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}.

Truncated square tiling
Type	Semiregular tiling
Vertex configuration	4.8.8
Schläfli symbol	t{4,4} tr{4,4} or ${\displaystyle t{\begin{Bmatrix}4\\4\end{Bmatrix}}}$
Wythoff symbol	2 | 4 4 4 4 2 |
Coxeter diagram	or
Symmetry	p4m, [4,4], (*442)
Rotation symmetry	p4, [4,4]+, (442)
Bowers acronym	Tosquat
Dual	Tetrakis square tiling
Properties	Vertex-transitive

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.