In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It...

grouped by the Wythoff symbol. All the faces are identical, each edge is identical and each vertex is identical. They all have a Wythoff symbol of the form...

A vertex is placed at the point A. This produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and...

There are different notations for expressing these uniform solutions, Wythoff symbol, Coxeter diagram, and Coxeter's t-notation. Simple tiles are generated...

figure. The nine forms, listed with their Wythoff symbols and vertex configurations are: Note that Wythoff's kaleidoscopic construction generates the nonorientable...

triangular faces. Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r. The Coxeter-Dynkin...

polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol | 3⁄2 5⁄3 3 5⁄2, relating it to a spherical...

plane. With orbifold notation symmetry of *n32 all of these tilings are Wythoff constructions within a fundamental domain of symmetry, with generator points...

only its Wythoff symbol. The 57 nonprismatic nonconvex forms, with exception of the great dirhombicosidodecahedron, are compiled by Wythoff constructions...

sections. All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon...

only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but that represents a double covering of the tetrahemihexahedron...

hexagon passing through the origin The Wythoff symbol relates the polyhedron to spherical triangles. Wythoff symbols are written p|q r, p q|r, p q r| where...

by the wallpaper group p6mm, (*632), and the tiling can be derived as a Wythoff construction within the reflectional fundamental domains of this group...

vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4⁄3 4 | 3, although that is a double-covering of this figure. A nonconvex...

domain mirrors meeting at 90 degrees generate no new mirrors). The Wythoff symbol takes the three integers and separates them by a vertical bar (|). If...

honeycomb Convex uniform honeycombs in hyperbolic space Wythoff construction and Wythoff symbol Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations:...

are all point groups. For example, the snub cube: Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3.p.3.q.3.r. Retrosnub...

equivalent constructions, for example Wythoff symbol 3 | 3 5⁄4 gives Coxeter diagram = . It has extended Schläfli symbol a{5⁄2,3} or c{3,5⁄2}, as an altered...

Schläfli symbol a{5,3}, as an altered dodecahedron, and Coxeter diagram or . It is constructed from Schwarz triangle (3 3 5⁄2) with Wythoff symbol 3 | 5⁄2...

notation eD or aaD Schläfli symbols rr{5,3} or r { 5 3 } {\displaystyle r{\begin{Bmatrix}5\\3\end{Bmatrix}}} t0,2{5,3} Wythoff symbol 3 5 | 2 Coxeter diagram...

It is also the only convex uniform tiling that can not be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal...

spherical icosahedron used in construction of geodesic domes. It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5...

colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions. The (h,k) represent the periodic repeat of one colored tile...

at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}. English mathematician John Conway called it a deltille, named...

triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}. Conway calls it a truncated hextille, constructed as a truncation...

square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it...

dihedron. A regular dihedron, with Schläfli symbol {n,2}, is made of two regular polygons, each with Schläfli symbol {n}. A spherical dihedron is made of two...

polytopes as well. Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions...

Schläfli symbol b{5,5⁄2}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol...

tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}. Conway calls it a truncated quadrille, constructed as a truncation...