Snub square tiling
| Snub square tiling | |
|---|---|
 This coloring has xx (pg) symmetry |
|
| Type | Semiregular tiling |
| Vertex configuration | 3.3.4.3.4 |
| Schläfli symbol | s{4,4} h0,1{4,4} |
| Wythoff symbol | | 4 4 2 |
| Coxeter-Dynkin |    |
| Symmetry | p4g, (4*2), [4+,4] p4, (442), [4,4]+ pg, (xx) [(∞,2)+,∞+] |
| Dual | Cairo pentagonal tiling |
| Properties | Vertex-transitive |
 Vertex figure: 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}.
Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille).
There are 3 regular and 8 semiregular tilings in the plane.
Contents |
[edit] Uniform colorings
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.)
| Coloring |  |
 |
|---|---|---|
| Symmetry | 4*2 (p4g) | 442 (p4) |
| Schläfli symbol | h0,1{4,4} | s{4,4} |
| Wythoff symbol | | 4 4 2 | |
| Coxeter-Dynkin diagram | |
|
[edit] Related tilings
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.
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 |
→
(Alternatetruncation) |
 Isosceles triangles (Nonuniform tiling) |
 Nonregular octagons alternately truncated |
→
(Alternatetruncation) |
Equilateral triangles |
[edit] See also
[edit] References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
- Richard Klitzing, 2D Euclidean tilings, s4s4s - snasquat - O10
- 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 (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p38
