In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the Wythoff symbol is a notation representing a
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
of a
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also ...
or plane tiling within a
Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
. It was first used by
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.
A Wythoff symbol consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular
cube can be represented by 3 , 2 4 with
Oh symmetry, and 2 4 , 2 as a square
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
with 2 colors and
D4h symmetry, as well as 2 2 2 , with 3 colors and D
2h symmetry.
With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space.
Description
The Wythoff construction begins by choosing a ''generator point'' on a fundamental triangle. This point must be chosen at equal distance from all edges that it does not lie on, and a perpendicular line is then dropped from it to each such edge.
The three numbers in Wythoff's symbol, ''p'', ''q'', and ''r'', represent the corners of the Schwarz triangle used in the construction, which are , , and
radians
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
respectively. The triangle is also represented with the same numbers, written (''p'' ''q'' ''r''). The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following:
* indicates that the generator lies on the corner ''p'',
* indicates that the generator lies on the edge between ''p'' and ''q'',
* indicates that the generator lies in the interior of the triangle.
In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The ''p'', ''q'', ''r'' values are listed ''before'' the bar if the corresponding mirror is active.
A special use is the symbol which is designated for the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.
The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2
3) possible forms, but the one where the generator point is on all the mirrors is impossible. The symbol that would normally refer to that is reused for the snub tilings.
The Wythoff symbol is functionally similar to the more general
Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.
Example spherical, euclidean and hyperbolic tilings on right triangles
The fundamental triangles are drawn in alternating colors as mirror images. The sequence of triangles (''p'' 3 2) change from spherical (''p'' = 3, 4, 5), to Euclidean (''p'' = 6), to hyperbolic (''p'' ≥ 7). Hyperbolic tilings are shown as a
Poincaré disk
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* L ...
projection.
See also
*
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
*
Regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
*
List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...
*
Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...
*
List of uniform polyhedra
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ...
*
List of uniform polyhedra by Schwarz triangle
There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non- dihedral a ...
*
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
References
*
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
''
Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', Third edition, (1973), Dover edition, (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
*
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
*
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, Longuet-Higgins, Miller, ''Uniform polyhedra'', Phil. Trans. 1954, 246 A, 401–50.
* pp. 9–10.
External links
*
The Wythoff symbol*
ttps://www.shadertoy.com/view/Md3yRB A Shadertoy renderization of Wythoff's construction methodbr>
KaleidoTile 3Free educational software for Windows by
Jeffrey Weeks that generated many of the images on the page.
* {{cite web , author = Hatch, Don , title = Hyperbolic Planar Tessellations , url = http://www.plunk.org/~hatch/HyperbolicTesselations
Polyhedra
Polytopes
Mathematical notation
ja:ワイソフ記号