In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Table of Geometry, from the 1728 Cyclopaedia. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A toy kaleidoscope tube Pattern as seen through a kaleidoscope tube Pattern as seen through a kaleidoscope tube Pattern as seen through a kaleidoscope tube Pattern as seen through a kaleidoscope tube The kaleidoscope is a toy containing small, brightly-colored tumbling objects, and a set of mirrors which reflect...

It is based on the idea of tiling a sphere, with spherical triangles. If three mirrors were to be arranged so that their planes intersected at a single point, then the mirrors would enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections would produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... Right spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that point produce a uniform polyhedron. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron:

1. 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 similarly for b and c.
2. A vertex is placed at a point on line AB so that it bisects the angle at C. This produces a polyhedron with Wythoff symbol a b|c.
3. A vertex is placed so that it is on the incentre of ABC. This produces a polyhedron with Wythoff symbol a b c|.
4. The vertex is at a point such that, when it is rotated around any of the triangle's corners by twice the angle at that point, it is displaced by the same distance for every angle. Only even-numbered reflections of the original vertex are used. The polyhedron has the Wythoff symbol |a b c.

The process in general also applies for higher dimensional regular polytopes, including the 4-dimensional uniform polychora. For the numerical analysis algorithm, see bisection method. ... A dodecahedron, one of the five Platonic solids. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...

Reflection symmetry triangles on the sphere

There are 4 classes of reflection symmetry triangles on the sphere with whole number reflection points:

1. (p 2 2) dihedral symmetry p=2,3,4... (Order 4p)
2. (3 3 2) tetrahedral symmetry (Order 24)
3. (4 3 2) octahedral symmetry (Order 48)
4. (5 3 2) icosahedral symmetry (Order 120)

D2h (*222)

D3h (*322)

Th (*332)

Oh (*432)

Ih (*532)

The list of Schwarz triangles includes rational numbers, and determine the full set of solutions of uniform polyhedrons. This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... In mathematics, a Schwarz triangle is a spherical triangle that can be used to tile a sphere. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...

Reflection symmetry triangles on the plane

There are 3 classes of reflection symmetry triangles on the plane

1. (3 3 3) p3m1
2. (4 4 2) p4m
3. (6 3 2) p6m

p3m1 (*333)

p4m (*442)

p6m (*632)

Image File history File links Tile_3,6. ... Image File history File links Tile_V488. ... Image File history File links Tile_V46b. ...

References

• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff's Construction for Uniform Polytopes)

H.S.M. Coxeter. ... H.S.M. Coxeter. ...

External links

• Displays Uniform Polyhedra using Wythoff's construction method
• Description of Wythoff Constructions
• "Jenn", software that generates views of (spherical) polyhedra and polychora from symmetry groups