Petrial hexagonal tiling

Petrial hexagonal tiling
Rank	3
Type	Regular
Space	Euclidean
Notation
Schläfli symbol	$\{6,3\}^\pi$ $\{\infty,3\}_6$
Elements
Faces	$N$ zigzags
Edges	$N\times 3M$
Vertices	$N\times 2M$
Vertex figure	Triangle, edge length √3
Related polytopes
Army	Hexat
Regiment	Hexat
Petrie dual	Hexagonal tiling
Abstract properties
Flag count	$N\times 12M$
Schläfli type	{∞,3}
Topological properties
Orientable	Yes
Genus	∞
Properties
Convex	No

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

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

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.