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In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}.
Contents
- Uniform colorings
- Circle packing
- Wythoff construction
- Related tilings
- Related polyhedra and tilings
- References
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.
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.)
Circle packing
The snub square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).
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:
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 rhombi and then bisected into two triangles.
Related polyhedra and tilings
The snub square tiling is similar to the elongated triangular tiling with vertex configuration 3.3.3.4.4, and two 2-uniform dual tilings and 2 3-uniform duals which mix the two types of pentagons:
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.