Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19thcentury Swiss mathematician Ludwig Schläfli, who made important contributions in geometry and other areas.
Description
The Schläfli symbol is a recursive description, starting with {p} for a psided regular polygon that is convex. For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon and so on. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices. For example, {5/2} is a pentagram.
A regular polyhedron that has q regular psided polygon faces around each vertex is represented by {p,q}. For example, the cube has 3 squares around each vertex and is represented by {4,3}.
A regular 4dimensional polytope, with r {p,q} regular polyhedral cells around each edge is represented by {p,q,r}. For example a tesseract, {4,3,3}, has 3 cubes, {4,3}, around an edge.
In general a regular polytope {p,q,r,...,y,z} has z {p,q,r,...,y} facets around every peak, where a peak is a vertex in a polyhedron, an edge in a 4polytope, a face in a 5polytope, a cell in a 6polytope, and an (n3)face in an npolytope.
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 {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 polytope, represented by the Schläfli symbol elements in reverse order. A selfdual regular polytope will have a symmetric Schläfli symbol.
Cases
Symmetry groups
Schläfli symbols are closely related to (finite) reflection symmetry groups, which correspond precisely to the finite Coxeter groups 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, and [3,4] is reflective octahedral symmetry, and [3,5] is reflective icosahedral symmetry.
Regular polygons (plane)
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 polygons, the constructive notation p/s is used, where p is the number of vertices and s1 is the number skipped when drawing each edge of the star. For example, {5/2} represents the pentagram.
Regular polyhedra (3 dimensions)
The Schläfli symbol of a regular polyhedron is {p,q} if its faces are pgons, and each vertex is surrounded by q faces (the vertex figure is a qgon).
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 solids, the 4 nonconvex KeplerPoinsot polyhedra.
Topologically, a regular 2dimensional tessellation may be regarded as similar to a (3dimensional) polyhedron, but such that the angular defect is zero. Thus, Schläfli symbols may also be defined for regular tessellations of Euclidean or 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 4polytopes (4 dimensions)
The Schläfli symbol of a regular 4polytope is of the form {p,q,r}. Its (twodimensional) faces are regular pgons ({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 rgons (type {r}).
See the six convex regular and 10 regular star 4polytopes.
For example, the 120cell 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 3space: 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.
Regular npolytopes (higher dimensions)
For higherdimensional regular polytopes, the Schläfli symbol is defined recursively as {p_{1}, p_{2}, ..., p_{n − 1}} if the facets have Schläfli symbol {p_{1},p_{2}, ..., p_{n − 2}} and the vertex figures have Schläfli symbol {p_{2},p_{3}, ..., p_{n − 1}}.
A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: {p_{2},p_{3}, ..., p_{n − 2}}.
There are only 3 regular polytopes in 5 dimensions and above: the simplex, {3,3,3,...,3}; the crosspolytope, {3,3, ..., 3,4}; and the hypercube, {4,3,3,...,3}. There are no nonconvex regular polytopes above 4 dimensions.
Dual polytopes
If a polytope of dimension ≥ 2 has Schläfli symbol {p_{1},p_{2}, ..., p_{n − 1}} then its dual has Schläfli symbol {p_{n − 1}, ..., p_{2},p_{1}}.
If the sequence is palindromic, i.e. the same forwards and backwards, the polytope is selfdual. Every regular polytope in 2 dimensions (polygon) is selfdual.
Prismatic polytopes
Uniform prismatic polytopes can be defined and named as a Cartesian product (with operator "×") of lowerdimensional regular polytopes.
 In 0D, a 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
. Its symmetry is [ ].  In 2D, a rectangle is represented as { } × { }. Its Coxeter diagram is
. Its symmetry is [2].  In 3D, a pgonal prism is represented as { } × {p}. Its Coxeter diagram is
. Its symmetry is [2,p].  In 4D, a uniform {p,q}hedral prism is represented as { } × {p,q}. Its Coxeter diagram is
. Its symmetry is [2,p,q].  In 4D, a uniform pq duoprism is represented as {p} × {q}. Its Coxeter diagram is
. Its symmetry is [p,2,q].
The prismatic duals, or bipyramids can be represented as composite symbols, but with the addition operator, "+".
 In 2D, a rhombus is represented as { } + { }. Its Coxeter diagram is
. Its symmetry is [2].  In 3D, a pgonal bipyramid, is represented as { } + {p}. Its Coxeter diagram is
. Its symmetry is [2,p].  In 4D, a {p,q}hedral bipyramid is represented as { } + {p,q}. Its Coxeter diagram is
. Its symmetry is [p,q].  In 4D, a pq duopyramid is represented as {p} + {q}. Its Coxeter diagram is
. 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 pgonal pyramid is represented as ( ) ∨ {p}.
In 4D:
 A pqhedral pyramid is represented as ( ) ∨ {p,q}.
 A 5cell 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 evensided regular 2ngon generates a star figure compound, 2{n}.
Form  Schläfli symbol  Symmetry  Coxeter diagram  Example, {6}  

Regular  {p}  [p]  Hexagon  
Truncated  t{p} = {2p}  [[p]] = [2p]  Truncated hexagon (Dodecagon)  
Altered and Holosnubbed 
a{2p} = β{p}  [2p]  Altered hexagon (Hexagram)  
Half and Snubbed 
h{2p} = s{p} = {p}  [1^{+},2p] = [p]  Half hexagon (Triangle) 
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 simplified the notation for vertical symbols with an r prefix. The tnotation is the most general, and directly corresponds to the rings of the Coxeter diagram. Symbols have a corresponding 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 evenordered 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 is a half form of a truncation, and a holosnub is both halves of an alternated truncation.
Form  Schläfli symbols  Symmetry  Coxeter diagram  Example, {4,3}  

Regular  {p,q}  t_{0}{p,q}  [p,q] or [(p,q,2)] 
Cube  
Truncated  t{p,q}  t_{0,1}{p,q}  Truncated cube  
Bitruncation (Truncated dual) 
2t{p,q}  t_{1,2}{p,q}  Truncated octahedron  
Rectified (Quasiregular) 
r{p,q}  t_{1}{p,q}  Cuboctahedron  
Birectification (Regular dual) 
2r{p,q}  t_{2}{p,q}  Octahedron  
Cantellated (Rectified rectified) 
rr{p,q}  t_{0,2}{p,q}  Rhombicuboctahedron  
Cantitruncated (Truncated rectified) 
tr{p,q}  t_{0,1,2}{p,q}  Truncated cuboctahedron 
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 doesn't imply which one. Quarter forms are shown here with a + inside a hollow ring to imply they are two independent alternations.
Form  Schläfli symbols  Symmetry  Coxeter diagram  Example, {4,3}  

Alternated (half) regular  h{2p,q}  ht_{0}{2p,q}  [1^{+},2p,q]  Demicube (Tetrahedron)  
Snub regular  s{p,2q}  ht_{0,1}{p,2q}  [p^{+},2q]  
Snub dual regular  s{q,2p}  ht_{1,2}{2p,q}  [2p,q^{+}]  Snub octahedron (Icosahedron)  
Alternated rectified (p and q are even) 
hr{p,q}  ht_{1}{p,q}  [p,1^{+},q]  
Alternated rectified rectified (p and q are even) 
hrr{p,q}  ht_{0,2}{p,q}  [(p,q,2^{+})]  
Quartered (p and q are even) 
q{p,q}  ht_{0}ht_{2}{p,q}  [1^{+},p,q,1^{+}]  
Snub rectified Snub quasiregular 
sr{p,q}  ht_{0,1,2}{p,q}  [p,q]^{+}  Snub cuboctahedron (Snub cube) 
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.
Form  Schläfli symbols  Symmetry  Coxeter diagram  Example, {4,3}  

Altered regular  a{p,q}  at_{0}{p,q}  [p,q]  Stellated octahedron  
Holosnub dual regular  ß{ q, p }  ß{q,p}  at_{0,1}{q,p}  [p,q]  Compound of two icosahedra 
Polychora and honeycombs
Form  Schläfli symbol  Coxeter diagram  Example, {4,3,3}  

Regular  {p,q,r}  t_{0}{p,q,r}  Tesseract  
Truncated  t{p,q,r}  t_{0,1}{p,q,r}  Truncated tesseract  
Rectified  r{p,q,r}  t_{1}{p,q,r}  Rectified tesseract  
Bitruncated  2t{p,q,r}  t_{1,2}{p,q,r}  Bitruncated tesseract  
Birectified (Rectified dual) 
2r{p,q,r} = r{r,q,p}  t_{2}{p,q,r}  Rectified 16cell  
Tritruncated (Truncated dual) 
3t{p,q,r} = t{r,q,p}  t_{2,3}{p,q,r}  Bitruncated tesseract  
Trirectified (Dual) 
3r{p,q,r} = {r,q,p}  t_{3}{p,q,r} = {r,q,p}  16cell  
Cantellated  rr{p,q,r}  t_{0,2}{p,q,r}  Cantellated tesseract  
Cantitruncated  tr{p,q,r}  t_{0,1,2}{p,q,r}  Cantitruncated tesseract  
Runcinated (Expanded) 
e_{3}{p,q,r}  t_{0,3}{p,q,r}  Runcinated tesseract  
Runcitruncated  t_{0,1,3}{p,q,r}  Runcitruncated tesseract  
Omnitruncated  t_{0,1,2,3}{p,q,r}  Omnitruncated tesseract 
Alternations, quarters and snubs
Form  Schläfli symbol  Coxeter diagram  Example, {4,3,3}  

Alternations  
Half p even 
h{p,q,r}  ht_{0}{p,q,r}  16cell  
Quarter p and r even 
q{p,q,r}  ht_{0}ht_{3}{p,q,r}  
Snub q even 
s{p,q,r}  ht_{0,1}{p,q,r}  Snub 24cell  
Snub rectified r even 
sr{p,q,r}  ht_{0,1,2}{p,q,r}  Snub 24cell  
Alternated duoprism  s{p}s{q}  ht_{0,1,2,3}{p,2,q}  Great duoantiprism 
Bifurcating families
Form  Extended Schläfli symbol  Coxeter diagram  Examples  

Quasiregular  {p,q^{1,1}}  t_{0}{p,q^{1,1}}  16cell  
Truncated  t{p,q^{1,1}}  t_{0,1}{p,q^{1,1}}  Truncated 16cell  
Rectified  r{p,q^{1,1}}  t_{1}{p,q^{1,1}}  24cell  
Cantellated  rr{p,q^{1,1}}  t_{0,2,3}{p,q^{1,1}}  Cantellated 16cell  
Cantitruncated  tr{p,q^{1,1}}  t_{0,1,2,3}{p,q^{1,1}}  Cantitruncated 16cell  
Snub rectified  sr{p,q^{1,1}}  ht_{0,1,2,3}{p,q^{1,1}}  Snub 24cell  
Quasiregular  {r,/q\,p}  t_{0}{r,/q\,p}  
Truncated  t{r,/q\,p}  t_{0,1}{r,/q\,p}  
Rectified  r{r,/q\,p}  t_{1}{r,/q\,p}  
Cantellated  rr{r,/q\,p}  t_{0,2,3}{r,/q\,p}  
Cantitruncated  tr{r,/q\,p}  t_{0,1,2,3}{r,/q\,p}  
Snub rectified  sr{p,/q,\r}  ht_{0,1,2,3}{p,/q\,r} 
Tessellations


Regular


Semiregular
Hyperbolic







See also
References
Sources
 Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. pp. 14, 69, 149. ISBN 0486614808. OCLC 798003.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995,
ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]