Blended hexagonal tiling

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 < ${\sqrt {3}}$
Army	Triangular tiling antiprism
Abstract & topological properties
Schläfli type	{6,3}
Orientable	Yes
Genus	0
Properties
Convex	No

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 hexagonal tiling and a dyad, 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.

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.