Blended cube

Blended cube
Rank	3
Space	Spherical, 4-dimensional space
Notation
Schläfli symbol	$\{4,3\}\#\{\}$
Elements
Faces	6 skew squares
Edges	12
Vertices	8
Related polytopes
Army	Hexadecachoron
Halving	Tetrahedron
Abstract & topological properties
Euler characteristic	2
Schläfli type	{4,3}
Orientable	Yes
Genus	0
Properties
Symmetry	(A3×2×A1)/2, order 48
Convex	No

The blended cube or skew cube is a regular skew polyhedron in 4D Euclidean space. It can be constructed by taking a cube and skewing it in the 4th dimension. It is the blend of a cube and the digon. Since the cube is 2-colorable it is abstractly equivalent to the cube. The cube is the only planar regular polyhedron that has this property, although the square tiling and hexagonal tiling are also 2-colorable and thus also have abstractly equivalent blends.

Vertex coordinates

For a skew cube with edge length 1 and skew distance $0<x<1$ its vertex coordinates can be given by

• $\left(\pm\frac{\sqrt{1-x^2}}{2},\pm\frac{\sqrt{1-x^2}}{2},\pm\frac{\sqrt{1-x^2}}{2},\pm \frac{x}{2}\right)$,

where the total number of negative coordinates is odd.

The vertex coordinates of a skew cube with skew distance $\frac{\sqrt{2}}{2}$ and edge length 1 can be given by all permutations of:

• $\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$,
• $\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$.

These correspond to the vertices of a hexadecachoron with unit edge length.

Gallery

• 

  A net.

Related polytopes

The blended cube has the same faces and vertex figure as the blended Petrial tetrahedron, and the two are abstractly equivalent.