Truncated square tiling

Truncated square tiling
Rank	3
Type	Uniform
Space	Euclidean
Notation
Bowers style acronym	Tosquat
Coxeter diagram	x4x4o ()
Elements
Faces	N squares, N octagons
Edges	2N+4N
Vertices	4N
Vertex figure	Isosceles triangle, edge lengths √2, √2+√2, √2+√2
Measures (edge length 1)
Vertex density	$4(3-2\sqrt2) \approx 0.68629$
Related polytopes
Army	Tosquat
Regiment	Tosquat
Dual	Tetrakis square tiling
Conjugate	Quasitruncated square tiling
Abstract & topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	R3
Convex	Yes

The truncated square tiling, or tosquat, is one of the eleven convex uniform tilings of the Euclidean plane. 1 square and 2 octagons join at each vertex of this tiling. As its name suggests, it is the truncation of the regular square tiling.

Vertex coordinates

Coordinates for the vertices of a truncated square tiling of edge length 1 are given by all permutations of

• $\left(±\frac12+(1+\sqrt2)i,\,±\frac{1+\sqrt2}{2}+j(1+\sqrt2)\right),$

where i and j range over the integers.

Representations

A truncated square tiling has the following Coxeter diagrams:

• x4x4o (regular)
• x4x4x (as omnitruncated square tiling)
• s4o4x (as alternated facting)
• s4x4x

Related tilings

o4o4o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Square tiling	squat	{4,4}	x4o4o
Truncated square tiling	tosquat	t{4,4}	x4x4o
Rectified square tiling = Square tiling	squat	r{4,4}	o4x4o
Truncated square tiling	tosquat	t{4,4}	o4x4x
Square tiling	squat	{4,4}	o4o4x
Cantellated square tiling = Square tiling	squat	rr{4,4}	x4o4x
Omnitruncated square tiling = Truncated square tiling	tosquat	tr{4,4}	x4x4x
Snub square tiling	snasquat	sr{4,4}	s4s4s

External links

• Klitzing, Richard. "tosquat".

• Wikipedia Contributors. "Truncated square tiling".