Blended triangular tiling

Blended triangular tiling
Rank	3
Dimension	3
Type	Regular
Space	Euclidean
Notation
Schläfli symbol	{3,6}#{}
Elements
Faces	N  skew triangles
Edges	3N
Vertices	N
Vertex figure	Hexagon, 0 < edge length < 1
Petrie polygons	Zigzags
Holes	Skew hexagons
Related polytopes
Army	Triangular tilingprism
Dual	{6,6 | 2} (degenerate)
Petrie dual	Petrial blended triangular tiling
φ 2	Blended hexagonal tiling
Convex hull	Triangular tiling prism
Abstract & topological properties
Schläfli type	{6,6}
Orientable	Yes
Genus	∞
Properties
Symmetry	V3×A1
Convex	No
Dimension vector	(1,2,2)

The blended triangular tiling is a regular skew polyhedron that contains an infinite number of skew triangles, with 6 at each vertex. It can be obtained by blending the triangular tiling with a dyad (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 ordinarily 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).

Unlike the blended square tiling and the blended hexagonal tiling the blended triangular tiling is not abstractly equivalent to its non-blended version.

Vertex coordinates

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

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

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.