Schläfli symbol

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Short description: Notation that defines regular polytopes and tessellations
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 [math]\displaystyle{ \{p,q,r, ...\} }[/math] that defines regular polytopes and tessellations.

The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli,[1]:143 who generalized Euclidean geometry 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,[1]:129 starting with {p} for a p-sided regular polygon that is convex. For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon, etc.

Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices, and q is their turning number. Equivalently, {p/q} is created from the vertices of {p}, connected every q. For example, {​52} is a pentagram; {​51} is a pentagon.

A regular polyhedron that 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}. For example, a tesseract, {4,3,3}, has 3 cubes, {4,3}, around an edge.

In general, a regular polytope {p,q,r,...,y,z} has z {p,q,r,...,y} 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. The vertex figure of a regular polytope {p,q,r,...,y,z} is {q,r,...,y,z}.

Regular polytopes can have star polygon elements, like the pentagram, with symbol {​52}, represented by the vertices of a pentagon but connected alternately.

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 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.[1]:138

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.[1]:144

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 groups, 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, [3,3] is the Coxeter group for reflective tetrahedral symmetry, [3,4] is reflective octahedral symmetry, and [3,5] is reflective icosahedral symmetry.

Regular polygons (plane)

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols

The Schläfli symbol of a convex regular polygon with p edges is {p}. For example, a regular pentagon is represented by {5}.

For nonconvex star polygons, the constructive notation {​pq} is used, where p is the number of vertices and q−1 is the number of vertices skipped when drawing each edge of the star. For example, {​52} represents the pentagram.

Regular polyhedra (3 dimensions)

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.

Topologically, a regular 2-dimensional tessellation may be regarded as similar to a (3-dimensional) polyhedron, but such that the angular defect 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 {6,3}.

Regular 4-polytopes (4 dimensions)

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 regular star 4-polytopes.

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 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 compact hyperbolic tessellations including {5,3,4}, the hyperbolic small dodecahedral honeycomb, which fills space with dodecahedron cells.

If a 4-polytope's symbol is palindromic (e.g. {3,3,3} or {3,4,3}), its bitruncation will only have truncated forms of the vertex figure as cells.

Regular n-polytopes (higher dimensions)

For higher-dimensional regular 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}.

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.

Schläfli symbol of regular polytopes.gif


Dual polytopes

If a polytope of dimension n2 has Schläfli symbol {p1, p2, ..., pn1} then its dual has Schläfli symbol {pn1, ..., 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.

Prismatic polytopes

Uniform prismatic polytopes can be defined and named as a Cartesian product (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 CDel node 1.png. Its symmetry is [ ].
  • In 2D, a rectangle is represented as { } × { }. Its Coxeter diagram is CDel node 1.pngCDel 2.pngCDel node 1.png. Its symmetry is [2].
  • In 3D, a p-gonal prism is represented as { } × {p}. Its Coxeter diagram is CDel node 1.pngCDel 2.pngCDel node 1.pngCDel p.pngCDel node.png. Its symmetry is [2,p].
  • In 4D, a uniform {p,q}-hedral prism is represented as { } × {p,q}. Its Coxeter diagram is CDel node 1.pngCDel 2.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png. Its symmetry is [2,p,q].
  • In 4D, a uniform p-q duoprism is represented as {p} × {q}. Its Coxeter diagram is CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png. Its symmetry is [p,2,q].

The prismatic duals, or bipyramids can be represented as composite symbols, but with the addition operator, "+".

  • In 2D, a rhombus is represented as { } + { }. Its Coxeter diagram is CDel node f1.pngCDel 2x.pngCDel node f1.png. Its symmetry is [2].
  • In 3D, a p-gonal bipyramid, is represented as { } + {p}. Its Coxeter diagram is CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel p.pngCDel node.png. Its symmetry is [2,p].
  • In 4D, a {p,q}-hedral bipyramid is represented as { } + {p,q}. Its Coxeter diagram is CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png. Its symmetry is [p,q].
  • In 4D, a p-q duopyramid is represented as {p} + {q}. Its Coxeter diagram is CDel node f1.pngCDel p.pngCDel node.pngCDel 2x.pngCDel node f1.pngCDel q.pngCDel node.png. 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 can be represented as ( ) ∨ { } = ( ) ∨ [( ) ∨ ( )].

In 3D:

  • A digonal disphenoid can be represented as { } ∨ { } = [( ) ∨ ( )] ∨ [( ) ∨ ( )].
  • A p-gonal pyramid is represented as ( ) ∨ {p}.

In 4D:

  • A p-q-hedral pyramid is represented as ( ) ∨ {p,q}.
  • A 5-cell is represented as ( ) ∨ [( ) ∨ {3}] or [( ) ∨ ( )] ∨ {3} = { } ∨ {3}.
  • A square pyramidal pyramid is represented as ( ) ∨ [( ) ∨ {4}] or [( ) ∨ ( )] ∨ {4} = { } ∨ {4}.

When mixing operators, the order of operations from highest to lowest is ×, +, ∨.

Axial polytopes containing vertices on parallel offset hyperplanes can be represented by the || operator. A uniform prism is {n}||{n} and antiprism {n}||r{n}.

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{n}.

Form Schläfli symbol Symmetry Coxeter diagram Example, {6}
Regular {p} [p] CDel node 1.pngCDel p.pngCDel node.png Regular polygon 6 annotated.svg Hexagon CDel node 1.pngCDel 6.pngCDel node.png
Truncated t{p} = {2p} p = [2p] CDel node 1.pngCDel p.pngCDel node 1.png = CDel node 1.pngCDel 2x.pngCDel p.pngCDel node.png Regular polygon 12 annotated.svg Truncated hexagon
(Dodecagon)
CDel node 1.pngCDel 6.pngCDel node 1.png = CDel node 1.pngCDel 12.pngCDel node.png
Altered and
Holosnubbed
a{2p} = β{p} [2p] CDel node h3.pngCDel p.pngCDel node h3.png = CDel node h3.pngCDel 2x.pngCDel p.pngCDel node.png Hexagram.svg Altered hexagon
(Hexagram)
CDel node h3.pngCDel 3.pngCDel node h3.png = CDel node h3.pngCDel 6.pngCDel node.png
Half and
Snubbed
h{2p} = s{p} = {p} [1+,2p] = [p] CDel node h.pngCDel 2x.pngCDel p.pngCDel node.png = CDel node h.pngCDel p.pngCDel node h.png = CDel node 1.pngCDel p.pngCDel node.png Regular polygon 3 annotated.svg Half hexagon
(Triangle)
CDel node h.pngCDel 6.pngCDel node.png = CDel node h.pngCDel 3.pngCDel node h.png = CDel node 1.pngCDel 3.pngCDel node.png

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 [math]\displaystyle{ \begin{Bmatrix} p , q \end{Bmatrix} }[/math] {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 [math]\displaystyle{ t\begin{Bmatrix} p , q \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ t\begin{Bmatrix} q , p \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ \begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ \begin{Bmatrix} q , p \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ r\begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ t\begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] 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, 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.

Alternations
Form Schläfli symbols Symmetry Coxeter diagram Example, {4,3}
Alternated (half) regular [math]\displaystyle{ h \begin{Bmatrix} 2p , q \end{Bmatrix} }[/math] h{2p,q} ht0{2p,q} [1+,2p,q] CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png = CDel labelp.pngCDel branch 10ru.pngCDel split2-qq.pngCDel node.png Tetrahedron.png Demicube
(Tetrahedron)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Snub regular [math]\displaystyle{ s\begin{Bmatrix} p , 2q \end{Bmatrix} }[/math] s{p,2q} ht0,1{p,2q} [p+,2q] CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node.png
Snub dual regular [math]\displaystyle{ s \begin{Bmatrix} q , 2p \end{Bmatrix} }[/math] s{q,2p} ht1,2{2p,q} [2p,q+] CDel node h.pngCDel q.pngCDel node h.pngCDel 2x.pngCDel p.pngCDel node.png CDel node.pngCDel 2x.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 rectified
(p and q are even)
[math]\displaystyle{ h \begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ hr \begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] hrr{p,q} ht0,2{p,q} [(p,q,2+)] CDel node.pngCDel split1-pq.pngCDel branch hh.pngCDel label2.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node h.png
Quartered
(p and q are even)
[math]\displaystyle{ q\begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] 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
[math]\displaystyle{ s\begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] 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

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.

Altered and holosnubbed
Form Schläfli symbols Symmetry Coxeter diagram Example, {4,3}
Altered regular [math]\displaystyle{ a \begin{Bmatrix} p , q \end{Bmatrix} }[/math] a{p,q} at0{p,q} [p,q] CDel node h3.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png = CDel labelp-2.pngCDel branch 10ru.pngCDel split2-qq.pngCDel node.pngCDel labelp-2.pngCDel branch 01rd.pngCDel split2-qq.pngCDel node.png Compound of two tetrahedra.png Stellated octahedron CDel node h3.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Holosnub dual regular ß{q, p} ß{q,p} at0,1{q,p} [p,q] CDel node h3.pngCDel q.pngCDel node h3.pngCDel p.pngCDel node.png CDel node.pngCDel p.pngCDel node h3.pngCDel q.pngCDel node h3.png UC46-2 icosahedra.png Compound of two icosahedra CDel node.pngCDel 4.pngCDel node h3.pngCDel 3.pngCDel node h3.png
ß, looking similar to the greek letter beta (β), is the German alphabet letter eszett.

Polychora and honeycombs

Linear families
Form Schläfli symbol Coxeter diagram Example, {4,3,3}
Regular [math]\displaystyle{ \begin{Bmatrix} p, q , r \end{Bmatrix} }[/math] {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 [math]\displaystyle{ t\begin{Bmatrix} p, q , r \end{Bmatrix} }[/math] 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 [math]\displaystyle{ \left\{\begin{array}{l}p\\q,r\end{array}\right\} }[/math] 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)
[math]\displaystyle{ \left\{\begin{array}{l}q,p\\r\end{array}\right\} }[/math] 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)
[math]\displaystyle{ t\begin{Bmatrix} r, q , p \end{Bmatrix} }[/math] 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)
[math]\displaystyle{ \begin{Bmatrix} r, q , p \end{Bmatrix} }[/math] 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 [math]\displaystyle{ r\left\{\begin{array}{l}p\\q,r\end{array}\right\} }[/math] 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 [math]\displaystyle{ t\left\{\begin{array}{l}p\\q,r\end{array}\right\} }[/math] 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)
[math]\displaystyle{ e_3\begin{Bmatrix} p, q , r \end{Bmatrix} }[/math] e3{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, quarters and snubs

Alternations
Form Schläfli symbol Coxeter diagram Example, {4,3,3}
Alternations
Half
p even
[math]\displaystyle{ h\begin{Bmatrix} p, q , r \end{Bmatrix} }[/math] 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
[math]\displaystyle{ q\begin{Bmatrix} p, q , r \end{Bmatrix} }[/math] 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
[math]\displaystyle{ s\begin{Bmatrix} p, q , r \end{Bmatrix} }[/math] 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
[math]\displaystyle{ s\left\{\begin{array}{l}p\\q,r\end{array}\right\} }[/math] 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 duoprism s{p}s{q} ht0,1,2,3{p,2,q} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel q.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

Bifurcating families
Form Extended Schläfli symbol Coxeter diagram Examples
Quasiregular [math]\displaystyle{ \left\{p,{q\atop q}\right\} }[/math] {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 Demitesseract
(16-cell)
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
Truncated [math]\displaystyle{ t\left\{p,{q\atop q}\right\} }[/math] 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 demitesseract
(Truncated 16-cell)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
Rectified [math]\displaystyle{ \left\{\begin{array}{l}p\\q\\q\end{array}\right\} }[/math] 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 Rectified demitesseract
(24-cell)
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
Cantellated [math]\displaystyle{ r\left\{\begin{array}{l}p\\q\\q\end{array}\right\} }[/math] 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 demitesseract
(Cantellated 16-cell)
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
Cantitruncated [math]\displaystyle{ t\left\{\begin{array}{l}p\\q\\q\end{array}\right\} }[/math] 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 demitesseract
(Cantitruncated 16-cell)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
Snub rectified [math]\displaystyle{ s\left\{\begin{array}{l}p\\q\\q\end{array}\right\} }[/math] 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 demitesseract
(Snub 24-cell)
CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
Quasiregular [math]\displaystyle{ \left\{r,{p\atop q}\right\} }[/math] {r,/q\,p} t0{r,/q\,p} CDel node 1.pngCDel r.pngCDel node.pngCDel split1-pq.pngCDel nodes.png Tetrahedral-octahedral honeycomb.png Tetrahedral-octahedral honeycomb CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png
Truncated [math]\displaystyle{ t\left\{r,{p\atop q}\right\} }[/math] t{r,/q\,p} t0,1{r,/q\,p} CDel node 1.pngCDel r.pngCDel node 1.pngCDel split1-pq.pngCDel nodes.png Truncated Alternated Cubic Honeycomb.svg Truncated tetrahedral-octahedral honeycomb CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes.png
Rectified [math]\displaystyle{ \left\{\begin{array}{l}r\\p\\q\end{array}\right\} }[/math] r{r,/q\,p} t1{r,/q\,p} CDel node.pngCDel r.pngCDel node 1.pngCDel split1-pq.pngCDel nodes.png Rectified cubic honeycomb4.png Rectified tetrahedral-octahedral honeycomb
(Rectified cubic honeycomb)
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes.png
Cantellated [math]\displaystyle{ r\left\{\begin{array}{l}r\\p\\q\end{array}\right\} }[/math] rr{r,/q\,p} t0,2,3{r,/q\,p} CDel node 1.pngCDel r.pngCDel node.pngCDel split1-pq.pngCDel nodes 11.png Cantellated cubic honeycomb.jpg Cantellated cubic honeycomb CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes 11.png
Cantitruncated [math]\displaystyle{ t\left\{\begin{array}{l}r\\p\\q\end{array}\right\} }[/math] 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 Cantitruncated Cubic Honeycomb.svg Cantitruncated cubic honeycomb CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes 11.png
Snub rectified [math]\displaystyle{ s\left\{\begin{array}{l}p\\q\\r\end{array}\right\} }[/math] 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 Alternated cantitruncated cubic honeycomb.png Snub rectified cubic honeycomb (non-uniform, but near miss) CDel node h.pngCDel 3.pngCDel node h.pngCDel split1-43.pngCDel nodes hh.png

Tessellations

Spherical

Regular

Semi-regular

Hyperbolic

References

  1. 1.0 1.1 1.2 1.3 Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover. 

Sources

External links