Schläfli symbol

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Schläfli symbols are a notation used to represent regular polytopes. They succinctly describe regular polytopes including regular Euclidean tilings and regular hyperbolic tilings, as well as the more ordinary spherical polytopes. Various extensions to Schläfli symbols exist to represent wider arrays of polytopes. Depending on one's theory of polytopes certain Schläfli symbols may not produce polytopes. For example, the results of a Schläfli symbol can be infinite, hyperbolic, and dense; some or all of which may not meet the definition of polytope within an area of study.

Description[edit | edit source]

A Schläfli symbol consists of several numbers in sequence (usually separated by commas) enclosed within curly brackets ({}). The numbers can be positive integers, fully reduced positive fractions, or infinity. The simplest Schläfli symbols just use positive integers. These simple Schläfli symbols can be defined recursively. As a base case, is the dyad, and is a regular n -gon. Then for rank 3 and higher the a symbol is a tiling of polytopes with the symbol with of them placed around each element. For example, the dodecahedron is . That means its faces are , i.e. pentagons, and there are 3  of them around each vertex. describes a regular polychoron with 4 cubes around each edge, i.e. a cubic honeycomb.

Although useful for picturing things, instead of counting the number of facets around elements this can be done in terms of vertex figures. As before is the dyad, and is a regular n -gon, however additionally is a star polygon with n  vertices with each vertex connected by an edge to the vertices m  steps away. Note that is exactly equivalent to . Then for higher dimensions the symbol represents the polytope with faces of and a vertex figure of . That is the first value gives the face of the polytope and the rest of the values are the vertex figure. This allows for polytopes like the great dodecahedron which has the symbol .

Conversion to Coxeter-Dynkin diagrams[edit | edit source]

Schläfli symbols can be easily converted to a Coxeter-Dynkin diagram. A regular polytope has linear Coxeter-Dynkin diagram with the only the first node ringed, so to convert a Schläfli symbol to a Coxeter-Dynkin diagram you create a linear diagram with the first node ringed and then label the edges in between with the values of the Schläfli symbol in order. The Schläfli symbol becomes the diagram .... This includes cases where the Schläfli symbol has fractional or infinite values.

Extended Schläfli symbols[edit | edit source]

Many extensions to the basic Schläfli symbols exist. These extensions allow symbols to represent more regular polytopes or other classes of polytope altogether.

Coxeter's extension[edit | edit source]

Coxeter extended Schläfli symbols to represent quasiregular polytopes as well. Coxeter's extension allows Schläfli symbols to have up to two rows of values between the curly brackets. A single row still represents the same polytope. If the symbol has two rows it is converted to a Coxeter-Dynkin diagram with a single ringed node. The ringed node is placed on the left and two linear branches extend off to the right. Each value in the symbol is placed between two nodes, with values from the top row being placed on the top branch in order and values from the bottom row being placed on the bottom branch in order.

= =

Coxeter further even allowed thereby for bifurcation nodes in the diagram (as being used for the Gosset polytopes). Then any of the left-aligned number sequences might split up into two (or more) lines from some point on.

Wythoffian prefixes[edit | edit source]

Schläfli symbols have also been extended to represent Wythoffian polytopes using prefixes representing Wythoffian operations. In general any Wythoffian operation is assigned the prefix t  with some number of subscripts representing the indices of ringed nodes in the Coxeter-Dynkin diagram. Indices begin at 0 so rings the first node.

For example the cuboctahedron (CDD: ) has the symmetry group . So we start with the symbol (CDD: ) which has the same symmetry group, and add a prefix indicating that the second node of the Coxeter-Dynkin Diagram should be ringed, . These prefixes easily allow representing any linear Coxeter-Dynkin diagram as an extended Schläfli symbol.

For convenience a number of specific prefixes have shorter names.

Short names for Wythoffian prefixes
Short prefix Long prefix Meaning Example
Polytope Extended Schläfli symbol CDD
2r  Birectification Rectified cubic honeycomb
r  Rectification Rectified cubic honeycomb
rr  Cantellation Cantellated cubic honeycomb
t  Truncation Truncated cubic honeycomb
tr  Cantitruncation Cantitruncated cubic honeycomb

Regular maps[edit | edit source]

As regular maps extend the notion of regular polyhedra, some extensions for Schläfli symbols exist for encoding regular maps. Since many skew polytopes are realizations of maps in Euclidean space (or occasionally hyperbolic space) these extensions can also allow these to be encoded with extended Schläfli symbols.

Petrie polygons[edit | edit source]

A diagram showing a section of the universal cover of the Petrial cube. The vertices drawn in red are all identified to the central point in the Petrial cube. The Petrie polygons which pass through the central point are drawn in solid black. Each other red vertex is 4 steps along a Petrie polygon from the central red vertex. The Petrie polygons in the hexagonal tiling become Petrie polygons in the Petrial cube. Thus each path of length 4 is a "square" in the Petrial cube.

Many regular maps can be written as indicating the polytope is with points that are r  steps apart along a Petrie polygon identified.[1] For example the Petrial cube has the symbol because its universal cover is and it can be obtained by identifying points which are 4 steps along its Petrie polygons, which are in this case zigzags.

When the number of steps r  evenly divides the size of the Petrie polygon (or it is infinite), the resulting polytope has a Petrie polygon with size r . For clarity subscripts are never written with subscripts that don't evenly divide, so you can think of the subscript as the size of the Petrie polygon.

k -holes[edit | edit source]

Additional notation can be used to specify maps in terms of their "k -holes". A face of a map is a path along the edges of a map always taking the rightmost (or alternatively always taking the leftmost) edge. k -holes generalize this notion. A k -hole is a path along the edges of a map such that it always takes the k th rightmost (or alternative always takes the k th leftmost) edge. Faces are thus 1 -holes.[2] Like the extension which denotes the size of the Petrie polygon, we also can extend Schläfli symbols to denote the size of the 2 -holes. The extended Schläfli symbol indicates a symbol with Schläfli type and 2 -holes of .[3] Like earlier the map is constructed by taking the universal cover and identifying vertices that are distance r  apart along the 2 -holes. For example the mucube has the extended symbol so it is constructed from by identifying vertices that are 4  steps apart along the 2 -holes.

This notation can be generalized even further with multiple values after the divider. The expression indicates a map with type and k -holes of for k  from 2  to n .[4] Constructing these maps follows the same process as before except it is done over multiple k -holes.

In McMullen & Schulte (1997) this notation is extended further. They prefix the size of the 2 -holes in the usual notation with a *  to indicate instead size of the 2 -holes in the dual map.[5] This means that it is the size of the k -holes following the faces rather than the vertices. For example the hemicube is a quotient of the cube which identifies opposite faces and thus can be represented as . There is no way to represent the hemicube with the normal notation.

Hemi-polytopes[edit | edit source]

Hemi-polytopes are a quotient of spherical polytope identifying opposite points. If that spherical polytope has a Schläfli symbol, the hemi-polytope can be represented with a /2  after its symbol. Additionally for regular hemi-polytopes the extension described above for regular maps can be used. The Schläfli symbol of the hemi-polytope is the symbol of its double cover with a subscript equal to half the number of vertices in the Petrie polygon.[6]

Extended Schläfli symbols of regular hemipolyhedra
Extended Schläfli symbols Name
[6] Hemicube
[6] Hemioctahedron
[7] [6] Hemidodecahedron
[6] Hemiicosahedron
[6] Hemiicositetrachoron
[6] Hemihexacosichoron
[6] Hemihecatonicosachoron

Petrie dual[edit | edit source]

It may be desirable to represent the regular Petrials with Schläfli symbol. The Petrie dual of a regular polytope can be represented with a superscript pi as .[8] However this is not the only way to represent Petrials. Many Petrials can be written with the extensions for regular maps. For example the Petrial cube is and the Petrial tetrahedron is . For Petrials we can generalize this to say that the symbol represents a polytope with p -gonal faces, a q -gonal vertex figure and an r -gonal Petrie polygon.[7] Thus the Petrial great stellated dodecahedron is since it's Petrie polygons are pentagrams. With this notation the Petrie dual of a polyhedron is , exchanging the faces with the Petrie polygons.[7]

Regular complex polytopes[edit | edit source]

For a regular complex polytope, 𝓟, with generators ,

  • let be the order of . That is the smallest positive integer such that ;
  • let be the smallest positive integer such that

then 𝓟 is given the Schäfli symbol:

For example the Möbius-Kantor polygon has the symbol .

Schläfli type[edit | edit source]

A Schläfli symbol indicates a regular polyhedron with p -gonal faces and a q -gonal vertex figure. However there may be multiple abstract regular polyhedra with p -gonal facets and a q -gonal vertex figure. The idea of the Schläfli type of a polytope is to classify these polytopes with similar structures together. is the Schläfli type of all regular polyhedra with p -gonal faces and a q -gonal vertex figure although it is the Schläfli symbol for only one of them.[9]

This idea can be made into a precise definition:

Let 𝓟 be a regular polytope of rank n , let be the distinguished generators of the automorphism group of 𝓟. Let be the the order of . The Schläfli type of 𝓟 is then .[10]

These two definitions are equivalent.

In ordinary Euclidean space the Schläfli symbol and Schläfli type are always the same. That is, if a polytope has a Schläfli type that is also its Schläfli symbol.[11]

The Schläfli type of a polyhedron is the Schläfli symbol of its universal cover.[1]

Non-regular polytopes[edit | edit source]

Schläfli type is also often used for equivelar polyhedra, including noble polyhedra. In these cases the Schläfli type just gives the number of edges of each face and vertex. The Schläfli type for prientable equivelar polyhedra may be extended to also include the genus. In these cases the extended type is written {p,q;g} for any orientable equivelar polyhedron with p -gonal faces, degree q  vertices, and genus g 

The Schläfli type does not generalize well for higher rank noble polytopes since their facets do not need to be isotopic.

External resources[edit | edit source]

References[edit | edit source]

Bibliography[edit | edit source]

  • Coxeter, Harold Scott MacDonald; Moser, William Oscar Jules (1972). Generators and Relations for Discrete groups (4 ed.). Springer-Verlag. doi:10.1007/978-3-662-21946-1.
  • McMullen, Peter (2007). "Four-Dimensional Regular Polyhedra" (PDF). Discrete & Computational Geometry.
  • McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.
  • McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0
  • Wills, Jörg (1987). "The combinatorially regular polyhedra of index 2". Aequationes Mathematicae: 206–220. doi:10.1007/BF01830672.