Blended square tiling

Blended square tiling
Rank	3
Type	Regular
Space	Euclidean
Notation
Schläfli symbol	$\{4,4\}\#\{\}$
Elements
Faces	N skew squares
Edges	2N
Vertices	2N
Vertex figure	Square, 0 < edge length < $\sqrt{2}$
Related polytopes
Dual	Only exists abstractly
Petrie dual	Petrial blended square tiling
Abstract properties
Schläfli type	{4,4}
Topological properties
Orientable	Yes
Genus	0
Properties
Convex	No

The blended square tiling is a regular skew polyhedron that has an infinite amount of skew squares as faces joined together 4 at a vertex. It can be obtained by blending the square tiling with a line segment (hence the name). It can be represented as the Schläfli symbol {4,4}#{}. The actual height of the blended square tiling can vary, but it is considered to still be one polyhedron, just like how the skew square can vary in height but it is still considered the same polygon.

Vertex coordinates

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

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

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).

Bibliography

• jan Misali (2020). "there are 48 regular polyhedra"
• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.