In geometry, 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 Tessellations 5 See also 6 References 7 Sources 8 External links Description[edit]
The
Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols The
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[edit] Polygons and circle tilings[edit] 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
Regular p [p] Hexagon Truncated t p = 2p [[p]] = [2p] = Truncated hexagon (Dodecagon) = Altered and Holosnubbed a 2p = β p [2p] = Altered hexagon (Hexagram) = Half and Snubbed h 2p = s p = p [1+,2p] = [p] = = Half hexagon (Triangle) = = Polyhedra and tilings[edit]
Form
Schläfli symbols
Symmetry
Regular p , q displaystyle begin Bmatrix p,qend Bmatrix p,q t0 p,q [p,q] or [(p,q,2)] Cube Truncated t p , q displaystyle t begin Bmatrix p,qend Bmatrix t p,q t0,1 p,q Truncated cube Bitruncation (Truncated dual) t q , p displaystyle t begin Bmatrix q,pend Bmatrix 2t p,q t1,2 p,q Truncated octahedron Rectified (Quasiregular) p q displaystyle begin Bmatrix p\qend Bmatrix r p,q t1 p,q Cuboctahedron Birectification (Regular dual) q , p displaystyle begin Bmatrix q,pend Bmatrix 2r p,q t2 p,q Octahedron Cantellated (Rectified rectified) r p q displaystyle r begin Bmatrix p\qend Bmatrix rr p,q t0,2 p,q Rhombicuboctahedron Cantitruncated (Truncated rectified) t p q displaystyle t begin Bmatrix p\qend Bmatrix tr p,q t0,1,2 p,q Truncated cuboctahedron Alternations, quarters and snubs[edit]
Alternations have half the symmetry of the
Alternations Form
Schläfli symbols
Symmetry
Alternated (half) regular h 2 p , q displaystyle h begin Bmatrix 2p,qend Bmatrix h 2p,q ht0 2p,q [1+,2p,q] = Demicube (Tetrahedron) Snub regular s p , 2 q displaystyle s begin Bmatrix p,2qend Bmatrix s p,2q ht0,1 p,2q [p+,2q] Snub dual regular s q , 2 p displaystyle s begin Bmatrix q,2pend Bmatrix s q,2p ht1,2 2p,q [2p,q+] Snub octahedron (Icosahedron) Alternated rectified (p and q are even) h p q displaystyle h begin Bmatrix p\qend Bmatrix hr p,q ht1 p,q [p,1+,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 [(p,q,2+)] Quartered (p and q are even) q p q displaystyle q begin Bmatrix p\qend Bmatrix q p,q ht0ht2 p,q [1+,p,q,1+] Snub rectified Snub quasiregular s p q displaystyle s begin Bmatrix p\qend Bmatrix sr p,q ht0,1,2 p,q [p,q]+ Snub cuboctahedron (Snub cube) Altered and holosnubbed[edit] 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
Altered regular a p , q displaystyle a begin Bmatrix p,qend Bmatrix a p,q at0 p,q [p,q] = ∪ Stellated octahedron Holosnub dual regular ß q , p displaystyle begin Bmatrix q,pend Bmatrix
Compound of two icosahedra ß, looking similar to the greek letter beta (β), is the German alphabet letter eszett. Polychora and honeycombs[edit] Linear families Form
Schläfli symbol
Regular p , q , r displaystyle begin Bmatrix p,q,rend Bmatrix p,q,r t0 p,q,r Tesseract Truncated t p , q , r displaystyle t begin Bmatrix p,q,rend Bmatrix t p,q,r t0,1 p,q,r Truncated tesseract Rectified p q , r displaystyle left begin array l p\q,rend array right r p,q,r t1 p,q,r Rectified tesseract = Bitruncated 2t p,q,r t1,2 p,q,r Bitruncated tesseract Birectified (Rectified dual) 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 Rectified 16-cell = 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 Bitruncated tesseract 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 16-cell Cantellated r p q , r displaystyle rleft begin array l p\q,rend array right rr p,q,r t0,2 p,q,r Cantellated tesseract = 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 Cantitruncated tesseract = 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 Runcinated tesseract Runcitruncated t0,1,3 p,q,r Runcitruncated tesseract Omnitruncated t0,1,2,3 p,q,r Omnitruncated tesseract Alternations, quarters and snubs[edit] Alternations Form
Schläfli symbol
Alternations Half p even h p , q , r displaystyle h begin Bmatrix p,q,rend Bmatrix h p,q,r ht0 p,q,r 16-cell 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 24-cell 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 Snub 24-cell = Alternated duoprism s p s q ht0,1,2,3 p,2,q Great duoantiprism Bifurcating families[edit] Bifurcating families Form
Extended Schläfli symbol
Quasiregular p , q q displaystyle left p, q atop q right p,q1,1 t0 p,q1,1 16-cell Truncated t p , q q displaystyle tleft p, q atop q right t p,q1,1 t0,1 p,q1,1 Truncated 16-cell Rectified p q q displaystyle left begin array l p\q\qend array right r p,q1,1 t1 p,q1,1 24-cell 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 16-cell 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 16-cell 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 Snub 24-cell 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 array right rr r,/q,p t0,2,3 r,/q,p Cantitruncated t r p q displaystyle tleft begin array l r\p\qend array right tr r,/q,p t0,1,2,3 r,/q,p Snub rectified s p q r displaystyle sleft begin array l p\q\rend array right sr p,/q,r ht0,1,2,3 p,/q,r Tessellations[edit] Spherical 2,n s 2,2n ??? t 2, n + n Regular 2,∞ 3,6 4,4 6,3 Semi-regular s 4,4 e 3,6 sr 2,∞ sr 6,3 rr 6,3 r 6,3 t 6,3 t 2,∞ tr 6,3 tr 4,4 Hyper-bolic sr 5,4 sr 6,4 sr 7,4 sr 8,4 sr ∞,4 s 5,4 sr 6,5 s 6,4 sr 8,6 sr 7,7 s 8,4 s 4,6 sr ∞,∞ sr 7,3 sr 8,3 sr ∞,3 s 3,8 3,7 3,8 3,∞ h 8,3 h 6,4 q 4,6 rr 7,3 rr 8,3 rr ∞,3 h2 8,3 r 7,3 r 8,3 t 7,3 t 8,3 r ∞,3 t ∞,3 rr 5,4 rr 6,4 rr 7,4 rr 8,4 rr ∞,4 4,5 4,6 4,7 4,8 4,∞ r 5,4 r 6,4 tr 6,3 tr 7,3 ??? tr 8,3 ??? tr ∞,3 r 7,4 r 8,4 tr 5,4 ??? tr 6,4 tr 7,4 tr 8,4 tr ∞,4 t 5,4 tr 6,5 t 6,4 tr 8,6 t 7,4 t 8,4 t ∞,4 r ∞,4 5,4 5,5 5,6 5,∞ rr 6,5 r 6,5 t 4,5 t 5,5 t 6,5 r ∞,5 6,4 6,5 6,6 6,8 rr 8,6 r 8,6 t 4,6 t 5,6 t 6,6 t 8,6 7,3 7,4 7,7 t 3,7 t 4,7 t 7,7 8,3 8,4 8,6 8,8 t 6,8 t 3,8 t 8,8 ∞,3 ∞,4 ∞,5 ∞,∞ t 3,∞ t 4,∞ See also[edit] Regular Polytopes, by Harold Scott MacDonald Coxeter References[edit] Sources[edit] Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. pp. 14, 69, 149. ISBN 0-486-61480-8. OCLC 798003. 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] Weisstein, Eric W. "Schläfli symbol". MathWorld. Polyhedral |