Schläfli symbol

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Mar 13, 2025 / 20:55

In geometry the Schläfli symbol is a notation of the form p q r displaystyle p q r that defines regular polytopes and te

Schläfli symbol
Schläfli symbol
Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

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

The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli,: 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,: 129  starting with image for a image for-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 image for contain irreducible fractions image, where image is the number of vertices, and image is their turning number. Equivalently, image is created from the vertices of {p}, connected every q. For example, image is a pentagram; image is a pentagon.

A regular polyhedron that has image regular image-sided polygon faces around each vertex is represented by image. For example, the cube has 3 squares around each vertex and is represented by {4,3}.

A regular 4-dimensional polytope, with image regular polyhedral cells around each edge is represented by image. For example, a tesseract, {4,3,3}, has 3 cubes, {4,3}, around an edge.

In general, a regular polytope image has image 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.: 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.: 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)

image
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.

image

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 image. Its symmetry is [ ].
  • In 2D, a rectangle is represented as { } × { }. Its Coxeter diagram is imageimageimage. Its symmetry is [2].
  • In 3D, a p-gonal prism is represented as { } × {p}. Its Coxeter diagram is imageimageimageimageimage. Its symmetry is [2,p].
  • In 4D, a uniform {p,q}-hedral prism is represented as { } × {p,q}. Its Coxeter diagram is imageimageimageimageimageimageimage. Its symmetry is [2,p,q].
  • In 4D, a uniform p-q duoprism is represented as {p} × {q}. Its Coxeter diagram is imageimageimageimageimageimageimage. 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 imageimageimage. Its symmetry is [2].
  • In 3D, a p-gonal bipyramid, is represented as { } + {p}. Its Coxeter diagram is imageimageimageimageimage. Its symmetry is [2,p].
  • In 4D, a {p,q}-hedral bipyramid is represented as { } + {p,q}. Its Coxeter diagram is imageimageimageimageimageimageimage. Its symmetry is [p,q].
  • In 4D, a p-q duopyramid is represented as {p} + {q}. Its Coxeter diagram is imageimageimageimageimageimageimage. 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] imageimageimage image Hexagon imageimageimage
Truncated t{p} = {2p} [[p]] = [2p] imageimageimage = imageimageimageimage image Truncated hexagon
(Dodecagon)
imageimageimage = imageimageimage
Altered and
Holosnubbed
a{2p} = β{p} [2p] imageimageimage = imageimageimageimage image Altered hexagon
(Hexagram)
imageimageimage = imageimageimage
Half and
Snubbed
h{2p} = s{p} = {p} [1+,2p] = [p] imageimageimageimage = imageimageimage = imageimageimage image Half hexagon
(Triangle)
imageimageimage = imageimageimage = imageimageimage

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 image {p,q} t0{p,q} [p,q]
or
[(p,q,2)]
imageimageimageimageimage image Cube imageimageimageimageimage
Truncated image t{p,q} t0,1{p,q} imageimageimageimageimage image Truncated cube imageimageimageimageimage
Bitruncation
(Truncated dual)
image 2t{p,q} t1,2{p,q} imageimageimageimageimage imageimageimageimageimage image Truncated octahedron imageimageimageimageimage
Rectified
(Quasiregular)
image r{p,q} t1{p,q} imageimageimage imageimageimageimageimage image Cuboctahedron imageimageimageimageimage
Birectification
(Regular dual)
image 2r{p,q} t2{p,q} imageimageimageimageimage imageimageimageimageimage image Octahedron imageimageimageimageimage
Cantellated
(Rectified rectified)
image rr{p,q} t0,2{p,q} imageimageimage imageimageimageimageimage image Rhombicuboctahedron imageimageimageimageimage
Cantitruncated
(Truncated rectified)
image tr{p,q} t0,1,2{p,q} imageimageimage imageimageimageimageimage image Truncated cuboctahedron imageimageimageimageimage

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 image h{2p,q} ht0{2p,q} [1+,2p,q] imageimageimageimageimageimage = imageimageimageimage image Demicube
(Tetrahedron)
imageimageimageimageimage
Snub regular image s{p,2q} ht0,1{p,2q} [p+,2q] imageimageimageimageimageimage
Snub dual regular image s{q,2p} ht1,2{2p,q} [2p,q+] imageimageimageimageimageimage imageimageimageimageimageimage image Snub octahedron
(Icosahedron)
imageimageimageimageimage
Alternated rectified
(p and q are even)
image hr{p,q} ht1{p,q} [p,1+,q] imageimageimage imageimageimageimageimage
Alternated rectified rectified
(p and q are even)
image hrr{p,q} ht0,2{p,q} [(p,q,2+)] imageimageimageimage imageimageimageimageimage
Quartered
(p and q are even)
image q{p,q} ht0ht2{p,q} [1+,p,q,1+] imageimageimage imageimageimageimageimage
Snub rectified
Snub quasiregular
image sr{p,q} ht0,1,2{p,q} [p,q]+ imageimageimage imageimageimageimageimage image Snub cuboctahedron
(Snub cube)
imageimageimageimageimage

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 image a{p,q} at0{p,q} [p,q] imageimageimageimageimage = imageimageimageimageimageimageimageimage image Stellated octahedron imageimageimageimageimage
Holosnub dual regular ß{q, p} ß{q,p} at0,1{q,p} [p,q] imageimageimageimageimage imageimageimageimageimage image Compound of two icosahedra imageimageimageimageimage
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 image {p,q,r} t0{p,q,r} imageimageimageimageimageimageimage image Tesseract imageimageimageimageimageimageimage
Truncated image t{p,q,r} t0,1{p,q,r} imageimageimageimageimageimageimage image Truncated tesseract imageimageimageimageimageimageimage
Rectified image r{p,q,r} t1{p,q,r} imageimageimageimageimageimageimage image Rectified tesseract imageimageimageimageimageimageimage = imageimageimageimageimage
Bitruncated 2t{p,q,r} t1,2{p,q,r} imageimageimageimageimageimageimage image Bitruncated tesseract imageimageimageimageimageimageimage
Birectified
(Rectified dual)
image 2r{p,q,r} = r{r,q,p} t2{p,q,r} imageimageimageimageimageimageimage image Rectified 16-cell imageimageimageimageimageimageimage = imageimageimageimageimage
Tritruncated
(Truncated dual)
image 3t{p,q,r} = t{r,q,p} t2,3{p,q,r} imageimageimageimageimageimageimage image Bitruncated tesseract imageimageimageimageimageimageimage
Trirectified
(Dual)
image 3r{p,q,r} = {r,q,p} t3{p,q,r} = {r,q,p} imageimageimageimageimageimageimage image 16-cell imageimageimageimageimageimageimage
Cantellated image rr{p,q,r} t0,2{p,q,r} imageimageimageimageimageimageimage image Cantellated tesseract imageimageimageimageimageimageimage = imageimageimageimageimage
Cantitruncated image tr{p,q,r} t0,1,2{p,q,r} imageimageimageimageimageimageimage image Cantitruncated tesseract imageimageimageimageimageimageimage = imageimageimageimageimage
Runcinated
(Expanded)
image e3{p,q,r} t0,3{p,q,r} imageimageimageimageimageimageimage image Runcinated tesseract imageimageimageimageimageimageimage
Runcitruncated t0,1,3{p,q,r} imageimageimageimageimageimageimage image Runcitruncated tesseract imageimageimageimageimageimageimage
Omnitruncated t0,1,2,3{p,q,r} imageimageimageimageimageimageimage image Omnitruncated tesseract imageimageimageimageimageimageimage

Alternations, quarters and snubs

Alternations
Form Schläfli symbol Coxeter diagram Example, {4,3,3}
Alternations
Half
p even
image h{p,q,r} ht0{p,q,r} imageimageimageimageimageimageimage image 16-cell imageimageimageimageimageimageimage
Quarter
p and r even
image q{p,q,r} ht0ht3{p,q,r} imageimageimageimageimageimageimage
Snub
q even
image s{p,q,r} ht0,1{p,q,r} imageimageimageimageimageimageimage image Snub 24-cell imageimageimageimageimageimageimage
Snub rectified
r even
image sr{p,q,r} ht0,1,2{p,q,r} imageimageimageimageimageimageimage image Snub 24-cell imageimageimageimageimageimageimage = imageimageimageimageimage
Alternated duoprism s{p}s{q} ht0,1,2,3{p,2,q} imageimageimageimageimageimageimage image Great duoantiprism imageimageimageimageimageimageimageimageimage

Bifurcating families

Bifurcating families
Form Extended Schläfli symbol Coxeter diagram Examples
Quasiregular image {p,q1,1} t0{p,q1,1} imageimageimageimageimage image Demitesseract
(16-cell)
imageimageimageimageimage
Truncated image t{p,q1,1} t0,1{p,q1,1} imageimageimageimageimage image Truncated demitesseract
(Truncated 16-cell)
imageimageimageimageimage
Rectified image r{p,q1,1} t1{p,q1,1} imageimageimageimageimage image Rectified demitesseract
(24-cell)
imageimageimageimageimage
Cantellated image rr{p,q1,1} t0,2,3{p,q1,1} imageimageimageimageimage image Cantellated demitesseract
(Cantellated 16-cell)
imageimageimageimageimage
Cantitruncated image tr{p,q1,1} t0,1,2,3{p,q1,1} imageimageimageimageimage image Cantitruncated demitesseract
(Cantitruncated 16-cell)
imageimageimageimageimage
Snub rectified image sr{p,q1,1} ht0,1,2,3{p,q1,1} imageimageimageimageimage image Snub demitesseract
(Snub 24-cell)
imageimageimageimageimage
Quasiregular image {r,/q\,p} t0{r,/q\,p} imageimageimageimageimage image Tetrahedral-octahedral honeycomb imageimageimageimageimage
Truncated image t{r,/q\,p} t0,1{r,/q\,p} imageimageimageimageimage image Truncated tetrahedral-octahedral honeycomb imageimageimageimageimage
Rectified image r{r,/q\,p} t1{r,/q\,p} imageimageimageimageimage image Rectified tetrahedral-octahedral honeycomb
(Rectified cubic honeycomb)

In geometry the Schlafli symbol is a notation of the form p q r displaystyle p q r that defines regular polytopes and tessellations The dodecahedron is a regular polyhedron with Schlafli symbol 5 3 having 3 pentagons around each vertex The Schlafli symbol is named after the 19th century Swiss mathematician Ludwig Schlafli 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 DefinitionThe Schlafli symbol is a recursive description 129 starting with p displaystyle p for a p displaystyle p for 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 Schlafli symbols p q displaystyle p q for contain irreducible fractions p q displaystyle p q where p displaystyle p is the number of vertices and q displaystyle q is their turning number Equivalently p q displaystyle p q is created from the vertices of p connected every q For example 5 2 displaystyle 5 2 is a pentagram 5 1 displaystyle 5 1 is a pentagon A regular polyhedron that has q displaystyle q regular p displaystyle p sided polygon faces around each vertex is represented by p q displaystyle 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 displaystyle r p q regular polyhedral cells around each edge is represented by p q r displaystyle 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 displaystyle p q r dots y z has z p q r y displaystyle z p q r dots 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 PropertiesA 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 5 2 represented by the vertices of a pentagon but connected alternately The Schlafli 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 Schlafli symbol elements in reverse order A self dual regular polytope will have a symmetric Schlafli symbol In addition to describing Euclidean polytopes Schlafli symbols can be used to describe spherical polytopes or spherical honeycombs 138 History and variationsSchlafli 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 Schlafli symbol which he wrote as p q r z rather than with brackets and commas as Schlafli did 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 CasesSymmetry groups Schlafli 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 Schlafli symbols The Schlafli 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 p q 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 5 2 represents the pentagram Regular polyhedra 3 dimensions The Schlafli 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 Schlafli 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 Schlafli 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 Schlafli 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 Schlafli symbol is defined recursively as p1 p2 pn 1 if the facets have Schlafli symbol p1 p2 pn 2 and the vertex figures have Schlafli 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 Dual polytopes If a polytope of dimension n 2 has Schlafli symbol p1 p2 pn 1 then its dual has Schlafli 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 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 Its symmetry is In 2D a rectangle is represented as Its Coxeter diagram is Its symmetry is 2 In 3D a p gonal prism is represented as p Its Coxeter diagram is Its symmetry is 2 p In 4D a uniform p q hedral prism is represented as p q Its Coxeter diagram is Its symmetry is 2 p q In 4D a uniform p q duoprism is represented as p q Its Coxeter diagram is 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 Its symmetry is 2 In 3D a p gonal bipyramid is represented as p Its Coxeter diagram is Its symmetry is 2 p In 4D a p q hedral bipyramid is represented as p q Its Coxeter diagram is Its symmetry is p q In 4D a p q duopyramid is represented as p q Its Coxeter diagram is 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 Schlafli symbolsPolygons 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 Schlafli symbol Symmetry Coxeter diagram Example 6 Regular p p HexagonTruncated t p 2p p 2p Truncated hexagon Dodecagon Altered and Holosnubbed a 2p b p 2p Altered hexagon Hexagram Half and Snubbed h 2p s p p 1 2p p Half hexagon Triangle Polyhedra and tilings Coxeter expanded his usage of the Schlafli 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 Schlafli symbols Symmetry Coxeter diagram Example 4 3 Regular p q displaystyle begin Bmatrix p q end Bmatrix p q t0 p q p q or p q 2 CubeTruncated t p q displaystyle t begin Bmatrix p q end Bmatrix t p q t0 1 p q Truncated cubeBitruncation Truncated dual t q p displaystyle t begin Bmatrix q p end Bmatrix 2t p q t1 2 p q Truncated octahedronRectified Quasiregular pq displaystyle begin Bmatrix p q end Bmatrix r p q t1 p q CuboctahedronBirectification Regular dual q p displaystyle begin Bmatrix q p end Bmatrix 2r p q t2 p q OctahedronCantellated Rectified rectified r pq displaystyle r begin Bmatrix p q end Bmatrix rr p q t0 2 p q RhombicuboctahedronCantitruncated Truncated rectified t pq displaystyle t begin Bmatrix p q end Bmatrix tr p q t0 1 2 p q Truncated cuboctahedronAlternations 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 Schlafli symbols Symmetry Coxeter diagram Example 4 3 Alternated half regular h 2p q displaystyle h begin Bmatrix 2p q end Bmatrix h 2p q ht0 2p q 1 2p q Demicube Tetrahedron Snub regular s p 2q displaystyle s begin Bmatrix p 2q end Bmatrix s p 2q ht0 1 p 2q p 2q Snub dual regular s q 2p displaystyle s begin Bmatrix q 2p end Bmatrix s q 2p ht1 2 2p q 2p q Snub octahedron Icosahedron Alternated rectified p and q are even h pq displaystyle 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 pq displaystyle hr begin Bmatrix p q end Bmatrix hrr p q ht0 2 p q p q 2 Quartered p and q are even q pq displaystyle q begin Bmatrix p q end Bmatrix q p q ht0ht2 p q 1 p q 1 Snub rectified Snub quasiregular s pq displaystyle s begin Bmatrix p q end Bmatrix sr p q ht0 1 2 p q p q Snub cuboctahedron Snub cube 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 Schlafli symbols Symmetry Coxeter diagram Example 4 3 Altered regular a p q displaystyle a begin Bmatrix p q end Bmatrix a p q at0 p q p q Stellated octahedronHolosnub dual regular ss q p ss q p at0 1 q p p q Compound of two icosahedrass looking similar to the greek letter beta b is the German alphabet letter eszett Polychora and honeycombs Linear families Form Schlafli symbol Coxeter diagram Example 4 3 3 Regular p q r displaystyle begin Bmatrix p q r end Bmatrix p q r t0 p q r TesseractTruncated t p q r displaystyle t begin Bmatrix p q r end Bmatrix t p q r t0 1 p q r Truncated tesseractRectified pq r displaystyle 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 tesseractBirectified Rectified dual q pr displaystyle 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 r q p displaystyle t begin Bmatrix r q p end Bmatrix 3t p q r t r q p t2 3 p q r Bitruncated tesseractTrirectified Dual r q p displaystyle begin Bmatrix r q p end Bmatrix 3r p q r r q p t3 p q r r q p 16 cellCantellated r pq r displaystyle r left begin array l p q r end array right rr p q r t0 2 p q r Cantellated tesseract Cantitruncated t pq r displaystyle 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 e3 p q r displaystyle e 3 begin Bmatrix p q r end Bmatrix e3 p q r t0 3 p q r Runcinated tesseractRuncitruncated t0 1 3 p q r Runcitruncated tesseractOmnitruncated t0 1 2 3 p q r Omnitruncated tesseractAlternations quarters and snubs Alternations Form Schlafli symbol Coxeter diagram Example 4 3 3 AlternationsHalf p even h p q r displaystyle h begin Bmatrix p q r end Bmatrix h p q r ht0 p q r 16 cellQuarter p and r even q p q r displaystyle q begin Bmatrix p q r end Bmatrix q p q r ht0ht3 p q r Snub q even s p q r displaystyle s begin Bmatrix p q r end Bmatrix s p q r ht0 1 p q r Snub 24 cellSnub rectified r even s pq r displaystyle s left begin array l p q r end array right sr p q r ht0 1 2 p q r Snub 24 cell Alternated duoprism s p s q ht0 1 2 3 p 2 q Great duoantiprismBifurcating families Bifurcating families Form Extended Schlafli symbol Coxeter diagram ExamplesQuasiregular p qq displaystyle left p q atop q right p q1 1 t0 p q1 1 Demitesseract 16 cell Truncated t p qq displaystyle t left p q atop q right t p q1 1 t0 1 p q1 1 Truncated demitesseract Truncated 16 cell Rectified pqq displaystyle left begin array l p q q end array right r p q1 1 t1 p q1 1 Rectified demitesseract 24 cell Cantellated r pqq displaystyle r left begin array l p q q end array right rr p q1 1 t0 2 3 p q1 1 Cantellated demitesseract Cantellated 16 cell Cantitruncated t pqq displaystyle t left begin array l p q q end array right tr p q1 1 t0 1 2 3 p q1 1 Cantitruncated demitesseract Cantitruncated 16 cell Snub rectified s pqq displaystyle s left begin array l p q q end array right sr p q1 1 ht0 1 2 3 p q1 1 Snub demitesseract Snub 24 cell Quasiregular r pq displaystyle left r p atop q right r q p t0 r q p Tetrahedral octahedral honeycombTruncated t r pq displaystyle t left r p atop q right t r q p t0 1 r q p Truncated tetrahedral octahedral honeycombRectified rpq displaystyle left begin array l r p q end array right r r q p t1 r q p Rectified tetrahedral octahedral honeycomb Rectified cubic honeycomb img

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