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Concept
Snub square tiling
Summary
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}.
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.
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 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).
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, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces.
Example:
File:Snub snub square tiling.svg|A [[Conway snub operator|snub operator]] applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons.
File:Isogonal snub square tiling-8x8.svg|A related [[isogonal tiling]] that combines pairs of triangles into rhombi
File:Triangular heptagonal tiling.svg|A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons.
File:P2_dual.png|The [[Cairo pentagonal tiling]] is dual to the snub square tiling.
Official source
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