Төртбұрыштың қиылып жасалған мозаикасы

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.

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).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling, snub square tiling.

The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. 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
Conway calls it a kisquadrille, 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.