Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.

Conway calls it a truncated quadrille, constructed as a truncation operation applied to a square tiling (quadrille).

Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges.

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

There are two distinct uniform colorings of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)

The truncated 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 3 other circles in the packing (kissing number). Since there is an even number of sides of all the polygons, the circles can be alternately colored as shown below.

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