
In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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,
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, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
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 description,
starting with
for a
-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 ...
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 polytop ...
. For example, is an
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, is a
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, a convex
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
, 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
, where
is the number of vertices, and
is their
turning number. Equivalently,
is created from the vertices of
, connected every
. For example,
is a
pentagram;
is a
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
.
A
regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...
that has
regular
-sided
polygon faces around each
vertex is represented by
. For example, the
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
has 3 squares around each vertex and is represented by .
A regular
4-dimensional polytope, with
regular
polyhedral cells around each edge is represented by
. For example, a
tesseract
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...
, , has 3
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
s, , around an edge.
In general, a
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
has
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, 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 general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
. The vertex figure of a regular polytope is .
Regular polytopes can have
star polygon elements, like the
pentagram, with symbol , represented by the vertices of a
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
but connected alternately.
The Schläfli symbol can represent a finite
convex polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
, an infinite
tessellation of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, 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 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 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 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 amb ...
s, 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,
,3is the Coxeter group for reflective
tetrahedral symmetry,
,4is reflective
octahedral symmetry, 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 polyhedr ...
.
Regular polygons (plane)
The Schläfli symbol of a convex
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 ...
with ''p'' edges is . For example, a regular
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
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
pentagram.
Regular polyhedra (3 dimensions)
The Schläfli symbol of a regular
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
is if its
faces are ''p''-gons, and each vertex is surrounded by ''q'' faces (the
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is a ''q''-gon).
For example, is the regular
dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
. 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 polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s, the 4 nonconvex
Kepler-Poinsot polyhedra.
Topologically, a regular 2-dimensional
tessellation may be regarded as similar to a (3-dimensional) polyhedron, but such that the
angular defect In geometry, the angular defect or simply defect (also called deficit or deficiency) is 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 ''exces ...
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 .
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 is represented by . It is made of
dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
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 (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...
, 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, which fills space with
dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
cells.
If a 4-polytope's symbol is palindromic (e.g. or ), its bitruncation will only have truncated forms of the vertex figure as cells.
Regular ''n''-polytopes (higher dimensions)
For higher-dimensional
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
s, the Schläfli symbol is defined recursively as if the
facets have Schläfli symbol and the
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
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, ; the
cross-polytope, ; and the
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
, . 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, 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 and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
(with operator "×") of lower-dimensional 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
* In 3D, a ''p''-gonal
prism 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 is represented as × . Its Coxeter diagram is . Its symmetry is
'p'',2,''q''
The prismatic duals, or
bipyramid
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...
s can be represented as composite symbols, but with the ''addition'' operator, "+".
* In 2D, a
rhombus
In plane Euclidean geometry, a rhombus (: 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 rhom ...
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 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 Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
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 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 is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.
These rules are formalized with a ...
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 compound, 2.
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 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 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 does not 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 ; or ) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Ancient Greek, beta represented the voiced bilabial plosive . In Modern Greek, it represe ...
(β), is the German alphabet letter
eszett.
Polychora and honeycombs
Alternations, quarters and snubs
Bifurcating families
Tessellations
Spherical
*
*
s
*
⊕
*
t
*
+
Regular
*
*
*
*
Semi-regular
*
s
*
e
*
sr
*
sr
*
rr
*
r
*
t
*
t
*
tr
*
tr
Hyperbolic
*
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
Polytope notation systems