Blended triangular tiling

Blended triangular tiling
File:Blended triangular tiling.png
Rank	3
Type	Regular
Space	Euclidean
Notation
Schläfli symbol	{3,6}#{}
Elements
Faces	∞ skew hexagons
Edges	∞
Vertices	∞
Vertex figure	Hexagon, 0 < edge length < 1
Related polytopes
Dual	Only exists abstractly
Petrie dual	Petrial blended triangular tiling
Topological properties
Orientable	Yes
Properties
Convex	No

The blended triangular tiling is a regular skew polyhedron that contains an infinite number of skew triangles, with 4 at each vertex. It can be obtained by blending the triangular tiling with a line segment (hence the name). It can be represented as the Schläfli symbol {3,6}#{}. The actual height of the blended triangular tiling can vary, however these are all considered to still be one polyhedron (just like how the skew square can vary in height but it is still considered the same regular polygon).

Vertex coordinates

Vertex coordinates of a blended triangular tiling centered at the origin with edge length 1 and height h are given by

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

where i and j range over the integers, and H is $\sqrt{1-h^2}$ (Note that $0<h<1$ must always be true for H to be a real number and for the blend to be non-degenerate).

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.