Snub square tiling

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Snub square tiling

Type	Uniform tiling
Vertex figure	3.3.4.3.4
Schläfli symbol	s{4,4}
Wythoff symbol	\| 4 4 2
Coxeter-Dynkin
Symmetry	p4g
Dual	Cairo pentagonal tiling
Properties	Vertex-transitive
3.3.4.3.4

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It has Schläfli symbol of s{4,4}.

There are 3 regular and 8 semiregular tilings in the plane.

This tiling is related to the elongated triangular tiling which also has 3 triangles and two squares on a vertex, but in a different order.

There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.) The coloring shown 12313 is not uniform.

The snub square tiling can be seen related to this 3-colored square tiling, with the yellow and red squares being twisted rigidly and the blue tiles being distorted into rhombus and then bisected into two triangles.

[edit] Wythoff construction

The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling.

An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.

If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths will produce a snub tiling with perfect equilateral triangle faces.

Example:

Regular octagons alternately truncated	--> (Alternate truncation)	Isosceles triangles (Nonuniform tiling)
Nonregular octagons alternately truncated	--> (Alternate truncation)	Equilateral triangles

[edit] References

Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65)
Williams, Robert The Geometrical Foundation of Natural Structure: A Source Book of Design New York: Dover, 1979. p38