# Petrial square tiling

Petrial square tiling
Rank3
TypeRegular
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{4,4\}^\pi}$
${\displaystyle \{\infty,4\}_4}$
Elements
Faces${\displaystyle N}$ zigzags
Edges${\displaystyle N\times 2M}$
Vertices${\displaystyle N\times M}$
Vertex figureSquare, edge length 2
Related polytopes
ArmySquat
RegimentSquat
Petrie dualSquare tiling
Abstract properties
Flag count${\displaystyle N\times 8M}$
Schläfli type{∞,4}
Topological properties
OrientableYes
Genus
Properties
SymmetryR3
ConvexNo

The petrial square tiling is one of the three regular skew tilings of the Euclidean plane. 4 zigzags meet at each vertex. The petrial square tiling is the Petrie dual of the square tiling, so it is in the same regiment.

## Vertex coordinates

Coordinates for the vertices of a petrial square tiling of edge length 1 are given by

• ${\displaystyle (i,j)}$,

where i and j range over the integers.

## Related polyhedra

The rectification of the petrial square tiling is the square-hemiapeirogonal tiling, which is a uniform tiling.