Blended cube

Blended cube
Rank	3
Dimension	4
Notation
Schläfli symbol
Elements
Faces	6 skew squares
Edges	12
Vertices	8
Petrie polygons	4 skew hexagons
Related polytopes
Army	Hexadecachoron
Petrie dual	Blended Petrial cube
Halving	Tetrahedron
Convex hull	Tetrahedral antiprism
Abstract & topological properties
Euler characteristic	2
Schläfli type	{4,3}
Orientable	Yes
Genus	0
Properties
Symmetry	(A3×2×A1)/2, order 48
Convex	No
Dimension vector	(2,3,3)

The blended cube or skew cube is a regular skew polyhedron in 4D Euclidean space. It can be constructed by taking the 3-dimensional cube and skewing it in the 4th dimension. It is the blend of a cube and the digon. Since the 3-dimensional cube is 2-colorable its blend does not duplicate any of its vertices and the result is abstractly equivalent. The cube is the only finite 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[edit | edit source]

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:

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