# 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 ${\displaystyle \{6,3\}\#\{\}}$. 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
Rank3
TypeRegular
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{6,3\}\#\{\}}$
Elements
FacesN skew hexagons
Edges3N
Vertices2N
Vertex figureTriangle, 0 < edge length < ${\displaystyle \sqrt{3}}$
Abstract & topological properties
OrientableYes
Genus0
Properties
ConvexNo

## Vertex coordinates

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

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

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