Wythoff symbol

Wythoff symbol

In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a kaleidoscopic construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane.

The Wythoff symbol gives 3 numbers "p,q,r" and a positional vertical bar () which separate the numbers before or after it. Each number represents the order of mirrors at a vertex of the fundamental triangle.

Each symbol 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 | 4 2 with Oh symmetry, and 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D2h symmetry.

Summary table

There are 7 generator points with each set of p,q,r: (And a few special forms)

"In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections."

Summary spherical and plane tilings

"Selected tilings created by the Wythoff construction are given below."


= Spherical tilings (r=2) =

Planar tilings (r>2)

The Coxeter-Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.


= Overlapping spherical tilings (r=2) =

"Tilings are shown as polyhedra." Some of the forms are degenerate, given with brackets for vertex figures, with overlapping edges or verices.

See also

*Regular polytope
*Regular polyhedron
*List of uniform tilings
*List of uniform polyhedra

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)
* Coxeter, Longuet-Higgins, Miller, "Uniform polyhedra", Phil. Trans. 1954, 246 A, 401-50.
* pp. 9-10.

External links

*
* [http://www.mathconsult.ch/showroom/unipoly/wythoff.html The Wythoff symbol]
* [http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2788&langcode=en Wythoff symbol]
* [http://gregegan.customer.netspace.net.au/APPLETS/26/26.html Displays Uniform Polyhedra using Wythoff's construction method]
* [http://gregegan.customer.netspace.net.au/APPLETS/26/WythoffNotes.html Description of Wythoff Constructions]
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3] Free educational software for Windows by Jeffrey Weeks that generated many of the images on the page.
*


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