In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The
CONTENTS * 1 Description * 2 Cases * 2.1 Symmetry groups * 2.2 Regular polygons (plane) * 2.3 Regular polyhedra (3 dimensions) * 2.4 Regular 4-polytopes (4 dimensions) * 2.5 Regular n-polytopes (higher dimensions) * 2.6 Dual polytopes * 2.7 Prismatic polytopes * 3 Extension of Schläfli symbols * 3.1 Polygons and circle tilings * 3.2 Polyhedra and tilings * 3.2.1 Alternations, quarters and snubs * 3.2.2 Altered and holosnubbed * 3.3 Polychora and honeycombs * 3.3.1 Alternations, quarters and snubs * 3.3.2 Bifurcating families * 4 See also * 5 References * 6 Sources * 7 External links DESCRIPTION The
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, a cell in a 6-polytope, and an (n-3)-face in an n-polytope. 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 {5/2}, represented by the vertices of a pentagon but connected alternately. The
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
CASES SYMMETRY GROUPS Schläfli symbols are closely related to (finite) reflection symmetry
groups , which correspond precisely to the finite
REGULAR POLYGONS (PLANE) Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols The
For (nonconvex) star polygons , the constructive notation p/s is used, where p is the number of vertices and s-1 is the number skipped when drawing each edge of the star. For example, {5/2} represents the pentagram . REGULAR POLYHEDRA (3 DIMENSIONS) The
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
See the six convex regular and 10 regular star 4-polytopes . For example, the
There is one regular tessellation of Euclidean 3-space: the cubic
honeycomb , with a
There are also 4 regular compact hyperbolic tessellations including {5,3,4}, the hyperbolic small dodecahedral honeycomb , which fills space with dodecahedron cells. REGULAR N-POLYTOPES (HIGHER DIMENSIONS) For higher-dimensional regular polytopes , the
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
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
The prismatic duals, or bipyramids can be represented as composite symbols, but with the addition operator, "+". * In 2D, a rhombus is represented as { } + { }. Its
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
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} TRUNCATED t{p} = {2p} ] = = Truncated hexagon (Dodecagon) = Altered and Holosnubbed a{2p} = β{p} = Altered hexagon (Hexagram) = Half and Snubbed h{2p} = s{p} = {p} = = = Half hexagon (Triangle) = = POLYHEDRA AND TILINGS
FORM SCHLäFLI SYMBOLS SYMMETRY COXETER DIAGRAM EXAMPLE, {4,3} REGULAR { p , q } {displaystyle {begin{Bmatrix}p,qend{Bmatrix}}} {p,q} t0{p,q} or TRUNCATED t { p , q } {displaystyle t{begin{Bmatrix}p,qend{Bmatrix}}} t{p,q} t0,1{p,q}
Rectified (Quasiregular ) { p q } {displaystyle {begin{Bmatrix}p\qend{Bmatrix}}} r{p,q} t1{p,q} Birectification (Regular dual) { q , p } {displaystyle {begin{Bmatrix}q,pend{Bmatrix}}} 2r{p,q} t2{p,q} Cantellated (Rectified rectified ) r { p q } {displaystyle r{begin{Bmatrix}p\qend{Bmatrix}}} rr{p,q} t0,2{p,q} Cantitruncated (Truncated rectified) t { p q } {displaystyle t{begin{Bmatrix}p\qend{Bmatrix}}} tr{p,q} t0,1,2{p,q} Alternations, Quarters And Snubs Alternations have half the symmetry of the
Alternations FORM SCHLäFLI SYMBOLS SYMMETRY COXETER DIAGRAM EXAMPLE, {4,3} ALTERNATED (HALF) REGULAR h { 2 p , q } {displaystyle h{begin{Bmatrix}2p,qend{Bmatrix}}} h{2p,q} ht0{2p,q} = Demicube
(
SNUB REGULAR s { p , 2 q } {displaystyle s{begin{Bmatrix}p,2qend{Bmatrix}}} s{p,2q} ht0,1{p,2q} SNUB DUAL REGULAR s { q , 2 p } {displaystyle s{begin{Bmatrix}q,2pend{Bmatrix}}} s{q,2p} ht1,2{2p,q} Snub octahedron
(
Alternated rectified (p and q are even) h { p q } {displaystyle h{begin{Bmatrix}p\qend{Bmatrix}}} hr{p,q} ht1{p,q} Alternated rectified rectified (p and q are even) h r { p q } {displaystyle hr{begin{Bmatrix}p\qend{Bmatrix}}} hrr{p,q} ht0,2{p,q} Quartered (p and q are even) q { p q } {displaystyle q{begin{Bmatrix}p\qend{Bmatrix}}} q{p,q} ht0ht2{p,q} Snub rectified Snub quasiregular s { p q } {displaystyle s{begin{Bmatrix}p\qend{Bmatrix}}} sr{p,q} ht0,1,2{p,q} +
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 a { p , q } {displaystyle a{begin{Bmatrix}p,qend{Bmatrix}}} a{p,q} at0{p,q} = ∪ HOLOSNUB DUAL REGULAR
POLYCHORA AND HONEYCOMBS Linear families FORM SCHLäFLI SYMBOL COXETER DIAGRAM EXAMPLE, {4,3,3} REGULAR { p , q , r } {displaystyle {begin{Bmatrix}p,q,rend{Bmatrix}}} {p,q,r} t0{p,q,r} TRUNCATED t { p , q , r } {displaystyle t{begin{Bmatrix}p,q,rend{Bmatrix}}} t{p,q,r} t0,1{p,q,r} RECTIFIED { p q , r } {displaystyle left{{begin{array}{l}p\q,rend{array}}right}} r{p,q,r} t1{p,q,r} BITRUNCATED 2t{p,q,r} t1,2{p,q,r} Birectified (Rectified dual) { q , p r } {displaystyle left{{begin{array}{l}q,p\rend{array}}right}} 2r{p,q,r} = r{r,q,p} t2{p,q,r} Tritruncated (Truncated dual) t { r , q , p } {displaystyle t{begin{Bmatrix}r,q,pend{Bmatrix}}} 3t{p,q,r} = t{r,q,p} t2,3{p,q,r} Trirectified (Dual) { r , q , p } {displaystyle {begin{Bmatrix}r,q,pend{Bmatrix}}} 3r{p,q,r} = {r,q,p} t3{p,q,r} = {r,q,p} CANTELLATED r { p q , r } {displaystyle rleft{{begin{array}{l}p\q,rend{array}}right}} rr{p,q,r} t0,2{p,q,r} CANTITRUNCATED t { p q , r } {displaystyle tleft{{begin{array}{l}p\q,rend{array}}right}} tr{p,q,r} t0,1,2{p,q,r} Runcinated (Expanded ) e 3 { p , q , r } {displaystyle e_{3}{begin{Bmatrix}p,q,rend{Bmatrix}}} e3{p,q,r} t0,3{p,q,r} RUNCITRUNCATED t0,1,3{p,q,r} OMNITRUNCATED t0,1,2,3{p,q,r} Alternations, Quarters And Snubs Alternations FORM SCHLäFLI SYMBOL COXETER DIAGRAM EXAMPLE, {4,3,3} ALTERNATIONS Half p even h { p , q , r } {displaystyle h{begin{Bmatrix}p,q,rend{Bmatrix}}} h{p,q,r} ht0{p,q,r} Quarter p and r even q { p , q , r } {displaystyle q{begin{Bmatrix}p,q,rend{Bmatrix}}} q{p,q,r} ht0ht3{p,q,r} Snub q even s { p , q , r } {displaystyle s{begin{Bmatrix}p,q,rend{Bmatrix}}} s{p,q,r} ht0,1{p,q,r} Snub rectified r even s { p q , r } {displaystyle sleft{{begin{array}{l}p\q,rend{array}}right}} sr{p,q,r} ht0,1,2{p,q,r} ALTERNATED DUOPRISM s{p}s{q} ht0,1,2,3{p,2,q} Bifurcating Families Bifurcating families FORM EXTENDED SCHLäFLI SYMBOL COXETER DIAGRAM EXAMPLES QUASIREGULAR { p , q q } {displaystyle left{p,{q atop q}right}} {p,q1,1} t0{p,q1,1} TRUNCATED t { p , q q } {displaystyle tleft{p,{q atop q}right}} t{p,q1,1} t0,1{p,q1,1} Truncated
RECTIFIED { p q q } {displaystyle left{{begin{array}{l}p\q\qend{array}}right}} r{p,q1,1} t1{p,q1,1} CANTELLATED r { p q q } {displaystyle rleft{{begin{array}{l}p\q\qend{array}}right}} rr{p,q1,1} t0,2,3{p,q1,1} Cantellated
CANTITRUNCATED t { p q q } {displaystyle tleft{{begin{array}{l}p\q\qend{array}}right}} tr{p,q1,1} t0,1,2,3{p,q1,1} Cantitruncated
SNUB RECTIFIED s { p q q } {displaystyle sleft{{begin{array}{l}p\q\qend{array}}right}} sr{p,q1,1} ht0,1,2,3{p,q1,1} QUASIREGULAR { r , p q } {displaystyle left{r,{p atop q}right}} {r,/q,p} t0{r,/q,p} TRUNCATED t { r , p q } {displaystyle tleft{r,{p atop q}right}} t{r,/q,p} t0,1{r,/q,p} RECTIFIED { r p q } {displaystyle left{{begin{array}{l}r\p\qend{array}}right}} r{r,/q,p} t1{r,/q,p} CANTELLATED r { r p q } {displaystyle rleft{{begin{array}{l}r\p\qend |