Truncated square tiling

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.

The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane: The 3-dimensional bitruncated cubic honeycomb projected into the plane shows two copies of a truncated tiling.

It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2.

Conway calls it a kisquadrille,[2] represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille).

It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.

The squares from the truncation can be alternately sized. In the limit, half of the vertices can remain untruncated, leading to a chamfered square tiling .
A skew equilateral form with squares into rhombi, and flattened octagons.
The truncated square tiling is used in an optical illusion with truncated vertices divides and colored alternately, seeming to twist the grid.
The tetrakis square tiling