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| [[File:Dodecahedron.png|thumb|The [[dodecahedron]] is a regular polyhedron with Schläfli symbol {5,3}, having 3 [[pentagon]]s around each [[Vertex (geometry)|vertex]].]]
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| In [[geometry]], the '''Schläfli symbol''' is a notation of the form {p,q,r,...} that defines [[list of regular polytopes|regular polytopes and tessellations]].
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| The Schläfli symbol is named after the 19th-century mathematician [[Ludwig Schläfli]] who made important contributions in [[geometry]] and other areas.
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| == Description ==
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| The Schläfli symbol is a [[Recursive definition|recursive]] description, starting with a ''p''-sided [[regular polygon]] as ''{p}''. For example, {3} is an [[equilateral triangle]], {4} is a [[Square (geometry)|square]] and so on.
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| A regular polyhedron which has ''q'' regular p-sided [[Face (geometry)|polygon faces]] around each [[Vertex (geometry)|vertex]] is represented by {p,q}. For example, the [[cube]] has 3 squares around each vertex and is represented by {4,3}.
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| A regular 4-dimensional polytope, with ''r'' {p,q} regular [[Face (geometry)|polyhedral cells]] around each edge is represented by {p,q,r}, and so on.
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| Regular polytopes can have [[star polygon]] elements, like the [[pentagram]], with symbol {5/2}, represented by the vertices of a [[pentagon]] but connected alternately.
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| A [[Facet (mathematics)|facet]] of a regular polytope {p,q,r,...,y,z} is {p,q,r,...,y}.
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| 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}.
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| 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 will tessellate space of the same dimension as the facets. A negative angle defect can't exist in ordinary space, but can be constructed in hyperbolic space.
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| Usually a vertex figure is assumed to be a finite polytope, but can sometimes be considered a tessellation itself.
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| 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.
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| == Symmetry groups ==
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| A Schläfli symbol is closely related to [[Reflection symmetry|reflection]] [[symmetry group]]s, also called [[Coxeter group]]s, given 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]], and [3,4] is reflective [[octahedral symmetry]], and [3,5] is reflective [[icosahedral symmetry]].
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| == Regular polygons (plane) ==
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| The Schläfli symbol of a regular [[polygon]] with ''n'' edges is {''n''}.
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| For example, a regular [[pentagon]] is represented by {5}.
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| See the convex [[regular polygon]] and nonconvex [[star polygon]].
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| For example, {5/2} is the [[pentagram]].
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| == Regular polyhedra (3-space) ==
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| 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).
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| For example {5,3} is the regular [[dodecahedron]]. It has pentagonal (5 edges) faces, and 3 pentagons around each vertex.
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| See the 5 convex [[Platonic solid]]s, the 4 nonconvex [[Kepler-Poinsot polyhedra]].
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| Schläfli symbols may also be defined for regular [[tessellation]]s of [[Euclidean geometry|Euclidean]] or [[hyperbolic geometry|hyperbolic]] space in a similar way.
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| For example, the [[hexagonal tiling]] is represented by {6,3}. | |
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| == Regular polychora (4-space) ==
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| The Schläfli symbol of a regular [[polychoron]] 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''}).
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| See the six [[convex regular 4-polytope|convex regular]] and 10 [[List of regular polytopes#Nonconvex forms (4D)|nonconvex polychora]].
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| 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.
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| There is also 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.
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| There are also 4 regular hyperbolic tessellations including {5,3,4}, the [[Hyperbolic small dodecahedral honeycomb]], which fills space with [[dodecahedron]] cells.
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| == Higher dimensions ==
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| For higher dimensional [[polytope]]s, the Schläfli symbol is defined recursively as {''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n'' − 1</sub>} if the [[facet (mathematics)|facet]]s have Schläfli symbol {''p''<sub>1</sub>,''p''<sub>2</sub>, ..., ''p''<sub>''n'' − 2</sub>} and the
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| [[vertex figure]]s have Schläfli symbol {''p''<sub>2</sub>,''p''<sub>3</sub>, ..., ''p''<sub>''n'' − 1</sub>}.
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| 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: {''p''<sub>2</sub>,''p''<sub>3</sub>, ..., ''p''<sub>''n'' − 2</sub>}.
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| 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.
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| == Dual polytopes ==
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| If a polytope of dimension ≥ 2 has Schläfli symbol {''p''<sub>1</sub>,''p''<sub>2</sub>, ..., ''p''<sub>''n'' − 1</sub>} then its [[dual polyhedron|dual]] has Schläfli symbol {''p''<sub>''n'' − 1</sub>, ..., ''p''<sub>2</sub>,''p''<sub>1</sub>}.
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| If the sequence is [[palindromic]], i.e. the same forwards and backwards, the polytope is ''self-dual''. Every regular polytope in 2 dimensions (polygon) is self-dual.
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| == Uniform prismatic polytopes ==
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| [[Prism (geometry)#Uniform prismatic polytope|Uniform prismatic polytopes]] can be defined and named as a [[Cartesian product]] of lower dimensional regular polytopes:
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| * A '''''p''-gonal [[Prism (geometry)|prism]]''', with vertex figure ''p''.4.4 as { } × {p}. The symbol { } means a [[digon]] or [[line segment]].
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| * A uniform '''{p,q}-hedral prism''' as { } × {p,q}.
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| * A uniform '''p-q [[duoprism]]''' as {p} × {q}.
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| == Extension of Schläfli symbols ==
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| === Polyhedra and tilings ===
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| [[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-Dynkin 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-Dynkin diagram]]. All of the symbols have a corresponding [[Alternation (geometry)|alternation]], replacing ''rings'' with ''holes'' in a Coxeter diagram and ''h'' prefix, construction limited by the requirement that neighboring branches must be even-ordered.
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| {| class=wikitable
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| !Form
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| !colspan=3|Extended Schläfli symbols
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| ![[Coxeter notation|Symmetry]]
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| !colspan=2|[[Coxeter diagram]]
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| !colspan=3|Example, {4,3}
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| |- align=center
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| ![[Regular polyhedron|Regular]]
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| |<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>||{p,q}||t<sub>0</sub>{p,q}
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| |rowspan=7|[p,q]<BR>or<BR>[(p,q,2)]
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| |colspan=2|{{CDD|node_1|p|node|q|node}}
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| |[[File:Hexahedron.png|40px]]||[[Cube]]||{{CDD|node_1|4|node|3|node}}
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| |- align=center
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| ![[Truncation (geometry)|Truncated]]
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| | <math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>||t{p,q}||t<sub>0,1</sub>{p,q}
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| |colspan=2| {{CDD|node_1|p|node_1|q|node}}
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| |[[File:Truncated hexahedron.png|40px]]||[[Truncated cube]]||{{CDD|node_1|4|node_1|3|node}}
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| |- align=center
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| ![[Bitruncation]]<BR>(Truncated dual)
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| | <math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>||2t{p,q}||t<sub>1,2</sub>{p,q}
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| |{{CDD|node_1|q|node_1|p|node}}||{{CDD|node|p|node_1|q|node_1}}
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| |[[File:Truncated octahedron.png|40px]]||[[Truncated octahedron]]||{{CDD|node|4|node_1|3|node_1}}
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| |- align=center
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| ![[Rectification (geometry)|Rectified]]<BR>([[Quasiregular polyhedron|Quasiregular]])
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| | <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||r{p,q}||t<sub>1</sub>{p,q}
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| ||{{CDD|node_1|split1-pq|nodes}}|| {{CDD|node|p|node_1|q|node}}
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| |[[File:Cuboctahedron.png|40px]]||[[Cuboctahedron]]||{{CDD|node|4|node_1|3|node}}
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| |- align=center
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| !Birectification<BR>(Regular dual)
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| |<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>||2r{p,q}||t<sub>2</sub>{p,q}
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| |{{CDD|node_1|q|node|p|node}}||{{CDD|node|p|node|q|node_1}}
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| |[[File:Octahedron.png|40px]]||[[Octahedron]]||{{CDD|node|4|node|3|node_1}}
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| |- align=center
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| ![[Cantellation (geometry)|Cantellated]]<BR>([[Expansion (geometry)|Rectified rectified]])
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| | <math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||rr{p,q}||t<sub>0,2</sub>{p,q}
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| ||{{CDD|node|split1-pq|nodes_11}}|| {{CDD|node_1|p|node|q|node_1}}
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| |[[File:Small rhombicuboctahedron.png|40px]]||[[Rhombicuboctahedron]]||{{CDD|node_1|4|node|3|node_1}}
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| |- align=center
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| ![[Omnitruncation|Cantitruncated]]<BR>(Truncated rectified)
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| | <math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||tr{p,q}||t<sub>0,1,2</sub>{p,q}
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| ||{{CDD|node_1|split1-pq|nodes_11}}|| {{CDD|node_1|p|node_1|q|node_1}}
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| |[[File:Great rhombicuboctahedron.png|40px]]||[[Truncated cuboctahedron]]||{{CDD|node_1|4|node_1|3|node_1}}
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| |- align=center
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| !colspan=10|[[Alternation (geometry)|Alternations]]
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| |- align=center
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| !Alternated regular<BR>(p is even)
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| | <math>h \begin{Bmatrix} p , q \end{Bmatrix}</math>||h{p,q}||ht<sub>0</sub>{p,q}||[1<sup>+</sup>,p,q]
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| |colspan=2| {{CDD|node_h1|p|node|q|node}}
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| |[[File:Tetrahedron.png|40px]]||Demicube<BR>([[Tetrahedron]])||{{CDD|node_h1|4|node|3|node}}
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| |- align=center
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| !Snub regular<BR>(q is even)
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| | <math>s\begin{Bmatrix} p , q \end{Bmatrix}</math>||s{p,q}||ht<sub>0,1</sub>{p,q}||[p<sup>+</sup>,q]
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| |colspan=2| {{CDD|node_h|p|node_h|q|node}}
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| ||| ||
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| |- align=center
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| !Snub dual regular<BR>(p is even)
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| | <math>s \begin{Bmatrix} q , p \end{Bmatrix}</math>||s{q,p}||ht<sub>1,2</sub>{p,q}||[p,q<sup>+</sup>]
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| |{{CDD|node_h|q|node_h|p|node}}||{{CDD|node|p|node_h|q|node_h}}
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| |[[File:Uniform_polyhedron-43-h01.svg|40px]]||Snub octahedron<BR>([[Icosahedron]])||{{CDD|node|4|node_h|3|node_h}}
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| |- align=center
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| !Alternated dual regular<BR>(q is even)
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| | <math>h \begin{Bmatrix} q , p \end{Bmatrix}</math>||h{q,p}||ht<sub>2</sub>{p,q}||[p,q,1<sup>+</sup>]
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| |{{CDD|node_h1|q|node|p|node}}||{{CDD|node|p|node|q|node_h1}}
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| |- align=center
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| !Alternated rectified<BR>(p and q are even)
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| | <math>h \begin{Bmatrix} p \\ q \end{Bmatrix}</math>||hr{p,q}||ht<sub>1</sub>{p,q}||[p,1<sup>+</sup>,q]
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| ||{{CDD|node_h1|split1-pq|nodes}}|| {{CDD|node|p|node_h1|q|node}}
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| |- align=center
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| !Alternated rectified rectified<BR>(p and q are even)
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| | <math>hr\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||hrr{p,q}||ht<sub>0,2</sub>{p,q}||[(p,q,2<sup>+</sup>)]
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| ||{{CDD|node|split1-pq|nodes_hh}}|| {{CDD|node_h|p|node|q|node_h}}
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| |- align=center
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| !Quartered<BR>(p and q are even)
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| | <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>]
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| ||{{CDD|node|split1-pq|nodes_h1h1}}|| {{CDD|node_h1|p|node|q|node_h1}}
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| |- align=center
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| !Snub rectified<BR>Snub quasiregular
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| | <math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||sr{p,q}||ht<sub>0,1,2</sub>{p,q}||[p,q]<sup>+</sup>
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| ||{{CDD|node_h|split1-pq|nodes_hh}}|| {{CDD|node_h|p|node_h|q|node_h}}
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| |[[File:Snub hexahedron.png|40px]]||[[Snub cuboctahedron]]<BR>(Snub cube)||{{CDD|node_h|4|node_h|3|node_h}}
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| |}
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| === Polychora and honeycombs ===
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| {| class="wikitable"
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| |+ Linear families
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| |-
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| !Form
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| !colspan=3|Extended Schläfli symbol
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| !colspan=2|[[Coxeter diagram]]
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| !colspan=3|Example, {4,3,3}
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| |- align=center
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| !Regular
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| | <math>\begin{Bmatrix} p, q , r \end{Bmatrix}</math>||{p,q,r}
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| |t<sub>0</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node|q|node|r|node}}
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| |[[File:Schlegel_wireframe_8-cell.png|40px]]||[[Tesseract]]||{{CDD|node_1|4|node|3|node|3|node}}
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| |- align=center
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| !Truncated
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| | <math>t\begin{Bmatrix} p, q , r \end{Bmatrix}</math> ||t{p,q,r}
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| |t<sub>0,1</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node_1|q|node|r|node}}
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| |[[File:Schlegel_half-solid_truncated_tesseract.png|40px]]||[[Truncated tesseract]]||{{CDD|node_1|4|node_1|3|node|3|node}}
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| |- align=center
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| !Rectified
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| | <math>\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math>
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| ||r{p,q,r}
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| |t<sub>1</sub>{p,q,r}
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| |colspan=2|{{CDD|node|p|node_1|q|node|r|node}}
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| |[[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}}
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| |- align=center
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| !Bitruncated
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| | ||2t{p,q,r}
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| |t<sub>1,2</sub>{p,q,r}
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| |colspan=2|{{CDD|node|p|node_1|q|node_1|r|node}}
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| |[[File:Schlegel half-solid bitruncated 16-cell.png|40px]]||[[Bitruncated tesseract]]||{{CDD|node|4|node_1|3|node_1|3|node}}
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| |- align=center
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| !Birectified<BR>(Rectified dual)
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| |<!--<math>\begin{Bmatrix} q , p \\ r \end{Bmatrix}</math><BR>--><math>\left\{\begin{array}{l}q,p\\r\end{array}\right\}</math>
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| ||2r{p,q,r} = r{r,q,p}
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| |t<sub>2</sub>{p,q,r}
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| |colspan=2|{{CDD|node|p|node|q|node_1|r|node}}
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| |[[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}}
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| |- align=center
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| !Tritruncated<BR>(Truncated dual)
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| | <math>t\begin{Bmatrix} r, q , p \end{Bmatrix}</math> ||3t{p,q,r} = t{r,q,p}
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| |t<sub>2,3</sub>{p,q,r}
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| |colspan=2|{{CDD|node|p|node|q|node_1|r|node_1}}
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| |[[File:Schlegel half-solid truncated 16-cell.png|40px]]||[[Bitruncated tesseract]]||{{CDD|node|4|node|3|node_1|3|node_1}}
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| |- align=center
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| !Trirectified<BR>(Dual)
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| | <math>\begin{Bmatrix} r, q , p \end{Bmatrix}</math>||3r{p,q,r} = {r,q,p}
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| |t<sub>3</sub>{p,q,r} = {r,q,p}
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| |colspan=2|{{CDD|node|p|node|q|node|r|node_1}}
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| |[[File:Schlegel_wireframe_16-cell.png|40px]]||[[16-cell]]||{{CDD|node|4|node|3|node|3|node_1}}
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| |- align=center
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| !Cantellated
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| | <math>r\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math> ||rr{p,q,r}
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| |t<sub>0,2</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node|q|node_1|r|node}}
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| |[[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}}
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| |- align=center
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| !Cantitruncated
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| | <math>t\left\{\begin{array}{l}p\\q,r\end{array}\right\}</math> ||tr{p,q,r}
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| |t<sub>0,1,2</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node_1|q|node_1|r|node}}
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| |[[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}}
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| |- align=center
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| !Runcinated<BR>([[Expansion (geometry)|Expanded]])
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| | <math>e\begin{Bmatrix} p, q , r \end{Bmatrix}</math> || e{p,q,r}
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| |t<sub>0,3</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node|q|node|r|node_1}}
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| |[[File:Schlegel_half-solid_runcinated_8-cell.png|40px]]||[[Runcinated tesseract]]||{{CDD|node_1|4|node|3|node|3|node_1}}
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| |- align=center
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| !Runcitruncated
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| |t<sub>0,1,3</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node_1|q|node|r|node_1}}
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| |[[File:Schlegel_half-solid_runcitruncated_8-cell.png|40px]]||[[Runcitruncated tesseract]]||{{CDD|node_1|4|node_1|3|node|3|node_1}}
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| |- align=center
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| !Omnitruncated
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| |t<sub>0,1,2,3</sub>{p,q,r}
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| |colspan=2|{{CDD|node_1|p|node_1|q|node_1|r|node_1}}
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| |[[File:Schlegel_half-solid_omnitruncated_8-cell.png|40px]]||[[Omnitruncated tesseract]]||{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
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| |- align=center
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| !colspan=10|[[Alternation (geometry)|Alternations]]
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| |- align=center
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| !Half<BR>p even
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| | <math>h\begin{Bmatrix} p, q , r \end{Bmatrix}</math>||h{p,q,r}
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| |ht<sub>0</sub>{p,q,r}
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| |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}}
| |
| |}
| |
| | |
| {| 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]]||[[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 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]]||[[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_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_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 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}}
| |
| | || || {{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}}
| |
| | || || {{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}}
| |
| | || || {{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}}
| |
| | || || {{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}}
| |
| | || || {{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}}
| |
| | || || {{CDD|node_h|3|node_h|split1-43|nodes_hh}}
| |
| | |
| |}
| |
| | |
| == References ==
| |
| * [[Coxeter|Coxeter, H.S.M.]]; ''[[Regular Polytopes (book)|Regular Polytopes]]'', (Methuen and Co., 1948). (pp. 14, 69, 149) [http://books.google.com/books?id=iWvXsVInpgMC&lpg=PP1&dq=Regular%20Polytopes&pg=PP1#v=onepage&q=&f=false]
| |
| * ''[http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Kaleidoscopes: Selected Writings of H.S.M. Coxeter]'', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
| |
| ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
| |
| ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
| |
| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
| |
| | |
| == External links ==
| |
| * {{mathworld | urlname = SchlaefliSymbol | title = Schläfli symbol}}
| |
| * [http://bbs.sachina.pku.edu.cn/Stat/Math_World/math/w/w166.htm Wythoff Symbol and generalized Schläfli Symbols]
| |
| * [http://www.ac-noumea.nc/maths/polyhedr/names_.htm polyhedral names et notations]
| |
| | |
| {{DEFAULTSORT:Schlafli symbol}}
| |
| [[Category:Polytopes]]
| |
| [[Category:Mathematical notation]]
| |