin space}}1</sub>{{thin space}}} then its [[dual polyhedron|dual]]  has Schläfli symbol {{{thin space}}''p''<sub>''n''{{thin space}}−{{thin space}}1</sub>, ..., ''p''<sub>2</sub>, ''p''<sub>1</sub>{{thin space}}}.

If the sequence is [[palindromic]], i.e. the same forwards and backwards, the polytope is ''self-dual''. Every regular polytope in 2&nbsp;dimensions (polygon) is self-dual.

=== Prismatic polytopes ===

[[Prism (geometry)#Uniform prismatic polytope|Uniform prismatic polytopes]] can be defined and named as a [[Cartesian product]] (with operator "×") of lower-dimensional regular polytopes.
* In 0D, a [[Point (geometry)|point]] is represented by ( ). Its [[Coxeter diagram]] is empty. Its [[Coxeter notation]] symmetry is ][.
* In 1D, a [[line segment]] is represented by { }. Its [[Coxeter diagram]] is {{CDD|node_1}}. Its symmetry is [ ].
* In 2D, a [[rectangle]] is represented as { } × { }. Its [[Coxeter diagram]] is {{CDD|node_1|2|node_1}}. Its symmetry is [2].
* In 3D, a ''p''-gonal [[Prism (geometry)|prism]] is represented as { } × {''p''}. Its Coxeter diagram is {{CDD|node_1|2|node_1|p|node}}. Its symmetry is [2,''p''].
* In 4D, a uniform {''p'',''q''}-hedral prism is represented as { } × {''p'',''q''}. Its Coxeter diagram is {{CDD|node_1|2|node_1|p|node|q|node}}. Its symmetry is [2,''p'',''q''].
* In 4D, a uniform ''p''-''q'' [[duoprism]] is represented as {''p''} × {''q''}. Its Coxeter diagram is {{CDD|node_1|p|node|2|node_1|q|node}}. Its symmetry is [''p'',2,''q''].

The prismatic duals, or [[bipyramid]]s can be represented as composite symbols, but with the ''addition'' operator, "+".
* In 2D, a [[rhombus]] is represented as { } + { }. Its Coxeter diagram is {{CDD|node_f1|2x|node_f1}}. Its symmetry is [2].
* In 3D, a ''p''-gonal bipyramid, is represented as { } + {''p''}. Its Coxeter diagram is {{CDD|node_f1|2x|node_f1|p|node}}. Its symmetry is [2,''p''].
* In 4D, a {''p'',''q''}-hedral bipyramid is represented as { } + {''p'',''q''}. Its Coxeter diagram is {{CDD|node_f1|2x|node_f1|p|node|q|node}}. Its symmetry is [''p'',''q''].
* In 4D, a ''p''-''q'' [[duopyramid]] is represented as {''p''} + {''q''}. Its Coxeter diagram is {{CDD|node_f1|p|node|2x|node_f1|q|node}}. Its symmetry is [''p'',2,''q''].

Pyramidal polytopes containing vertices orthogonally offset can be represented using a join operator, "∨". Every pair of vertices between joined figures are connected by edges.

In 2D, an [[isosceles triangle]] can be represented as ( ) ∨ { } = ( ) ∨ [( ) ∨ ( )].

In 3D:
* A [[digonal disphenoid]] can be represented as { } ∨ { } = [( ) ∨ ( )] ∨ [( ) ∨ ( )].
* A [[Pyramid (geometry)|''p-gonal pyramid'']] is represented as ( ) ∨ {''p''}.
In 4D:
* A [[Polyhedral pyramid|''p-q-hedral pyramid'']] is represented as ( ) ∨ {''p'',''q''}.
*A [[5-cell]] is represented as ( ) ∨ [( ) ∨ {3}] or [( ) ∨ ( )] ∨ {3} = { } ∨ {3}.
*A square pyramidal pyramid is represented as ( ) ∨ [( ) ∨ {4}] or [( ) ∨ ( )] ∨ {4} = { } ∨ {4}.
When mixing operators, the [[order of operations]] from highest to lowest is ×, +, ∨.

Axial polytopes containing vertices on parallel offset hyperplanes can be represented by the ‖ operator. A uniform prism is {''n''}‖{''n''} and antiprism {''n''}‖''r''{''n''}.

== Extension of Schläfli symbols ==

=== Polygons and circle tilings ===

A truncated regular polygon doubles in sides. A regular polygon with even sides can be halved. An altered even-sided regular 2n-gon generates a [[star figure]] compound, 2{n}.
{| class=wikitable
!Form
!Schläfli symbol
![[Coxeter notation|Symmetry]]
!colspan=2|[[Coxeter diagram]]
!colspan=3|Example, {6}
|- align=center
![[Regular polyhedron|Regular]]
|{p}
|[p]
|colspan=2|{{CDD|node_1|p|node}}
|[[File:Regular polygon 6 annotated.svg|60px]]||[[Hexagon]]||{{CDD|node_1|6|node}}
|- align=center
![[Truncation (geometry)|Truncated]]
|t{p} = {2p}||<nowiki>[[</nowiki>p]] = [2p]
|colspan=2| {{CDD|node_1|p|node_1}} = {{CDD|node_1|2x|p|node}}
|[[File:Regular polygon 12 annotated.svg|60px]]||[[dodecagon|Truncated hexagon]]<br>(Dodecagon)||{{CDD|node_1|6|node_1}} = {{CDD|node_1|12|node}}
|- align=center
!Altered and <BR>Holosnubbed
|a{2p} = β{p}||[2p]
|colspan=2| {{CDD|node_h3|p|node_h3}} = {{CDD|node_h3|2x|p|node}}
|[[File:Hexagram.svg|60px]]||[[Hexagram|Altered hexagon]]<br>(Hexagram)||{{CDD|node_h3|3|node_h3}} = {{CDD|node_h3|6|node}}
|- align=center
!Half and <BR>Snubbed
|h{2p} = s{p} = {p} ||[1<sup>+</sup>,2p] = [p]
|colspan=2| {{CDD|node_h|2x|p|node}} = {{CDD|node_h|p|node_h}} = {{CDD|node_1|p|node}}
|[[File:Regular polygon 3 annotated.svg|60px]]||[[triangle|Half hexagon]]<br>(Triangle)||{{CDD|node_h|6|node}} = {{CDD|node_h|3|node_h}} = {{CDD|node_1|3|node}}
|}

=== Polyhedra and tilings ===
[[Coxeter]] expanded his usage of the Schläfli symbol to [[quasiregular polyhedra]] by adding a vertical dimension to the symbol. It was a starting point toward the more general [[Coxeter diagram]]. [[Norman Johnson (mathematician)|Norman Johnson]] simplified the notation for vertical symbols with an ''r'' prefix. The t-notation is the most general, and directly corresponds to the rings of the Coxeter diagram. Symbols have a corresponding [[Alternation (geometry)|alternation]], replacing ''rings'' with  ''holes'' in a Coxeter diagram and ''h'' prefix standing for ''half'', construction limited by the requirement that neighboring branches must be even-ordered and cuts the symmetry order in half. A related operator, ''a'' for ''altered'', is shown with two nested holes, represents a compound polyhedra with both alternated halves, retaining the original full symmetry. A [[snub (geometry)|snub]] is a half form of a truncation, and a holosnub is both halves of an alternated truncation.

{| class=wikitable
!Form
!colspan=3|Schläfli symbols
![[Coxeter notation|Symmetry]]
!colspan=2|[[Coxeter diagram]]
!colspan=3|Example, {4,3}
|- align=center
![[Regular polyhedron|Regular]]
|<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>||{p,q}||t<sub>0</sub>{p,q}
|rowspan=7|[p,q]<BR>or<BR>[(p,q,2)]
|colspan=2|{{CDD|node_1|p|node|q|node}}
|[[File:Hexahedron.png|40px]]||[[Cube]]||{{CDD|node_1|4|node|3|node}}
|- align=center
![[Truncation (geometry)|Truncated]]
| <math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>||t{p,q}||t<sub>0,1</sub>{p,q}
|colspan=2| {{CDD|node_1|p|node_1|q|node}}
|[[File:Truncated hexahedron.png|40px]]||[[Truncated cube]]||{{CDD|node_1|4|node_1|3|node}}
|- align=center
![[Bitruncation]]<BR>(Truncated dual)
| <math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>||2t{p,q}||t<sub>1,2</sub>{p,q}
|{{CDD|node_1|q|node_1|p|node}}||{{CDD|node|p|node_1|q|node_1}}
|[[File:Truncated octahedron.png|40px]]||[[Truncated octahedron]]||{{CDD|node|4|node_1|3|node_1}}
|- align=center
![[Rectification (geometry)|Rectified]]<BR>([[Quasiregular polyhedron|Quasiregular]])
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||r{p,q}||t<sub>1</sub>{p,q}
||{{CDD|node_1|split1-pq|nodes}}|| {{CDD|node|p|node_1|q|node}}
|[[File:Cuboctahedron.png|40px]]||[[Cuboctahedron]]||{{CDD|node|4|node_1|3|node}}
|- align=center
!Birectification<BR>(Regular dual)
|<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>||2r{p,q}||t<sub>2</sub>{p,q}
|{{CDD|node_1|q|node|p|node}}||{{CDD|node|p|node|q|node_1}}
|[[File:Octahedron.png|40px]]||[[Octahedron]]||{{CDD|node|4|node|3|node_1}}
|- align=center
![[Cantellation (geometry)|Cantellated]]<BR>([[Expansion (geometry)|Rectified rectified]])
| <math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||rr{p,q}||t<sub>0,2</sub>{p,q}
||{{CDD|node|split1-pq|nodes_11}}|| {{CDD|node_1|p|node|q|node_1}}
|[[File:Small rhombicuboctahedron.png|40px]]||[[Rhombicuboctahedron]]||{{CDD|node_1|4|node|3|node_1}}
|- align=center
![[Omnitruncation|Cantitruncated]]<BR>(Truncated rectified)
| <math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||tr{p,q}||t<sub>0,1,2</sub>{p,q}
||{{CDD|node_1|split1-pq|nodes_11}}|| {{CDD|node_1|p|node_1|q|node_1}}
|[[File:Great rhombicuboctahedron.png|40px]]||[[Truncated cuboctahedron]]||{{CDD|node_1|4|node_1|3|node_1}}
|}

==== Alternations, quarters and snubs ====
Alternations have half the symmetry of the Coxeter groups and are represented by unfilled rings. There are two choices possible on which half of vertices are taken, but the symbol does not imply which one. Quarter forms are shown here with a + inside a hollow ring to imply they are two independent alternations.

{| class=wikitable
|+ [[Alternation (geometry)|Alternations]]
!Form
!colspan=3|Schläfli symbols
![[Coxeter notation|Symmetry]]
!colspan=2|[[Coxeter diagram]]
!colspan=3|Example, {4,3}
|- align=center
!Alternated (half) regular
| <math>h \begin{Bmatrix} 2p , q \end{Bmatrix}</math>||h{2p,q}||ht<sub>0</sub>{2p,q}||[1<sup>+</sup>,2p,q]
|colspan=2| {{CDD|node_h1|2x|p|node|q|node}} = {{CDD|labelp|branch_10ru|split2-qq|node}}
|[[File:Tetrahedron.png|40px]]||Demicube<BR>([[Tetrahedron]])||{{CDD|node_h1|4|node|3|node}}
|- align=center
!Snub regular
| <math>s\begin{Bmatrix} p , 2q \end{Bmatrix}</math>||s{p,2q}||ht<sub>0,1</sub>{p,2q}||[p<sup>+</sup>,2q]
|colspan=2| {{CDD|node_h|p|node_h|2x|q|node}}
||| ||
|- align=center
!Snub dual regular
| <math>s \begin{Bmatrix} q , 2p \end{Bmatrix}</math>||s{q,2p}||ht<sub>1,2</sub>{2p,q}||[2p,q<sup>+</sup>]
|{{CDD|node_h|q|node_h|2x|p|node}}||{{CDD|node|2x|p|node_h|q|node_h}}
|[[File:Uniform polyhedron-43-h01.svg|40px]]||Snub octahedron<BR>([[Icosahedron]])||{{CDD|node|4|node_h|3|node_h}}

|- align=center
!Alternated rectified<BR>(p and q are even)
| <math>h \begin{Bmatrix} p \\ q \end{Bmatrix}</math>||hr{p,q}||ht<sub>1</sub>{p,q}||[p,1<sup>+</sup>,q]
||{{CDD|node_h1|split1-pq|nodes}}|| {{CDD|node|p|node_h1|q|node}}
| || ||
|- align=center
!Alternated rectified rectified<BR>(p and q are even)
| <math>hr \begin{Bmatrix} p \\ q \end{Bmatrix}</math>||hrr{p,q}||ht<sub>0,2</sub>{p,q}||[(p,q,2<sup>+</sup>)]
||{{CDD|node|split1-pq|branch_hh|label2}}|| {{CDD|node_h|p|node|q|node_h}}
| || ||
|- align=center
!Quartered<BR>(p and q are even)
| <math>q\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||q{p,q}||ht<sub>0</sub>ht<sub>2</sub>{p,q}||[1<sup>+</sup>,p,q,1<sup>+</sup>]
||{{CDD|node|split1-pq|nodes_h1h1}}|| {{CDD|node_h1|p|node|q|node_h1}}
| || ||
|- align=center
!Snub rectified<BR>Snub quasiregular
| <math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||sr{p,q}||ht<sub>0,1,2</sub>{p,q}||[p,q]<sup>+</sup>
||{{CDD|node_h|split1-pq|nodes_hh}}|| {{CDD|node_h|p|node_h|q|node_h}}
|[[File:Snub hexahedron.png|40px]]||[[Snub cuboctahedron]]<BR>(Snub cube)||{{CDD|node_h|4|node_h|3|node_h}}
|}

====  Altered and holosnubbed ====
Altered and holosnubbed forms have the full symmetry of the Coxeter group, and are represented by double unfilled rings, but may be represented as compounds.
{| class=wikitable
|+ Altered and holosnubbed
!Form
!colspan=3|Schläfli symbols
![[Coxeter notation|Symmetry]]
!colspan=2|[[Coxeter diagram]]
!colspan=3|Example, {4,3}
|- align=center
!Altered regular
| <math>a \begin{Bmatrix} p , q \end{Bmatrix}</math>||a{p,q}||at<sub>0</sub>{p,q}||[p,q]
|colspan=2| {{CDD|node_h3|p|node|q|node}} = {{CDD|labelp-2|branch_10ru|split2-qq|node}} ∪ {{CDD|labelp-2|branch_01rd|split2-qq|node}}
|[[File:Compound of two tetrahedra.png|40px]]||[[Stellated octahedron]]||{{CDD|node_h3|4|node|3|node}}
|- align=center
!Holosnub dual regular
| {{math|ß{{mset|''q'', ''p''}}}}||ß{q,p}||at<sub>0,1</sub>{q,p}||[p,q]
|{{CDD|node_h3|q|node_h3|p|node}}||{{CDD|node|p|node_h3|q|node_h3}}
|[[File:UC46-2 icosahedra.png|40px]]||[[Compound of two icosahedra]]||{{CDD|node|4|node_h3|3|node_h3}}
|}

: [[ß]], looking similar to the greek letter [[beta]] (β), is the German alphabet letter [[eszett]].

=== Polychora and honeycombs ===
{| class="wikitable"
|+ Linear families
|-
!Form
!colspan=3|Schläfli symbol
!colspan=2|[[Coxeter diagram]]
!colspan=3|Example, {4,3,3}
|- align=center
!Regular
| <math>\begin{Bmatrix} p, q , r \end{Bmatrix}</math>||{p,q,r}
|t<sub>0</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node|q|node|r|node}}
|[[File:Schlegel wireframe 8-cell.png|40px]]||[[Tesseract]]||{{CDD|node_1|4|node|3|node|3|node}}
|- align=center
!Truncated
| <math>t\begin{Bmatrix} p, q , r \end{Bmatrix}</math> ||t{p,q,r}
|t<sub>0,1</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node_1|q|node|r|node}}
|[[File:Schlegel half-solid truncated tesseract.png|40px]]||[[Truncated tesseract]]||{{CDD|node_1|4|node_1|3|node|3|node}}
|- align=center
!Rectified
| <math>\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math>
||r{p,q,r}
|t<sub>1</sub>{p,q,r}
|colspan=2|{{CDD|node|p|node_1|q|node|r|node}}
|[[File:Schlegel half-solid rectified 8-cell.png|40px]]||[[Rectified tesseract]]||{{CDD|node|4|node_1|3|node|3|node}} = {{CDD|node_1|split1-43|nodes|3b|nodeb}}
|- align=center
!Bitruncated
| ||2t{p,q,r}
|t<sub>1,2</sub>{p,q,r}
|colspan=2|{{CDD|node|p|node_1|q|node_1|r|node}}
|[[File:Schlegel half-solid bitruncated 16-cell.png|40px]]||[[Bitruncated tesseract]]||{{CDD|node|4|node_1|3|node_1|3|node}}
|- align=center
!Birectified<BR>(Rectified dual)
|<!--<math>\begin{Bmatrix} q , p \\ r \end{Bmatrix}</math><BR>--><math>\left\{\begin{array}{l}q,p\\r\end{array}\right\}</math>
||2r{p,q,r} = r{r,q,p}
|t<sub>2</sub>{p,q,r}
|colspan=2|{{CDD|node|p|node|q|node_1|r|node}}
|[[File:Schlegel half-solid rectified 16-cell.png|40px]]||[[Rectified 16-cell]]||{{CDD|node|4|node|3|node_1|3|node}} = {{CDD|node_1|split1|nodes|4a|nodea}}
|- align=center
!Tritruncated<BR>(Truncated dual)
| <math>t\begin{Bmatrix} r, q , p \end{Bmatrix}</math> ||3t{p,q,r} = t{r,q,p}
|t<sub>2,3</sub>{p,q,r}
|colspan=2|{{CDD|node|p|node|q|node_1|r|node_1}}
|[[File:Schlegel half-solid truncated 16-cell.png|40px]]||[[Bitruncated tesseract]]||{{CDD|node|4|node|3|node_1|3|node_1}}
|- align=center
!Trirectified<BR>(Dual)
| <math>\begin{Bmatrix} r, q , p \end{Bmatrix}</math>||3r{p,q,r} = {r,q,p}
|t<sub>3</sub>{p,q,r} = {r,q,p}
|colspan=2|{{CDD|node|p|node|q|node|r|node_1}}
|[[File:Schlegel wireframe 16-cell.png|40px]]||[[16-cell]]||{{CDD|node|4|node|3|node|3|node_1}}
|- align=center
!Cantellated
| <math>r\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math> ||rr{p,q,r}
|t<sub>0,2</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node|q|node_1|r|node}}
|[[File:Schlegel half-solid cantellated 8-cell.png|40px]]||[[Cantellated tesseract]]||{{CDD|node_1|4|node|3|node_1|3|node}} = {{CDD|node|split1-43|nodes_11|3b|nodeb}}
|- align=center
!Cantitruncated
| <math>t\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math> ||tr{p,q,r}
|t<sub>0,1,2</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node_1|q|node_1|r|node}}
|[[File:Schlegel half-solid cantitruncated 8-cell.png|40px]]||[[Cantitruncated tesseract]]||{{CDD|node_1|4|node_1|3|node_1|3|node}} = {{CDD|node_1|split1-43|nodes_11|3b|nodeb}}
|- align=center
!Runcinated<BR>([[Expansion (geometry)|Expanded]])
| <math>e_3\begin{Bmatrix} p, q , r \end{Bmatrix}</math>  || e<sub>3</sub>{p,q,r}
|t<sub>0,3</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node|q|node|r|node_1}}
|[[File:Schlegel half-solid runcinated 8-cell.png|40px]]||[[Runcinated tesseract]]||{{CDD|node_1|4|node|3|node|3|node_1}}
|- align=center
!Runcitruncated
| || 
|t<sub>0,1,3</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node_1|q|node|r|node_1}}
|[[File:Schlegel half-solid runcitruncated 8-cell.png|40px]]||[[Runcitruncated tesseract]]||{{CDD|node_1|4|node_1|3|node|3|node_1}}
|- align=center
!Omnitruncated
| || 
|t<sub>0,1,2,3</sub>{p,q,r}
|colspan=2|{{CDD|node_1|p|node_1|q|node_1|r|node_1}}
|[[File:Schlegel half-solid omnitruncated 8-cell.png|40px]]||[[Omnitruncated tesseract]]||{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
|}

====Alternations, quarters and snubs====
{| class="wikitable"
|+ Alternations
|-
!Form
!colspan=3|Schläfli symbol
!colspan=2|[[Coxeter diagram]]
!colspan=3|Example, {4,3,3}
|- align=center
|- align=center
!colspan=10|[[Alternation (geometry)|Alternations]]
|- align=center
!Half<BR>p even
| <math>h\begin{Bmatrix} p, q , r \end{Bmatrix}</math>||h{p,q,r}
|ht<sub>0</sub>{p,q,r}
|colspan=2|{{CDD|node_h1|p|node|q|node|r|node}}
|[[File:Schlegel wireframe 16-cell.png|40px]]||[[16-cell]]||{{CDD|node_h1|4|node|3|node|3|node}}
|- align=center
!Quarter<BR>p and r even
| <math>q\begin{Bmatrix} p, q , r \end{Bmatrix}</math>||q{p,q,r}
|ht<sub>0</sub>ht<sub>3</sub>{p,q,r}
|colspan=2|{{CDD|node_h1|p|node|q|node|r|node_h1}}
| || || 
|- align=center
!Snub<BR>q even
| <math>s\begin{Bmatrix} p, q , r \end{Bmatrix}</math>||s{p,q,r}
|ht<sub>0,1</sub>{p,q,r}
|colspan=2|{{CDD|node_h|p|node_h|q|node|r|node}}
|[[File:Ortho solid 969-uniform polychoron 343-snub.png|40px]]||[[Snub 24-cell]]||{{CDD|node_h|3|node_h|4|node|3|node}}
|- align=center
!Snub rectified<BR>r even
| <math>s\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math>||sr{p,q,r}
|ht<sub>0,1,2</sub>{p,q,r}
|colspan=2|{{CDD|node_h|p|node_h|q|node_h|r|node}}
|[[File:Ortho solid 969-uniform polychoron 343-snub.png|40px]]||[[Snub 24-cell]]||{{CDD|node_h|3|node_h|3|node_h|4|node}} = {{CDD|node_h|split1|nodes_hh|4a|nodea}}
|- align=center
!Alternated duoprism
| || s{p}s{q}
|ht<sub>0,1,2,3</sub>{p,2,q}
|colspan=2|{{CDD|node_h|p|node_h|2x|node_h|q|node_h}}
|[[File:Great duoantiprism.png|40px]]||[[Great duoantiprism]] ||{{CDD|node_h|5|node_h|2x|node_h|5|rat|3x|node_h}}
|}

==== Bifurcating families ====

{| class="wikitable"
|+ Bifurcating families
|-
!Form
!colspan=3|Extended Schläfli symbol
!colspan=2|[[Coxeter diagram]]
!colspan=3|Examples
|- align=center
!Quasiregular
| <math>\left\{p,{q\atop q}\right\}</math>||{p,q<sup>1,1</sup>}
|t<sub>0</sub>{p,q<sup>1,1</sup>}
|colspan=2|{{CDD|node_1|p|node|split1-qq|nodes}}
|[[File:Schlegel wireframe 16-cell.png|40px]]||Demitesseract<br>([[16-cell]])||{{CDD|node_1|3|node|split1|nodes}}
|- align=center
!Truncated
| <math>t\left\{p,{q\atop q}\right\}</math>||t{p,q<sup>1,1</sup>}
|t<sub>0,1</sub>{p,q<sup>1,1</sup>}
|colspan=2|{{CDD|node_1|p|node_1|split1-qq|nodes}}
|[[File:Schlegel half-solid truncated 16-cell.png|40px]]||Truncated demitesseract<br>([[Truncated 16-cell]])||{{CDD|node_1|3|node_1|split1|nodes}}
|- align=center
!Rectified
| <math>\left\{\begin{array}{l}p\\q\\q\end{array}\right\}</math>||r{p,q<sup>1,1</sup>}
|t<sub>1</sub>{p,q<sup>1,1</sup>}
|colspan=2|{{CDD|node|p|node_1|split1-qq|nodes}}
|[[File:Schlegel wireframe 24-cell.png|40px]]||Rectified demitesseract<br>([[24-cell]])||{{CDD|node|3|node_1|split1|nodes}}
|- align=center
!Cantellated
| <math>r\left\{\begin{array}{l}p\\q\\q\end{array}\right\}</math>||rr{p,q<sup>1,1</sup>}
|t<sub>0,2,3</sub>{p,q<sup>1,1</sup>}
|colspan=2|{{CDD|node_1|p|node|split1-qq|nodes_11}}
|[[File:Schlegel half-solid cantellated 16-cell.png|40px]]||Cantellated demitesseract<br>([[Cantellated 16-cell]])||{{CDD|node_1|3|node|split1|nodes_11}}
|- align=center
!Cantitruncated
| <math>t\left\{\begin{array}{l}p\\q\\q\end{array}\right\}</math>||tr{p,q<sup>1,1</sup>}
|t<sub>0,1,2,3</sub>{p,q<sup>1,1</sup>}
|colspan=2|{{CDD|node_1|p|node_1|split1-qq|nodes_11}}
|[[File:Schlegel half-solid cantitruncated 16-cell.png|40px]]||Cantitruncated demitesseract<br>([[Cantitruncated 16-cell]])||{{CDD|node_1|3|node_1|split1|nodes_11}}
|- align=center
!Snub rectified
| <math>s\left\{\begin{array}{l}p\\q\\q\end{array}\right\}</math>||sr{p,q<sup>1,1</sup>}
|ht<sub>0,1,2,3</sub>{p,q<sup>1,1</sup>}
|colspan=2|{{CDD|node_h|p|node_h|split1-qq|nodes_hh}}
|[[File:Ortho solid 969-uniform polychoron 343-snub.png|40px]]||Snub demitesseract<br>([[Snub 24-cell]])||{{CDD|node_h|3|node_h|split1|nodes_hh}}

|- align=center
!Quasiregular
| <math>\left\{r,{p\atop q}\right\}</math>||{r,/q\,p}
| t<sub>0</sub>{r,/q\,p}
|colspan=2|{{CDD|node_1|r|node|split1-pq|nodes}}
|[[File:Tetrahedral-octahedral honeycomb.png|40px]] ||[[Tetrahedral-octahedral honeycomb]]|| {{CDD|node_1|3|node|split1-43|nodes}}
|- align=center
!Truncated
| <math>t\left\{r,{p\atop q}\right\}</math>|| t{r,/q\,p}
| t<sub>0,1</sub>{r,/q\,p}
|colspan=2|{{CDD|node_1|r|node_1|split1-pq|nodes}}
|[[File:Truncated Alternated Cubic Honeycomb.svg|40px]] ||[[Cantic cubic honeycomb|Truncated tetrahedral-octahedral honeycomb]]|| {{CDD|node_1|3|node_1|split1-43|nodes}}
|- align=center
!Rectified
| <math>\left\{\begin{array}{l}r\\p\\q\end{array}\right\}</math>|| r{r,/q\,p}
| t<sub>1</sub>{r,/q\,p}
|colspan=2|{{CDD|node|r|node_1|split1-pq|nodes}}
|[[File:Rectified cubic honeycomb4.png|40px]] || [[Rectified cubic honeycomb|Rectified tetrahedral-octahedral honeycomb]]<br>(Rectified cubic honeycomb) || {{CDD|node|3|node_1|split1-43|nodes}}
|- align=center
!Cantellated
| <math>r\left\{\begin{array}{l}r\\p\\q\end{array}\right\}</math>|| rr{r,/q\,p}
| t<sub>0,2,3</sub>{r,/q\,p}
|colspan=2|{{CDD|node_1|r|node|split1-pq|nodes_11}}
|[[File:Cantellated cubic honeycomb.jpg|40px]] || [[Cantellated cubic honeycomb]] || {{CDD|node_1|3|node|split1-43|nodes_11}}
|- align=center
!Cantitruncated
| <math>t\left\{\begin{array}{l}r\\p\\q\end{array}\right\}</math>|| tr{r,/q\,p}
| t<sub>0,1,2,3</sub>{r,/q\,p}
|colspan=2|{{CDD|node_1|r|node_1|split1-pq|nodes_11}}
|[[File:Cantitruncated Cubic Honeycomb.svg|40px]] || [[Cantitruncated cubic honeycomb]] || {{CDD|node_1|3|node_1|split1-43|nodes_11}}
|- align=center
!Snub rectified
| <math>s\left\{\begin{array}{l}p\\q\\r\end{array}\right\}</math>|| sr{p,/q,\r}
| ht<sub>0,1,2,3</sub>{p,/q\,r}
|colspan=2|{{CDD|node_h|r|node_h|split1-pq|nodes_hh}}
|[[File:Alternated cantitruncated cubic honeycomb.png|40px]] || [[Alternated cantitruncated cubic honeycomb|Snub rectified cubic honeycomb]] (non-uniform, but near miss) || {{CDD|node_h|3|node_h|split1-43|nodes_hh}}
|}

==Tessellations==
[[Spherical polyhedron|Spherical]]
{{Col-begin}}{{col-break}}
* [[Hosohedron|{2,n}]]
* [[Antiprism|s{2,2n}]]
* [[Trapezohedron|{ } ⊕ {n}]]
{{col-break}}
* [[Prism (geometry)|t{2, n}]]
* [[Bipyramid|{ } + {n}]]
{{Col-end}}

Regular
{{Col-begin}}{{col-break}}
* [[Apeirogonal hosohedron|{2,∞}]]
* [[Triangular tiling|{3,6}]]
{{col-break}}
* [[Square tiling|{4,4}]]
* [[Hexagonal tiling|{6,3}]]
{{Col-end}}

Semi-regular
{{Col-begin}}{{col-break}}
* [[Snub square tiling|s{4,4}]]
* [[Elongated triangular tiling|e{3,6}]]
* [[Apeirogonal antiprism|sr{2,∞}]]
* [[Snub trihexagonal tiling|sr{6,3}]]
* [[Rhombitrihexagonal tiling|rr{6,3}]]
{{col-break}}
* [[Trihexagonal tiling|r{6,3}]]
* [[Truncated hexagonal tiling|t{6,3}]]
* [[Apeirogonal prism|t{2,∞}]]
* [[Truncated trihexagonal tiling|tr{6,3}]]
* [[Truncated square tiling|tr{4,4}]]
{{Col-end}}

[[Uniform tilings in hyperbolic plane|Hyperbolic]]
{{Col-begin}}{{col-break}}
* [[Snub tetrapentagonal tiling|sr{5,4}]]
* [[Snub tetrahexagonal tiling|sr{6,4}]]
* [[Snub tetraheptagonal tiling|sr{7,4}]]
* [[Snub tetraoctagonal tiling|sr{8,4}]]
* [[Snub tetraapeirogonal tiling|sr{∞,4}]]
* [[Snub pentapentagonal tiling|s{5,4}]]
* [[Snub pentahexagonal tiling|sr{6,5}]]
* [[Snub hexahexagonal tiling|s{6,4}]]
* [[Snub hexaoctagonal tiling|sr{8,6}]]
* [[Snub heptaheptagonal tiling|sr{7,7}]]
* [[Snub octaoctagonal tiling|s{8,4}]]
* [[Snub order-6 square tiling|s{4,6}]]
* [[Snub apeiroapeirogonal tiling|sr{∞,∞}]]
* [[Snub triheptagonal tiling|sr{7,3}]]
* [[Snub trioctagonal tiling|sr{8,3}]]
* [[Snub triapeirogonal tiling|sr{∞,3}]]
{{col-break}}
* [[Snub order-8 triangular tiling|s{3,8}]]
* [[Order-7 triangular tiling|{3,7}]]
* [[Order-8 triangular tiling|{3,8}]]
* [[Infinite-order triangular tiling|{3,∞}]]
* [[Alternated octagonal tiling|h{8,3}]]
* [[Alternated order-4 hexagonal tiling|h{6,4}]]
* [[Quarter order-6 square tiling|q{4,6}]]
* [[Rhombitriheptagonal tiling|rr{7,3}]]
* [[Rhombitrioctagonal tiling|rr{8,3}]]
* [[Rhombitriapeirogonal tiling|rr{∞,3}]]
* [[Cantic octagonal tiling|h<sub>2</sub>{8,3}]]
* [[Triheptagonal tiling|r{7,3}]]
* [[Trioctagonal tiling|r{8,3}]]
* [[Truncated heptagonal tiling|t{7,3}]]
* [[Truncated octagonal tiling|t{8,3}]]
* r{∞,3}
{{col-break}}
* [[Truncated order-3 apeirogonal tiling|t{∞,3}]]
* [[Rhombitetrapentagonal tiling|rr{5,4}]]
* [[Rhombitetrahexagonal tiling|rr{6,4}]]
* [[Rhombitetraheptagonal tiling|rr{7,4}]]
* [[Rhombitetraoctagonal tiling|rr{8,4}]]
* [[Rhombitetraapeirogonal tiling|rr{∞,4}]]
* [[Order-5 square tiling|{4,5}]]
* [[Order-6 square tiling|{4,6}]]
* [[Order-7 square tiling|{4,7}]]
* [[Order-8 square tiling|{4,8}]]
* [[Infinite-order square tiling|{4,∞}]]
* [[Tetrapentagonal tiling|r{5,4}]]
* [[Tetrahexagonal tiling|r{6,4}]]
* [[Truncated trihexagonal tiling|tr{6,3}]]
* [[Truncated triheptagonal tiling|tr{7,3}]]
* [[Order 3-7 kisrhombille|???]]
{{col-break}}
* [[Truncated trioctagonal tiling|tr{8,3}]]
* [[Order 3-8 kisrhombille|???]]
* [[Truncated triapeirogonal tiling|tr{∞,3}]]
* [[Tetraheptagonal tiling|r{7,4}]]
* [[Tetraoctagonal tiling|r{8,4}]]
* [[Truncated tetrapentagonal tiling|tr{5,4}]]
* [[4-5 kisrhombille|???]]
* [[Truncated tetrahexagonal tiling|tr{6,4}]]
* [[Truncated tetraheptagonal tiling|tr{7,4}]]
* [[Truncated tetraoctagonal tiling|tr{8,4}]]
* [[Truncated tetraapeirogonal tiling|tr{∞,4}]]
* [[Truncated order-4 pentagonal tiling|t{5,4}]]
* [[Truncated pentahexagonal tiling|tr{6,5}]]
* [[Truncated order-4 hexagonal tiling|t{6,4}]]
* [[Truncated hexaoctagonal tiling|tr{8,6}]]
* [[Truncated order-4 heptagonal tiling|t{7,4}]]
{{col-break}}
* [[Truncated order-4 octagonal tiling|t{8,4}]]
* [[Truncated order-4 apeirogonal tiling|t{∞,4}]]
* r{∞,4}
* [[Order-4 pentagonal tiling|{5,4}]]
* [[Order-5 pentagonal tiling|{5,5}]]
* [[Order-6 pentagonal tiling|{5,6}]]
* [[Infinite-order pentagonal tiling|{5,∞}]]
* [[Rhombipentahexagonal tiling|rr{6,5}]]
* [[Pentahexagonal tiling|r{6,5}]]
* [[Truncated order-5 square tiling|t{4,5}]]
* [[Truncated order-5 pentagonal tiling|t{5,5}]]
* [[Truncated order-5 hexagonal tiling|t{6,5}]]
* r{∞,5}
* [[Order-4 hexagonal tiling|{6,4}]]
* [[Order-5 hexagonal tiling|{6,5}]]
* [[Order-6 hexagonal tiling|{6,6}]]
{{Col-break}}
* [[Order-8 hexagonal tiling|{6,8}]]
* [[Rhombihexaoctagonal tiling|rr{8,6}]]
* [[Hexaoctagonal tiling|r{8,6}]]
* [[Truncated order-6 square tiling|t{4,6}]]
* [[Truncated order-6 pentagonal tiling|t{5,6}]]
* [[Truncated order-6 hexagonal tiling|t{6,6}]]
* [[Truncated order-6 octagonal tiling|t{8,6}]]
* [[Heptagonal tiling|{7,3}]]
* [[Order-4 heptagonal tiling|{7,4}]]
* [[Order-7 heptagonal tiling|{7,7}]]
* [[Truncated order-7 triangular tiling|t{3,7}]]
* [[Truncated order-7 square tiling|t{4,7}]]
* [[Truncated order-7 heptagonal tiling|t{7,7}]]
* [[Octagonal tiling|{8,3}]]
* [[Order-4 octagonal tiling|{8,4}]]
* [[Order-6 octagonal tiling|{8,6}]]
* [[Order-8 octagonal tiling|{8,8}]]
{{col-break}}
* [[Truncated order-8 hexagonal tiling|t{6,8}]]
* [[Truncated order-8 triangular tiling|t{3,8}]]
* [[Truncated order-8 octagonal tiling|t{8,8}]]
* [[Order-3 apeirogonal tiling|{∞,3}]]
* [[Order-4 apeirogonal tiling|{∞,4}]]
* [[Order-5 apeirogonal tiling|{∞,5}]]
* [[Infinite-order apeirogonal tiling|{∞,∞}]]
* [[Truncated infinite-order triangular tiling|t{3,∞}]]
* [[Truncated infinite-order square tiling|t{4,∞}]]
{{Col-end}}

==References==
{{reflist|30em}}

== Sources ==

* {{cite book | first=Harold Scott MacDonald | last=Coxeter | author-link=Harold Scott MacDonald Coxeter | title=Regular Polytopes | publisher = Dover Publications | edition=3rd | oclc=798003 | date = 1973  | orig-year=1948 | isbn = 0-486-61480-8 | pages=[https://archive.org/details/regularpolytopes0000coxe/page/14 14], 69, 149 | url=https://archive.org/details/regularpolytopes0000coxe | url-access=registration | quote=Regular Polytopes. }}
*{{cite book |editor-first=F. Arthur |editor-last=Sherk |editor2-first=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivic |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley |year=1995 |isbn=978-0-471-01003-6 |url=https://archive.org/details/kaleidoscopessel0000coxe |url-access=registration }}
** (Paper 22) [https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA251 pp. 251–278] {{cite journal |first=H.S.M. |last=Coxeter |title=Regular and Semi Regular Polytopes I |journal=Math. Zeit. |volume=46  |pages=380–407 |year=1940 |doi=10.1007/BF01181449 |zbl=0022.38305|s2cid=186237114 }} MR 2,10
** (Paper 23) [https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA279 pp. 279–312] {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes II |journal=Math. Zeit. |volume=188 |issue=4 |pages=559–591 |year=1985 |doi=10.1007/BF01161657 |zbl=0547.52005|s2cid=120429557 }}
** (Paper 24) [https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313 pp. 313–358] {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes III |journal=Math. Zeit. |volume=200 |issue=1 |pages=3–45 |year=1988 |doi=10.1007/BF01161745 |zbl=0633.52006|s2cid=186237142 }}

== External links ==
* {{MathWorld | id = SchlaefliSymbol | title = Schläfli Symbol | access-date = December 28, 2019}}
* {{cite web | url = http://www.ac-noumea.nc/maths/polyhedr/names_.htm | title = Polyhedral Names and Notations | work = A Ride Through the Polyhedra World | first = Maurice | last = Starck | date = April 13, 2012 |  access-date = December 28, 2019}}

{{DEFAULTSORT:Schlafli symbol}}
[[Category:Polytope notation systems]]