# Petrial hexagonal tiling

Petrial hexagonal tiling
Rank3
TypeRegular
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{6,3\}^\pi}$
${\displaystyle \{\infty,3\}_6}$
Elements
Faces${\displaystyle N}$ zigzags
Edges${\displaystyle N\times 3M}$
Vertices${\displaystyle N\times 2M}$
Vertex figureTriangle, edge length 3
Related polytopes
ArmyHexat
RegimentHexat
Petrie dualHexagonal tiling
Abstract properties
Flag count${\displaystyle N\times 12M}$
Schläfli type{∞,3}
Topological properties
OrientableYes
Genus
Properties
ConvexNo

The petrial hexagonal tiling is one of the three regular skew tilings of the Euclidean plane. 3 zigzags meet at each vertex. The petrial hexagonal tiling is the Petrie dual of the hexagonal tiling.

## Vertex coordinates

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

• ${\displaystyle \left(3i\pm\frac12,\,\sqrt3j+\frac{\sqrt3}{2}\right)}$,
• ${\displaystyle \left(3i\pm1,\,\sqrt3j\right)}$,

where i and j range over the integers.

## Related polyhedra

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