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 19th-century mathematician Ludwig Schläfli who made important contributions in geometry and other areas.
Description
The Schläfli symbol is a recursive description, starting with a p-sided regular polygon as {p}. For example, {3} is an equilateral triangle, {4} is a square and so on.
A regular polyhedron which 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}, and so on.
Regular polytopes can have star polygon elements, like the pentagram, with symbol {5/2}, represented by the vertices of a pentagon but connected alternately.
A facet of a regular polytope {p,q,r,...,y,z} is {p,q,r,...,y}.
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}.
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 will tessellate space of the same dimension as the facets. A negative angle defect can't exist in ordinary space, but can be constructed in hyperbolic space.
Usually a vertex figure is assumed to be a finite polytope, but can sometimes be considered a tessellation itself.
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.
Symmetry groups
A Schläfli symbol is closely related to reflection symmetry groups, also called Coxeter groups, given 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 regular polygon with n edges is {n}.
For example, a regular pentagon is represented by {5}.
See the convex regular polygon and nonconvex star polygon.
For example, {5/2} is the pentagram.
Regular polyhedra (3-space)
The 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.
Schläfli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way.
For example, the hexagonal tiling is represented by {6,3}.
Regular polychora (4-space)
The Schläfli symbol of a regular 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 nonconvex polychora.
For example, the 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 also one regular tessellation of Euclidean 3-space: 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 hyperbolic tessellations including {5,3,4}, the Hyperbolic small dodecahedral honeycomb, which fills space with dodecahedron cells.
Higher dimensions
For higher-dimensional 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}}.
Notice that 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 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
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 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 of lower-dimensional regular polytopes:
- In 3D, a p-gonal prism is represented as { } × {p}. The symbol { } means a digon or line segment. Its Coxeter diagram is Template:CDD.
- In 4D, a uniform {p,q}-hedral prism as { } × {p,q}. Its Coxeter diagram is Template:CDD.
- In 4D, a uniform p-q duoprism as {p} × {q}. Its Coxeter diagram is Template:CDD.
The prismatic duals, or bipyramids can also be represented as composite symbols, but with the addition operator, +.
- In 3D, a p-gonal bipyramid, is represented as { } + {p}. Its Coxeter diagram is Template:CDD.
- In 4D, a {p,q}-hedral bipyramids as { } + {p,q}. Its Coxeter diagram is Template:CDD.
- In 4D, a p-q duopyramid as {p} + {q}. Its Coxeter diagram is Template:CDD.
Prismoid polyhedra and polytopes, containing vertices on two parallel hyperplanes. These can be represented in a join operator, ∨. A point is represented ( ).
In 2D:
- Isosceles triangle can be represented as ( ) ∨ { }.
In 3D:
- p-gonal pyramids are represented as ( ) ∨ {p}.
- p-gonal antiprism are represented as {p} ∨ r{p}. (Antiprisms constructed as alternated prisms, Coxeter diagram, Template:CDD, and an additional central inversion, are also represented by { } ⨂ {p}, a skew-rectangular product.)
- p-gonal cupola as {p} ∨ t{p}.
In 4D:
- p-q-hedral pyramids as ( ) ∨ {p,q}.
- p-q-hedral cupola as {p,q} ∨ rr{p,q}.
- p-q-hedral antiprisms as {p,q} ∨ 2r{p,q}. (An antiprism with Coxeter diagram Template:CDD, is also represented by { } ⨂ {2p,q}.)
- Regular and rectified prismoid as {p,q} ∨ r{p,q}.
- Truncated and dual truncated prismoid as t{p,q} ∨ 2t{p,q}.
When mixing operators, the order of operations from highest to lowest is: ×, +, and ∨.
Extension of Schläfli symbols
Polygons and circle tilings
A truncated regular polygon doubles in sides. A regular polygon with even sides can be halfed. An altered even-sided regular 2n-gon generates a star figure compound, 2{n}.
Form | Schläfli symbol | Symmetry | Coxeter diagram | Example, {6} | |||
---|---|---|---|---|---|---|---|
Regular | {p} | [p] | Template:CDD | Hexagon | Template:CDD | ||
Truncated | t{p} = {2p} | [[p]] = [2p] | Template:CDD = Template:CDD | Truncated hexagon | Template:CDD = Template:CDD | ||
Altered | a{2p} | [2p] | Template:CDD | Altered hexagon | Template:CDD | ||
Half | h{2p} = {p} | [1^{+},2p] = [p] | Template:CDD = Template:CDD | half hexagon | Template:CDD = Template:CDD |
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.
Form | Schläfli symbols | Symmetry | Coxeter diagram | Example, {4,3} | |||||
---|---|---|---|---|---|---|---|---|---|
Regular | {p,q} | t_{0}{p,q} | [p,q] or [(p,q,2)] |
Template:CDD | Cube | Template:CDD | |||
Truncated | t{p,q} | t_{0,1}{p,q} | Template:CDD | Truncated cube | Template:CDD | ||||
Bitruncation (Truncated dual) |
2t{p,q} | t_{1,2}{p,q} | Template:CDD | Template:CDD | Truncated octahedron | Template:CDD | |||
Rectified (Quasiregular) |
r{p,q} | t_{1}{p,q} | Template:CDD | Template:CDD | Cuboctahedron | Template:CDD | |||
Birectification (Regular dual) |
2r{p,q} | t_{2}{p,q} | Template:CDD | Template:CDD | Octahedron | Template:CDD | |||
Cantellated (Rectified rectified) |
rr{p,q} | t_{0,2}{p,q} | Template:CDD | Template:CDD | Rhombicuboctahedron | Template:CDD | |||
Cantitruncated (Truncated rectified) |
tr{p,q} | t_{0,1,2}{p,q} | Template:CDD | Template:CDD | Truncated cuboctahedron | Template:CDD | |||
Alternations | |||||||||
Alternated (half) regular | h{2p,q} | ht_{0}{2p,q} | [1^{+},2p,q] | Template:CDD or Template:CDD | Demicube (Tetrahedron) |
Template:CDD | |||
Snub regular | s{p,2q} | ht_{0,1}{p,2q} | [p^{+},2q] | Template:CDD | |||||
Snub dual regular | s{q,2p} | ht_{1,2}{2p,q} | [2p,q^{+}] | Template:CDD | Template:CDD | Snub octahedron (Icosahedron) |
Template:CDD | ||
Alternated dual regular | h{2q,p} | ht_{2}{p,2q} | [p,2q,1^{+}] | Template:CDD | Template:CDD | ||||
Alternated rectified (p and q are even) |
hr{p,q} | ht_{1}{p,q} | [p,1^{+},q] | Template:CDD | Template:CDD | ||||
Alternated rectified rectified (p and q are even) |
hrr{p,q} | ht_{0,2}{p,q} | [(p,q,2^{+})] | Template:CDD | Template:CDD | ||||
Quartered (p and q are even) |
q{p,q} | ht_{0}ht_{2}{p,q} | [1^{+},p,q,1^{+}] | Template:CDD | Template:CDD | ||||
Snub rectified Snub quasiregular |
sr{p,q} | ht_{0,1,2}{p,q} | [p,q]^{+} | Template:CDD | Template:CDD | Snub cuboctahedron (Snub cube) |
Template:CDD | ||
Altered and holosnubbed | |||||||||
Altered regular | a{p,q} | at_{0}{p,q} | [p,q] | Template:CDD | Stellated octahedron | Template:CDD | |||
Holosnub dual regular | ß | ß{q,p} | at_{0,1}{q,p} | [p,q] | Template:CDD | Template:CDD | Compound of two icosahedra | Template:CDD |
Polychora and honeycombs
Form | Schläfli symbol | Coxeter diagram | Example, {4,3,3} | ||||||
---|---|---|---|---|---|---|---|---|---|
Regular | {p,q,r} | t_{0}{p,q,r} | Template:CDD | Tesseract | Template:CDD | ||||
Truncated | t{p,q,r} | t_{0,1}{p,q,r} | Template:CDD | Truncated tesseract | Template:CDD | ||||
Rectified | r{p,q,r} | t_{1}{p,q,r} | Template:CDD | Rectified tesseract | Template:CDD = Template:CDD | ||||
Bitruncated | 2t{p,q,r} | t_{1,2}{p,q,r} | Template:CDD | Bitruncated tesseract | Template:CDD | ||||
Birectified (Rectified dual) |
2r{p,q,r} = r{r,q,p} | t_{2}{p,q,r} | Template:CDD | Rectified 16-cell | Template:CDD = Template:CDD | ||||
Tritruncated (Truncated dual) |
3t{p,q,r} = t{r,q,p} | t_{2,3}{p,q,r} | Template:CDD | Bitruncated tesseract | Template:CDD | ||||
Trirectified (Dual) |
3r{p,q,r} = {r,q,p} | t_{3}{p,q,r} = {r,q,p} | Template:CDD | 16-cell | Template:CDD | ||||
Cantellated | rr{p,q,r} | t_{0,2}{p,q,r} | Template:CDD | Cantellated tesseract | Template:CDD = Template:CDD | ||||
Cantitruncated | tr{p,q,r} | t_{0,1,2}{p,q,r} | Template:CDD | Cantitruncated tesseract | Template:CDD = Template:CDD | ||||
Runcinated (Expanded) |
e_{3}{p,q,r} | t_{0,3}{p,q,r} | Template:CDD | Runcinated tesseract | Template:CDD | ||||
Runcitruncated | t_{0,1,3}{p,q,r} | Template:CDD | Runcitruncated tesseract | Template:CDD | |||||
Omnitruncated | t_{0,1,2,3}{p,q,r} | Template:CDD | Omnitruncated tesseract | Template:CDD | |||||
Alternations | |||||||||
Half p even |
h{p,q,r} | ht_{0}{p,q,r} | Template:CDD | 16-cell | Template:CDD | ||||
Quarter p and r even |
q{p,q,r} | ht_{0}ht_{3}{p,q,r} | Template:CDD | ||||||
Snub q even |
s{p,q,r} | ht_{0,1}{p,q,r} | Template:CDD | Snub 24-cell | Template:CDD | ||||
Snub rectified r even |
sr{p,q,r} | ht_{0,1,2}{p,q,r} | Template:CDD | Snub 24-cell | Template:CDD = Template:CDD | ||||
Alternated duoprism | s{p}s{q} | ht_{0,1,2,3}{p,2,q} | Template:CDD | Great duoantiprism | Template:CDD |
Form | Extended Schläfli symbol | Coxeter diagram | Examples | |||||
---|---|---|---|---|---|---|---|---|
Quasiregular | {p,q^{1,1}} | t_{0}{p,q^{1,1}} | Template:CDD | 16-cell | Template:CDD | |||
Truncated | t{p,q^{1,1}} | t_{0,1}{p,q^{1,1}} | Template:CDD | Truncated 16-cell | Template:CDD | |||
Rectified | r{p,q^{1,1}} | t_{1}{p,q^{1,1}} | Template:CDD | 24-cell | Template:CDD | |||
Cantellated | rr{p,q^{1,1}} | t_{0,2,3}{p,q^{1,1}} | Template:CDD | Cantellated 16-cell | Template:CDD | |||
Cantitruncated | tr{p,q^{1,1}} | t_{0,1,2,3}{p,q^{1,1}} | Template:CDD | Cantitruncated 16-cell | Template:CDD | |||
Snub rectified | sr{p,q^{1,1}} | ht_{0,1,2,3}{p,q^{1,1}} | Template:CDD | Snub 24-cell | Template:CDD | |||
Quasiregular | {r,/q\,p} | t_{0}{r,/q\,p} | Template:CDD | Template:CDD | ||||
Truncated | t{r,/q\,p} | t_{0,1}{r,/q\,p} | Template:CDD | Template:CDD | ||||
Rectified | r{r,/q\,p} | t_{1}{r,/q\,p} | Template:CDD | Template:CDD | ||||
Cantellated | rr{r,/q\,p} | t_{0,2,3}{r,/q\,p} | Template:CDD | Template:CDD | ||||
Cantitruncated | tr{r,/q\,p} | t_{0,1,2,3}{r,/q\,p} | Template:CDD | Template:CDD | ||||
Snub rectified | sr{p,/q,\r} | ht_{0,1,2,3}{p,/q\,r} | Template:CDD | Template:CDD |
References
- Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (pp. 14, 69, 149) [1]
- 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]