Blended hexagonal tiling

The blended hexagonal tiling is a regular skew polyhedron consisting of an infinite amount of skew hexagons, with 3 at a vertex. It can be obtained as the blend of a line segment and a hexagonal tiling, and so it has a Schlafli symbol of $\{6,3\}\#\{\}$. It is abstractly identical to the hexagonal tiling. Just like the skew hexagon, the blended hexagonal tiling can vary in height but it is considered one polyhedron.

Blended hexagonal tiling
Rank	3
Type	Regular
Space	Euclidean
Notation
Schläfli symbol	$\{6,3\}\#\{\}$
Elements
Faces	N skew hexagons
Edges	3N
Vertices	2N
Vertex figure	Triangle, 0 < edge length < $\sqrt{3}$
Abstract & topological properties
Orientable	Yes
Genus	0
Properties
Convex	No

Vertex coordinates

The vertex coordinates of a blended hexagonal tiling centered at the origin with edge length 1 and height h are

• $(3Hi-\frac{1}{2},\sqrt{3}Hj+\frac{\sqrt{3}}{2},\frac{h}{2})$
• $(3Hi-1,\sqrt{3}Hj,-\frac{h}{2})$
• $(3Hi+\frac{1}{2},\sqrt{3}Hj+\frac{\sqrt{3}}{2},-\frac{h}{2})$
• $(3Hi+1,\sqrt{3}Hj,\frac{h}{2})$

where i and j range over the integers, and $H = \sqrt{1-h^2}$.

References

• jan Misali (2020). "there are 48 regular polyhedra"
• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space". Discrete Computational Geometry. 17: 449–478.