# Truncated square tiling

Truncated square tiling Rank3
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymTosquat
Coxeter diagramx4x4o (     )
Elements
FacesN squares, N octagons
Edges2N+4N
Vertices4N
Vertex figureIsosceles triangle, edge lengths 2, 2+2, 2+2
Measures (edge length 1)
Vertex density$4(3-2\sqrt2) \approx 0.68629$ Related polytopes
ArmyTosquat
RegimentTosquat
DualTetrakis square tiling
ConjugateQuasitruncated square tiling
Abstract & topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryR3
ConvexYes

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