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Schlafli symbol

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Schlafli 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 polychoron 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 {p1, p2, ..., pn − 1} if the facets have Schläfli symbol {p1,p2, ..., pn − 2} and the vertex figures have Schläfli symbol {p2,p3, ..., pn − 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: {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

If a polytope of dimension ≥ 2 has Schläfli symbol {p1,p2, ..., pn − 1} then its dual has 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.

Uniform prismatic polytopes

Uniform prismatic polytopes can be defined and named as a Cartesian product of lower dimensional regular polytopes:

  • A p-gonal prism, with vertex figure p.4.4 as { } × {p}. The symbol { } means a digon or line segment.
  • A uniform {p,q}-hedral prism as { } × {p,q}.
  • A uniform p-q duoprism as {p} × {q}.

Extension of Schläfli symbols

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-Dynkin 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-Dynkin diagram. All of the symbols have a corresponding alternation, replacing rings with holes in a Coxeter diagram and h prefix, construction limited by the requirement that neighboring branches must be even-ordered.

Form Extended Schläfli symbols Symmetry Coxeter diagram Example, {4,3}
Regular \begin{Bmatrix} p , q \end{Bmatrix} {p,q} t0{p,q} [p,q]
or
[(p,q,2)]
Cube
Truncated t\begin{Bmatrix} p , q \end{Bmatrix} t{p,q} t0,1{p,q} Truncated cube
Bitruncation
(Truncated dual)
t\begin{Bmatrix} q , p \end{Bmatrix} 2t{p,q} t1,2{p,q} Truncated octahedron
Rectified
(Quasiregular)
\begin{Bmatrix} p \\ q \end{Bmatrix} r{p,q} t1{p,q} Cuboctahedron
Birectification
(Regular dual)
\begin{Bmatrix} q , p \end{Bmatrix} 2r{p,q} t2{p,q} Octahedron
Cantellated
(Rectified rectified)
r\begin{Bmatrix} p \\ q \end{Bmatrix} rr{p,q} t0,2{p,q} Rhombicuboctahedron
Cantitruncated
(Truncated rectified)
t\begin{Bmatrix} p \\ q \end{Bmatrix} tr{p,q} t0,1,2{p,q} Truncated cuboctahedron
Alternations
Alternated regular
(p is even)
h \begin{Bmatrix} p , q \end{Bmatrix} h{p,q} ht0{p,q} [1+,p,q] Demicube
(Tetrahedron)
Snub regular
(q is even)
s\begin{Bmatrix} p , q \end{Bmatrix} s{p,q} ht0,1{p,q} [p+,q]
Snub dual regular
(p is even)
s \begin{Bmatrix} q , p \end{Bmatrix} s{q,p} ht1,2{p,q} [p,q+] Snub octahedron
(Icosahedron)
Alternated dual regular
(q is even)
h \begin{Bmatrix} q , p \end{Bmatrix} h{q,p} ht2{p,q} [p,q,1+]
Alternated rectified
(p and q are even)
h \begin{Bmatrix} p \\ q \end{Bmatrix} hr{p,q} ht1{p,q} [p,1+,q]
Alternated rectified rectified
(p and q are even)
hr\begin{Bmatrix} p \\ q \end{Bmatrix} hrr{p,q} ht0,2{p,q} [(p,q,2+)]
Quartered
(p and q are even)
q\begin{Bmatrix} p \\ q \end{Bmatrix} q{p,q} ht0ht2{p,q} [1+,p,q,1+]
Snub rectified
Snub quasiregular
s\begin{Bmatrix} p \\ q \end{Bmatrix} sr{p,q} ht0,1,2{p,q} [p,q]+ Snub cuboctahedron
(Snub cube)

Polychora and honeycombs

Linear families
Form Extended Schläfli symbol Coxeter diagram Example, {4,3,3}
Regular \begin{Bmatrix} p, q , r \end{Bmatrix} {p,q,r} t0{p,q,r} Tesseract
Truncated t\begin{Bmatrix} p, q , r \end{Bmatrix} t{p,q,r} t0,1{p,q,r} Truncated tesseract
Rectified \left\{\begin{array}{l}p\\q,r\end{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)
\left\{\begin{array}{l}q,p\\r\end{array}\right\} 2r{p,q,r} = r{r,q,p} t2{p,q,r} Rectified 16-cell =
Tritruncated
(Truncated dual)
t\begin{Bmatrix} r, q , p \end{Bmatrix} 3t{p,q,r} = t{r,q,p} t2,3{p,q,r} Bitruncated tesseract
Trirectified
(Dual)
\begin{Bmatrix} r, q , p \end{Bmatrix} 3r{p,q,r} = {r,q,p} t3{p,q,r} = {r,q,p} 16-cell
Cantellated r\left\{\begin{array}{l}p\\q,r\end{array}\right\} rr{p,q,r} t0,2{p,q,r} Cantellated tesseract =
Cantitruncated t\left\{\begin{array}{l}p\\q,r\end{array}\right\} tr{p,q,r} t0,1,2{p,q,r} Cantitruncated tesseract =
Runcinated
(Expanded)
e\begin{Bmatrix} p, q , r \end{Bmatrix} e{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
Half
p even
h\begin{Bmatrix} p, q , r \end{Bmatrix} h{p,q,r} ht0{p,q,r} 16-cell
Quarter
p and r even
q\begin{Bmatrix} p, q , r \end{Bmatrix} q{p,q,r} ht0ht3{p,q,r}
Snub
q even
s\begin{Bmatrix} p, q , r \end{Bmatrix} s{p,q,r} ht0,1{p,q,r} Snub 24-cell
Snub rectified
r even
s\left\{\begin{array}{l}p\\q,r\end{array}\right\} sr{p,q,r} ht0,1,2{p,q,r} Snub 24-cell =
Bifurcating families
Form Extended Schläfli symbol Coxeter diagram Examples
Quasiregular \left\{p,{q\atop q}\right\} {p,q1,1} t0{p,q1,1} 16-cell
Truncated t\left\{p,{q\atop q}\right\} t{p,q1,1} t0,1{p,q1,1} Truncated 16-cell
Rectified \left\{\begin{array}{l}p\\q\\q\end{array}\right\} r{p,q1,1} t1{p,q1,1} 24-cell
Cantellated r\left\{\begin{array}{l}p\\q\\q\end{array}\right\} rr{p,q1,1} t0,2,3{p,q1,1} Cantellated_16-cell
Cantitruncated t\left\{\begin{array}{l}p\\q\\q\end{array}\right\} tr{p,q1,1} t0,1,2,3{q,31,1} Cantitruncated_16-cell
Snub s\left\{\begin{array}{l}p\\q\\q\end{array}\right\} sr{p,q1,1} ht0,1,2,3{q,31,1} Snub 24-cell
Quasiregular \left\{r,{p\atop q}\right\} {r,/q\,p} t0{r,/q\,p}
Truncated t\left\{r,{p\atop q}\right\} t{r,/q\,p} t0,1{r,/q\,p}
Rectified \left\{\begin{array}{l}r\\p\\q\end{array}\right\} r{r,/q\,p} t1{r,/q\,p}
Cantellated r\left\{\begin{array}{l}r\\p\\q\end{array}\right\} rr{r,/q\,p} t0,2,3{r,/q\,p}
Cantitruncated t\left\{\begin{array}{l}r\\p\\q\end{array}\right\} tr{r,/q\,p} t0,1,2,3{r,/q\,p}

References

  • [1]
  • 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

  • MathWorld.
  • Wythoff Symbol and generalized Schläfli Symbols
  • polyhedral names et notations
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