# Blended square tiling

Blended square tiling
Rank3
TypeRegular
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{4,4\}\#\{\}}$[1]
Elements
FacesN skew squares
Edges2N
Vertices2N
Vertex figureSquare, 0 < edge length < ${\displaystyle \sqrt{2} }$
Related polytopes
DualOnly exists abstractly
Petrie dualPetrial blended square tiling
Abstract properties
Schläfli type{4,4}
Topological properties
OrientableYes
Genus0
Properties
ConvexNo

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

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

where i and j range over the integers, and H is ${\displaystyle \sqrt{1-h^2} }$ (Note that ${\displaystyle 0 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.