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.
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.
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:
- 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.
- 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.
- A vertex is placed so that it is on the incentre of ABC. This produces a polyhedron with Wythoff symbol a b c|.
- 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.
See also
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)
- W.A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences, 20 (1918) 966–970.
External links
- Weisstein, Eric W. "Wythoff construction". MathWorld.
- Olshevsky, George. "Wythoff construction". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- 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