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In geometry, the Schläfli symbol
Schläfli symbol
is a notation of the form p,q,r,... that defines regular polytopes and tessellations. The Schläfli symbol
Schläfli symbol
is named after the 19th-century Swiss mathematician Ludwig Schläfli, who made important contributions in geometry and other areas.

Contents

1 Description 2 Cases

2.1 Symmetry groups 2.2 Regular polygons (plane) 2.3 Regular polyhedra (3 dimensions) 2.4 Regular 4-polytopes (4 dimensions) 2.5 Regular n-polytopes (higher dimensions) 2.6 Dual polytopes 2.7 Prismatic polytopes

3 Extension of Schläfli symbols

3.1 Polygons and circle tilings 3.2 Polyhedra and tilings

3.2.1 Alternations, quarters and snubs 3.2.2 Altered and holosnubbed

3.3 Polychora and honeycombs

3.3.1 Alternations, quarters and snubs 3.3.2 Bifurcating families

4 Tessellations 5 See also 6 References 7 Sources 8 External links

Description[edit] The Schläfli symbol
Schläfli symbol
is a recursive description, starting with p for a p-sided 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 p-sided 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 4-dimensional 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 4-polytope, a face in a 5-polytope, a cell in a 6-polytope, and an (n-3)-face in an n-polytope. 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
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
Schläfli symbol
elements in reverse order. A self-dual regular polytope will have a symmetric Schläfli symbol. Cases[edit] Symmetry groups[edit] Schläfli symbols are closely related to (finite) reflection symmetry groups, which correspond precisely to the finite Coxeter
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
Coxeter group
for reflective tetrahedral symmetry, and [3,4] is reflective octahedral symmetry, and [3,5] is reflective icosahedral symmetry. Regular polygons (plane)[edit]

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols

The Schläfli symbol
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 s-1 is the number skipped when drawing each edge of the star. For example, 5/2 represents the pentagram. Regular polyhedra (3 dimensions)[edit] The Schläfli symbol
Schläfli symbol
of a regular polyhedron is p,q if its 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 solids, 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 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 4-polytopes (4 dimensions)[edit] The Schläfli symbol
Schläfli symbol
of a regular 4-polytope
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 and 10 regular star 4-polytopes. For example, the 120-cell
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
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 n-polytopes (higher dimensions)[edit] For higher-dimensional regular polytopes, the Schläfli symbol
Schläfli symbol
is defined recursively as p1, p2, ..., pn − 1 if the facets have Schläfli symbol
Schläfli symbol
p1,p2, ..., pn − 2 and the vertex figures have Schläfli symbol
Schläfli symbol
p2,p3, ..., pn − 1 . A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: p2,p3, ..., pn − 2 . 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. Dual polytopes[edit] If a polytope of dimension ≥ 2 has Schläfli symbol
Schläfli symbol
p1,p2, ..., pn − 1 then its dual has Schläfli symbol
Schläfli symbol
pn − 1, ..., p2,p1 . 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. Prismatic polytopes[edit] Uniform prismatic polytopes can be defined and named as a Cartesian product (with operator "×") of lower-dimensional regular polytopes.

In 0D, a point is represented by ( ). Its Coxeter diagram
Coxeter diagram
is empty. Its Coxeter notation
Coxeter notation
symmetry is ][. In 1D, a line segment is represented by . Its Coxeter diagram
Coxeter diagram
is . Its symmetry is [ ]. In 2D, a rectangle is represented as × . Its Coxeter
Coxeter
diagram is . Its symmetry is [2]. In 3D, a p-gonal 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
Coxeter diagram
is . Its symmetry is [2,p,q]. In 4D, a uniform p-q duoprism is represented as p × q . Its Coxeter diagram
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
Coxeter diagram
is . Its symmetry is [2]. In 3D, a p-gonal 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
Coxeter diagram
is . Its symmetry is [p,q]. In 4D, a p-q 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 p-gonal pyramid is represented as ( ) ∨ p .

In 4D:

A p-q-hedral pyramid is represented as ( ) ∨ p,q . A 5-cell
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[edit] Polygons and circle tilings[edit] 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 .

Form Schläfli symbol Symmetry Coxeter
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[edit] Coxeter
Coxeter
expanded his usage of the Schläfli symbol
Schläfli symbol
to quasiregular polyhedra by adding a vertical dimension to the symbol. It was a starting point toward the more general Coxeter
Coxeter
diagram. 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
Coxeter
diagram. Symbols have a corresponding alternation, replacing rings with holes in a Coxeter diagram
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 is a half form of a truncation, and a holosnub is both halves of an alternated truncation.

Form Schläfli symbols Symmetry Coxeter
Coxeter
diagram Example, 4,3

Regular

p , q

displaystyle begin Bmatrix p,qend Bmatrix

p,q t0 p,q [p,q] or [(p,q,2)]

Cube

Truncated

t

p , q

displaystyle t begin Bmatrix p,qend Bmatrix

t p,q t0,1 p,q

Truncated cube

Bitruncation (Truncated dual)

t

q , p

displaystyle t begin Bmatrix q,pend Bmatrix

2t p,q t1,2 p,q

Truncated octahedron

Rectified (Quasiregular)

p

q

displaystyle begin Bmatrix p\qend Bmatrix

r p,q t1 p,q

Cuboctahedron

Birectification (Regular dual)

q , p

displaystyle begin Bmatrix q,pend Bmatrix

2r p,q t2 p,q

Octahedron

Cantellated (Rectified rectified)

r

p

q

displaystyle r begin Bmatrix p\qend Bmatrix

rr p,q t0,2 p,q

Rhombicuboctahedron

Cantitruncated (Truncated rectified)

t

p

q

displaystyle t begin Bmatrix p\qend Bmatrix

tr p,q t0,1,2 p,q

Truncated cuboctahedron

Alternations, quarters and snubs[edit] Alternations have half the symmetry of the Coxeter
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.

Alternations

Form Schläfli symbols Symmetry Coxeter
Coxeter
diagram Example, 4,3

Alternated (half) regular

h

2 p , q

displaystyle h begin Bmatrix 2p,qend Bmatrix

h 2p,q ht0 2p,q [1+,2p,q] =

Demicube (Tetrahedron)

Snub regular

s

p , 2 q

displaystyle s begin Bmatrix p,2qend Bmatrix

s p,2q ht0,1 p,2q [p+,2q]

Snub dual regular

s

q , 2 p

displaystyle s begin Bmatrix q,2pend Bmatrix

s q,2p ht1,2 2p,q [2p,q+]

Snub octahedron (Icosahedron)

Alternated rectified (p and q are even)

h

p

q

displaystyle h begin Bmatrix p\qend Bmatrix

hr p,q ht1 p,q [p,1+,q]

Alternated rectified rectified (p and q are even)

h r

p

q

displaystyle hr begin Bmatrix p\qend Bmatrix

hrr p,q ht0,2 p,q [(p,q,2+)]

Quartered (p and q are even)

q

p

q

displaystyle q begin Bmatrix p\qend Bmatrix

q p,q ht0ht2 p,q [1+,p,q,1+]

Snub rectified Snub quasiregular

s

p

q

displaystyle s begin Bmatrix p\qend Bmatrix

sr p,q ht0,1,2 p,q [p,q]+

Snub cuboctahedron (Snub cube)

Altered and holosnubbed[edit] 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.

Altered and holosnubbed

Form Schläfli symbols Symmetry Coxeter
Coxeter
diagram Example, 4,3

Altered regular

a

p , q

displaystyle a begin Bmatrix p,qend Bmatrix

a p,q at0 p,q [p,q] = ∪

Stellated octahedron

Holosnub dual regular ß

q , p

displaystyle begin Bmatrix q,pend Bmatrix

ß
ß
q,p at0,1 q,p [p,q]

Compound of two icosahedra

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

Polychora and honeycombs[edit]

Linear families

Form Schläfli symbol Coxeter
Coxeter
diagram Example, 4,3,3

Regular

p , q , r

displaystyle begin Bmatrix p,q,rend Bmatrix

p,q,r t0 p,q,r

Tesseract

Truncated

t

p , q , r

displaystyle t begin Bmatrix p,q,rend Bmatrix

t p,q,r t0,1 p,q,r

Truncated tesseract

Rectified

p

q , r

displaystyle left begin array l p\q,rend array right

r p,q,r t1 p,q,r

Rectified tesseract =

Bitruncated

2t p,q,r t1,2 p,q,r

Bitruncated tesseract

Birectified (Rectified dual)

q , p

r

displaystyle left begin array l q,p\rend array right

2r p,q,r = r r,q,p t2 p,q,r

Rectified 16-cell =

Tritruncated (Truncated dual)

t

r , q , p

displaystyle t begin Bmatrix r,q,pend Bmatrix

3t p,q,r = t r,q,p t2,3 p,q,r

Bitruncated tesseract

Trirectified (Dual)

r , q , p

displaystyle begin Bmatrix r,q,pend Bmatrix

3r p,q,r = r,q,p t3 p,q,r = r,q,p

16-cell

Cantellated

r

p

q , r

displaystyle rleft begin array l p\q,rend array right

rr p,q,r t0,2 p,q,r

Cantellated tesseract =

Cantitruncated

t

p

q , r

displaystyle tleft begin array l p\q,rend array right

tr p,q,r t0,1,2 p,q,r

Cantitruncated tesseract =

Runcinated (Expanded)

e

3

p , q , r

displaystyle e_ 3 begin Bmatrix p,q,rend Bmatrix

e3 p,q,r t0,3 p,q,r

Runcinated tesseract

Runcitruncated

t0,1,3 p,q,r

Runcitruncated tesseract

Omnitruncated

t0,1,2,3 p,q,r

Omnitruncated tesseract

Alternations, quarters and snubs[edit]

Alternations

Form Schläfli symbol Coxeter
Coxeter
diagram Example, 4,3,3

Alternations

Half p even

h

p , q , r

displaystyle h begin Bmatrix p,q,rend Bmatrix

h p,q,r ht0 p,q,r

16-cell

Quarter p and r even

q

p , q , r

displaystyle q begin Bmatrix p,q,rend Bmatrix

q p,q,r ht0ht3 p,q,r

Snub q even

s

p , q , r

displaystyle s begin Bmatrix p,q,rend Bmatrix

s p,q,r ht0,1 p,q,r

Snub 24-cell

Snub rectified r even

s

p

q , r

displaystyle sleft begin array l p\q,rend array right

sr p,q,r ht0,1,2 p,q,r

Snub 24-cell =

Alternated duoprism

s p s q ht0,1,2,3 p,2,q

Great duoantiprism

Bifurcating families[edit]

Bifurcating families

Form Extended Schläfli symbol Coxeter
Coxeter
diagram Examples

Quasiregular

p ,

q q

displaystyle left p, q atop q right

p,q1,1 t0 p,q1,1

16-cell

Truncated

t

p ,

q q

displaystyle tleft p, q atop q right

t p,q1,1 t0,1 p,q1,1

Truncated 16-cell

Rectified

p

q

q

displaystyle left begin array l p\q\qend array right

r p,q1,1 t1 p,q1,1

24-cell

Cantellated

r

p

q

q

displaystyle rleft begin array l p\q\qend array right

rr p,q1,1 t0,2,3 p,q1,1

Cantellated 16-cell

Cantitruncated

t

p

q

q

displaystyle tleft begin array l p\q\qend array right

tr p,q1,1 t0,1,2,3 p,q1,1

Cantitruncated 16-cell

Snub rectified

s

p

q

q

displaystyle sleft begin array l p\q\qend array right

sr p,q1,1 ht0,1,2,3 p,q1,1

Snub 24-cell

Quasiregular

r ,

p q

displaystyle left r, p atop q right

r,/q,p t0 r,/q,p

Truncated

t

r ,

p q

displaystyle tleft r, p atop q right

t r,/q,p t0,1 r,/q,p

Rectified

r

p

q

displaystyle left begin array l r\p\qend array right

r r,/q,p t1 r,/q,p

Cantellated

r

r

p

q

displaystyle rleft begin array l r\p\qend array right

rr r,/q,p t0,2,3 r,/q,p

Cantitruncated

t

r

p

q

displaystyle tleft begin array l r\p\qend array right

tr r,/q,p t0,1,2,3 r,/q,p

Snub rectified

s

p

q

r

displaystyle sleft begin array l p\q\rend array right

sr p,/q,r ht0,1,2,3 p,/q,r

Tessellations[edit] Spherical

2,n s 2,2n ???

t 2, n + n

Regular

2,∞ 3,6

4,4 6,3

Semi-regular

s 4,4 e 3,6 sr 2,∞ sr 6,3 rr 6,3

r 6,3 t 6,3 t 2,∞ tr 6,3 tr 4,4

Hyper-bolic

sr 5,4 sr 6,4 sr 7,4 sr 8,4 sr ∞,4 s 5,4 sr 6,5 s 6,4 sr 8,6 sr 7,7 s 8,4 s 4,6 sr ∞,∞ sr 7,3 sr 8,3 sr ∞,3

s 3,8 3,7 3,8 3,∞ h 8,3 h 6,4 q 4,6 rr 7,3 rr 8,3 rr ∞,3 h2 8,3 r 7,3 r 8,3 t 7,3 t 8,3 r ∞,3

t ∞,3 rr 5,4 rr 6,4 rr 7,4 rr 8,4 rr ∞,4 4,5 4,6 4,7 4,8 4,∞ r 5,4 r 6,4 tr 6,3 tr 7,3 ???

tr 8,3 ??? tr ∞,3 r 7,4 r 8,4 tr 5,4 ??? tr 6,4 tr 7,4 tr 8,4 tr ∞,4 t 5,4 tr 6,5 t 6,4 tr 8,6 t 7,4

t 8,4 t ∞,4 r ∞,4 5,4 5,5 5,6 5,∞ rr 6,5 r 6,5 t 4,5 t 5,5 t 6,5 r ∞,5 6,4 6,5 6,6

6,8 rr 8,6 r 8,6 t 4,6 t 5,6 t 6,6 t 8,6 7,3 7,4 7,7 t 3,7 t 4,7 t 7,7 8,3 8,4 8,6 8,8

t 6,8 t 3,8 t 8,8 ∞,3 ∞,4 ∞,5 ∞,∞ t 3,∞ t 4,∞

See also[edit]

Regular Polytopes, by Harold Scott MacDonald Coxeter

References[edit]

Sources[edit]

Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. pp. 14, 69, 149. ISBN 0-486-61480-8. OCLC 798003.  Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited 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[edit]

Weisstein, Eric W. "Schläfli symbol". MathWorld.  Polyhedral

.