geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Schläfli symbol is a notation of the form

\

that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician

Ludwig Schläfli Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...

, who generalized

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.

Definition

The Schläfli symbol is a

recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

description, starting with for a ''p''-sided

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

that is

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

. For example, is an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

, is a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, a convex

regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

, etc. Regular star polygons are not convex, and their Schläfli symbols contain

irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). ...

s ^''p''/_''q'', where ''p'' is the number of vertices, and ''q'' is their

turning number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...

. Equivalently, is created from the vertices of , connected every ''q''. For example, is a

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

; is a

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

. A

regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

that has ''q'' regular ''p''-sided polygon faces around each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

is represented by . For example, the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

has 3 squares around each vertex and is represented by . A regular 4-dimensional polytope, with ''r'' regular polyhedral cells around each edge is represented by . For example, a

tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

, , has 3

s, , around an edge. In general, a

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

has ''z''

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

s around every

peak Peak or The Peak may refer to: Basic meanings Geology * Mountain peak ** Pyramidal peak, a mountaintop that has been sculpted by erosion to form a point Mathematics * Peak hour or rush hour, in traffic congestion * Peak (geometry), an (''n''-3)-di ...

, where a peak is a

in a polyhedron, an edge in a 4-polytope, a

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

in a 5-polytope, and an (''n''-3)-face in an ''n''-polytope.

Properties

A regular polytope has a regular

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

. The vertex figure of a regular polytope is . Regular polytopes can have star polygon elements, like the

, with symbol , represented by the vertices of a

but connected alternately. The Schläfli symbol can represent a finite

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

, an infinite

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, or an infinite tessellation of

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...

, depending on the

angle defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the de ...

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 In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

, 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.

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 John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...

rediscovered the Schläfli symbol which he wrote as , ''p'' , ''q'' , ''r'' , ... , ''z'' , rather than with brackets and commas as Schläfli did. 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 , ''p'' as an operator, which can be applied to , ''q'' , ... , ''z'' , to produce a polytope with ''p''-gonal faces whose vertex figure is , ''q'' , ... , ''z'' , .

Cases

Symmetry groups

Schläfli symbols are closely related to (finite)

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

s, which correspond precisely to the finite

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

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, ,3is the Coxeter group for reflective

tetrahedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...

, ,4is reflective

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

, and ,5is reflective

icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...

Regular polygons (plane)

The Schläfli symbol of a (convex)

with ''p'' edges is . For example, a regular

is represented by . For (nonconvex) star polygons, the constructive notation is used, where ''p'' is the number of vertices and is the number of vertices skipped when drawing each edge of the star. For example, represents the

Regular polyhedra (3 dimensions)

The Schläfli symbol of a regular

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

is if its

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

are ''p''-gons, and each vertex is surrounded by ''q'' faces (the

is a ''q''-gon). For example, is the regular

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

. It has pentagonal (5 edges) faces, and 3 pentagons around each vertex. See the 5 convex

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s, the 4 nonconvex Kepler-Poinsot polyhedra. Topologically, a regular 2-dimensional

may be regarded as similar to a (3-dimensional) polyhedron, but such that the

angular defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the defe ...

is zero. Thus, Schläfli symbols may also be defined for regular

s of Euclidean or

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

space in a similar way as for polyhedra. The analogy holds for higher dimensions. For example, the hexagonal tiling is represented by .

Regular 4-polytopes (4 dimensions)

The Schläfli symbol of a regular 4-polytope is of the form . Its (two-dimensional) faces are regular ''p''-gons (), the cells are regular polyhedra of type , the vertex figures are regular polyhedra of type , and the edge figures are regular ''r''-gons (type ). See the six convex regular and 10 regular star 4-polytopes. For example, the

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...

is represented by . It is made of

cells , and has 3 cells around each edge. There is one regular tessellation of Euclidean 3-space: the

cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a re ...

, with a Schläfli symbol of , made of cubic cells and 4 cubes around each edge. There are also 4 regular compact hyperbolic tessellations including , the

hyperbolic small dodecahedral honeycomb In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) of hyperbolic 3-space. With Schläfli symbol it has four regular dodeca ...

, which fills space with

cells.

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

For higher-dimensional

s, the Schläfli symbol is defined recursively as if the

s have Schläfli symbol and the

s have Schläfli symbol . A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: . There are only 3 regular polytopes in 5 dimensions and above: the

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

, ; the cross-polytope, ; and the

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...

, . There are no non-convex regular polytopes above 4 dimensions.

Dual polytopes

If a polytope of dimension n ≥ 2 has Schläfli symbol then its dual has Schläfli symbol . If the sequence is

palindromic A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Pana ...

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

Prismatic polytopes

Uniform prismatic polytopes can be defined and named as a

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

(with operator "×") of lower-dimensional regular polytopes. * In 0D, a

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

is represented by ( ). Its

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

is empty. Its

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

symmetry is ] * In 1D, a line segment is represented by . Its

is . Its symmetry is [ ]. * In 2D, a rectangle is represented as × . Its

is . Its symmetry is * In 3D, a ''p''-gonal

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

is represented as × . Its Coxeter diagram is . Its symmetry is ,''p'' * In 4D, a uniform -hedral prism is represented as × . Its Coxeter diagram is . Its symmetry is ,''p'',''q'' * In 4D, a uniform ''p''-''q''

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

is represented as × . 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 In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

is represented as + . Its Coxeter diagram is . Its symmetry is * In 3D, a ''p''-gonal bipyramid, is represented as + . Its Coxeter diagram is . Its symmetry is ,''p'' * In 4D, a -hedral bipyramid is represented as + . Its Coxeter diagram is . Its symmetry is 'p'',''q'' * In 4D, a ''p''-''q''

duopyramid In geometry of 4 dimensions or higher, a double pyramid or duopyramid or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhom ...

is represented as + . 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 In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

can be represented as ( ) ∨ = ( ) ∨ ) ∨ ( ) In 3D: * A digonal disphenoid can be represented as ∨ = ) ∨ ( )∨ ) ∨ ( ) * A ''p-gonal pyramid'' is represented as ( ) ∨ . In 4D: * A ''p-q-hedral pyramid'' is represented as ( ) ∨ . *A

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...

is represented as ( ) ∨ ) ∨ or ) ∨ ( )∨ = ∨ . *A square pyramidal pyramid is represented as ( ) ∨ ) ∨ or ) ∨ ( )∨ = ∨ . When mixing operators, the

order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For exampl ...

from highest to lowest is ×, +, ∨. Axial polytopes containing vertices on parallel offset hyperplanes can be represented by the , , operator. A uniform prism is , , and antiprism , , ''r''.

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 In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but can also include disconnected sets of edges, called a compound polygon. For example, a regular pentagram, ...

compound, 2.

Polyhedra and tilings

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

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

. 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, 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 A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub. In Awards and Lists For awards, the term "snub" ...

is a half form of a truncation, and a holosnub is both halves of an alternated truncation.

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.

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. : ß, looking similar to the greek letter

beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...

(β), is the German alphabet letter eszett.

Polychora and honeycombs

Alternations, quarters and snubs

Bifurcating families

Tessellations

Spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...

* * s * ⊕ * t * + Regular * * * * Semi-regular * s * e * sr * sr * rr * r * t * t * tr * tr

Hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

* sr * sr * sr * sr * sr * s * sr * s * sr * sr * s * s * sr * sr * sr * sr * s * * * * h * h * q * rr * rr * rr * h₂ * r * r * t * t * r * t * rr * rr * rr * rr * rr * * * * * * r * r * tr * tr * ??? * tr * ??? * tr * r * r * tr * ??? * tr * tr * tr * tr * t * tr * t * tr * t * t * t * r * * * * * rr * r * t * t * t * r * * * * * rr * r * t * t * t * t * * * * t * t * t * * * * * t * t * t * * * * * t * t

References

Sources

* * ** (Paper 22
pp. 251–278
MR 2,10 ** (Paper 23
pp. 279–312
** (Paper 24
pp. 313–358

External links

* * {{DEFAULTSORT:Schlafli symbol Polytopes Mathematical notation