Schläfli symbol

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The dodecahedron is a regular polyhedron with Schläfli symbol {5,3}, having 3 pentagons around each vertex.

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[edit]

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[edit]

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)[edit]

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)[edit]

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)[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Polyhedra and tilings[edit]

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)]
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png Hexahedron.png Cube CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Truncated t\begin{Bmatrix} p , q \end{Bmatrix} t{p,q} t0,1{p,q} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png Truncated hexahedron.png Truncated cube CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Bitruncation
(Truncated dual)
t\begin{Bmatrix} q , p \end{Bmatrix} 2t{p,q} t1,2{p,q} CDel node 1.pngCDel q.pngCDel node 1.pngCDel p.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png Truncated octahedron.png Truncated octahedron CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Rectified
(Quasiregular)
\begin{Bmatrix} p \\ q \end{Bmatrix} r{p,q} t1{p,q} CDel node 1.pngCDel split1-pq.pngCDel nodes.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png Cuboctahedron.png Cuboctahedron CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Birectification
(Regular dual)
\begin{Bmatrix} q , p \end{Bmatrix} 2r{p,q} t2{p,q} CDel node 1.pngCDel q.pngCDel node.pngCDel p.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png Octahedron.png Octahedron CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantellated
(Rectified rectified)
r\begin{Bmatrix} p \\ q \end{Bmatrix} rr{p,q} t0,2{p,q} CDel node.pngCDel split1-pq.pngCDel nodes 11.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png Small rhombicuboctahedron.png Rhombicuboctahedron CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantitruncated
(Truncated rectified)
t\begin{Bmatrix} p \\ q \end{Bmatrix} tr{p,q} t0,1,2{p,q} CDel node 1.pngCDel split1-pq.pngCDel nodes 11.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png Great rhombicuboctahedron.png Truncated cuboctahedron CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Alternations
Alternated regular
(p is even)
h \begin{Bmatrix} p , q \end{Bmatrix} h{p,q} ht0{p,q} [1+,p,q] CDel node h1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png Tetrahedron.png Demicube
(Tetrahedron)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Snub regular
(q is even)
s\begin{Bmatrix} p , q \end{Bmatrix} s{p,q} ht0,1{p,q} [p+,q] CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.png
Snub dual regular
(p is even)
s \begin{Bmatrix} q , p \end{Bmatrix} s{q,p} ht1,2{p,q} [p,q+] CDel node h.pngCDel q.pngCDel node h.pngCDel p.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png Uniform polyhedron-43-h01.svg Snub octahedron
(Icosahedron)
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Alternated dual regular
(q is even)
h \begin{Bmatrix} q , p \end{Bmatrix} h{q,p} ht2{p,q} [p,q,1+] CDel node h1.pngCDel q.pngCDel node.pngCDel p.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node h1.png
Alternated rectified
(p and q are even)
h \begin{Bmatrix} p \\ q \end{Bmatrix} hr{p,q} ht1{p,q} [p,1+,q] CDel node h1.pngCDel split1-pq.pngCDel nodes.png CDel node.pngCDel p.pngCDel node h1.pngCDel q.pngCDel node.png
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+)] CDel node.pngCDel split1-pq.pngCDel nodes hh.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node h.png
Quartered
(p and q are even)
q\begin{Bmatrix} p \\ q \end{Bmatrix} q{p,q} ht0ht2{p,q} [1+,p,q,1+] CDel node.pngCDel split1-pq.pngCDel nodes h1h1.png CDel node h1.pngCDel p.pngCDel node.pngCDel q.pngCDel node h1.png
Snub rectified
Snub quasiregular
s\begin{Bmatrix} p \\ q \end{Bmatrix} sr{p,q} ht0,1,2{p,q} [p,q]+ CDel node h.pngCDel split1-pq.pngCDel nodes hh.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png Snub hexahedron.png Snub cuboctahedron
(Snub cube)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png

Polychora and honeycombs[edit]

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} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Schlegel wireframe 8-cell.png Tesseract CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Truncated t\begin{Bmatrix} p, q , r \end{Bmatrix} t{p,q,r} t0,1{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Schlegel half-solid truncated tesseract.png Truncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Rectified \left\{\begin{array}{l}p\\q,r\end{array}\right\} r{p,q,r} t1{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Schlegel half-solid rectified 8-cell.png Rectified tesseract CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1-43.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
Bitruncated 2t{p,q,r} t1,2{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Schlegel half-solid bitruncated 16-cell.png Bitruncated tesseract CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Schlegel half-solid rectified 16-cell.png Rectified 16-cell CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a.pngCDel nodea.png
Tritruncated
(Truncated dual)
t\begin{Bmatrix} r, q , p \end{Bmatrix} 3t{p,q,r} = t{r,q,p} t2,3{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Schlegel half-solid truncated 16-cell.png Bitruncated tesseract CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Trirectified
(Dual)
\begin{Bmatrix} r, q , p \end{Bmatrix} 3r{p,q,r} = {r,q,p} t3{p,q,r} = {r,q,p} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Schlegel wireframe 16-cell.png 16-cell CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantellated r\left\{\begin{array}{l}p\\q,r\end{array}\right\} rr{p,q,r} t0,2{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Schlegel half-solid cantellated 8-cell.png Cantellated tesseract CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node.pngCDel split1-43.pngCDel nodes 11.pngCDel 3b.pngCDel nodeb.png
Cantitruncated t\left\{\begin{array}{l}p\\q,r\end{array}\right\} tr{p,q,r} t0,1,2{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Schlegel half-solid cantitruncated 8-cell.png Cantitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1-43.pngCDel nodes 11.pngCDel 3b.pngCDel nodeb.png
Runcinated
(Expanded)
e\begin{Bmatrix} p, q , r \end{Bmatrix} e{p,q,r} t0,3{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Schlegel half-solid runcinated 8-cell.png Runcinated tesseract CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated t0,1,3{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Schlegel half-solid runcitruncated 8-cell.png Runcitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Omnitruncated t0,1,2,3{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Schlegel half-solid omnitruncated 8-cell.png Omnitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Alternations
Half
p even
h\begin{Bmatrix} p, q , r \end{Bmatrix} h{p,q,r} ht0{p,q,r} CDel node h1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Schlegel wireframe 16-cell.png 16-cell CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Quarter
p and r even
q\begin{Bmatrix} p, q , r \end{Bmatrix} q{p,q,r} ht0ht3{p,q,r} CDel node h1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node h1.png
Snub
q even
s\begin{Bmatrix} p, q , r \end{Bmatrix} s{p,q,r} ht0,1{p,q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Ortho solid 969-uniform polychoron 343-snub.png Snub 24-cell CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
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} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node.png Ortho solid 969-uniform polychoron 343-snub.png Snub 24-cell CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png = CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel 4a.pngCDel nodea.png
Alternated omnitruncation ht0,1,2,3{p,q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node h.png Great duoantiprism.png Great duoantiprism CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel node h.png
Bifurcating families
Form Extended Schläfli symbol Coxeter diagram Examples
Quasiregular \left\{p,{q\atop q}\right\} {p,q1,1} t0{p,q1,1} CDel node 1.pngCDel p.pngCDel node.pngCDel split1-qq.pngCDel nodes.png Schlegel wireframe 16-cell.png 16-cell CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
Truncated t\left\{p,{q\atop q}\right\} t{p,q1,1} t0,1{p,q1,1} CDel node 1.pngCDel p.pngCDel node 1.pngCDel split1-qq.pngCDel nodes.png Schlegel half-solid truncated 16-cell.png Truncated 16-cell CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
Rectified \left\{\begin{array}{l}p\\q\\q\end{array}\right\} r{p,q1,1} t1{p,q1,1} CDel node.pngCDel p.pngCDel node 1.pngCDel split1-qq.pngCDel nodes.png Schlegel wireframe 24-cell.png 24-cell CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
Cantellated r\left\{\begin{array}{l}p\\q\\q\end{array}\right\} rr{p,q1,1} t0,2,3{p,q1,1} CDel node 1.pngCDel p.pngCDel node.pngCDel split1-qq.pngCDel nodes 11.png Schlegel half-solid cantellated 16-cell.png Cantellated 16-cell CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
Cantitruncated t\left\{\begin{array}{l}p\\q\\q\end{array}\right\} tr{p,q1,1} t0,1,2,3{p,q1,1} CDel node 1.pngCDel p.pngCDel node 1.pngCDel split1-qq.pngCDel nodes 11.png Schlegel half-solid cantitruncated 16-cell.png Cantitruncated 16-cell CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
Snub rectified s\left\{\begin{array}{l}p\\q\\q\end{array}\right\} sr{p,q1,1} ht0,1,2,3{p,q1,1} CDel node h.pngCDel p.pngCDel node h.pngCDel split1-qq.pngCDel nodes hh.png Ortho solid 969-uniform polychoron 343-snub.png Snub 24-cell CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
Quasiregular \left\{r,{p\atop q}\right\} {r,/q\,p} t0{r,/q\,p} CDel node 1.pngCDel r.pngCDel node.pngCDel split1-pq.pngCDel nodes.png CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png
Truncated t\left\{r,{p\atop q}\right\} t{r,/q\,p} t0,1{r,/q\,p} CDel node 1.pngCDel r.pngCDel node 1.pngCDel split1-pq.pngCDel nodes.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes.png
Rectified \left\{\begin{array}{l}r\\p\\q\end{array}\right\} r{r,/q\,p} t1{r,/q\,p} CDel node.pngCDel r.pngCDel node 1.pngCDel split1-pq.pngCDel nodes.png CDel node.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes.png
Cantellated r\left\{\begin{array}{l}r\\p\\q\end{array}\right\} rr{r,/q\,p} t0,2,3{r,/q\,p} CDel node 1.pngCDel r.pngCDel node.pngCDel split1-pq.pngCDel nodes 11.png CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes 11.png
Cantitruncated t\left\{\begin{array}{l}r\\p\\q\end{array}\right\} tr{r,/q\,p} t0,1,2,3{r,/q\,p} CDel node 1.pngCDel r.pngCDel node 1.pngCDel split1-pq.pngCDel nodes 11.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes 11.png
snub rectified s\left\{\begin{array}{l}p\\q\\r\end{array}\right\} sr{p,/q,\r} ht0,1,2,3{p,/q\,r} CDel node h.pngCDel r.pngCDel node h.pngCDel split1-pq.pngCDel nodes hh.png CDel node h.pngCDel 3.pngCDel node h.pngCDel split1-43.pngCDel nodes hh.png

References[edit]

  • 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]

External links[edit]