# Blended hexagonal tiling

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

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 ${\displaystyle \{6,3\}\#\{\}}$. 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.

## 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 ${\displaystyle H={\sqrt {1-h^{2}}}}$.