# 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${\displaystyle 4(3-2{\sqrt {2}})\approx 0.68629}$
Related polytopes
ArmyTosquat
RegimentTosquat
DualTetrakis square tiling
ConjugateQuasitruncated square tiling
Abstract & topological properties
Flag count24N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryR3
ConvexYes
NatureTame

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

• ${\displaystyle \left(\pm {\frac {1}{2}}+(1+{\sqrt {2}})i,\,\pm {\frac {1+{\sqrt {2}}}{2}}+j(1+{\sqrt {2}})\right)}$,

where i  and j  range over the integers.

## Representations

A truncated square tiling has the following Coxeter diagrams:

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