Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.

Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere.

Most uniform tilings can be made from a Wythoff construction starting with a symmetry groups and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name reprsented by the order of the mirrors in sequential vertices.

A fundamental domain triangle is ("p" "q" "r"), and a right triangle ("p" "q" 2), where "p", "q", "r" are whole numbers greater than 1. The triangle may exist as a spherical triangle, an Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of "p", "q" and "r".

There are a number of symbolic schemes for naming these figures, from a modified Schläfli symbol for right triangle domains: ("p" "q" 2) --> {"p", "q"}. The Coxeter-Dynkin diagram is a triangular graph with "p", "q", "r" labeled on the edges. If "r" = 2, the graph is linear since order-2 domain nodes generate no reflections. The Wythoff symbol takes the 3 integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite a domain node, it is given before the bar.

Finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex.

All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by Norman Johnson are called truncation (cutting vertices), rectification (cutting vertices until edges disappear), and Cantellation (cutting edges). Omnitruncation is an operation that combines truncation and cantellation. Snubbing is an operation of Alternate truncation of the omnitruncated form. (See Uniform_polyhedron#Definition_of_operations for more details.)

Coxeter groups

Coxeter groups for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams:

For groups with whole number orders, including:

Uniform tilings of the hyperbolic plane

See also: Uniform tilings in hyperbolic plane

There are infinitely many uniform tilings of convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r).

A sampling is shown here with a Poincaré disk projection.

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.

Further Symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains starting with (2 2 2 3), etc, that can generate new forms. As well there's fundamental domains that place vertices at infinity, such as (∞ 2 3), etc.

Right angle fundamental triangles: ("p" "q" 2)

See also

* Uniform tessellation
* Wythoff symbol
* List of uniform tilings
* Uniform tilings in hyperbolic plane
* Uniform polytope

References

* Norman Johnson "Uniform Polytopes", Manuscript (1991)
** N.W. Johnson: "The Theory of Uniform Polytopes and Honeycombs", Ph.D. Dissertation, University of Toronto, 1966
* (Star tilings section 12.3)
*H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, "Uniform polyhedra", Phil. Trans. 1954, 246 A, 401–50 JSTOR: [http://links.jstor.org/sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4] (Table 8)

External links

*
* [http://www2u.biglobe.ne.jp/~hsaka/mandara/ue2 Uniform Tessellations on the Euclid plane]
* [http://web.ukonline.co.uk/polyhedra/tessellations/tessel.htm Tessellations of the Plane]
* [http://www.tess-elation.co.uk/index.htm David Bailey's World of Tessellations]
* [http://www.uwgb.edu/dutchs/symmetry/uniftil.htm k-uniform tilings]
* [http://probabilitysports.com/tilings.html n-uniform tilings]