Tetrakis Square Tiling - Related Polyhedra and Tilings

Related Polyhedra and Tilings

It is topologically related to a series of polyhedra and tilings with face configuration Vn.6.6.

The symmetry type is:

• with the coloring: cmm; a primitive cell is 8 triangles, a fundamental domain 2 triangles (1/2 for each color)
• with the dark triangles in black and the light ones in white: p4g; a primitive cell is 8 triangles, a fundamental domain 1 triangle (1/2 each for black and white)
• with the edges in black and the interiors in white: p4m; a primitive cell is 2 triangles, a fundamental domain 1/2

The edges of the tetrakis square tiling form a simplicial arrangement of lines, a property it shares with the triangular tiling and the bisected hexagonal tiling. These lines form the axes of symmetry of a reflection group (the wallpaper group *442 or p4m), which has the triangles of the tiling as its fundamental domains. This group is isomorphic to, but not the same as, the group of automorphisms of the tiling, which has additional axes of symmetry bisecting the triangles and which has half-triangles as its fundamental domains.