# Blended cube

(Redirected from Skew cube)
Blended cube
Rank3
SpaceSpherical, 4-dimensional space
Notation
Schläfli symbol${\displaystyle \{4,3\}\#\{\}}$
Elements
Faces6 skew squares
Edges12
Vertices8
Related polytopes
HalvingTetrahedron
Abstract & topological properties
Euler characteristic2
Schläfli type{4,3}
OrientableYes
Genus0
Properties
Symmetry(A3×2×A1)/2, order 48
ConvexNo

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 ${\displaystyle 0 its vertex coordinates can be given by

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

where the total number of negative coordinates is odd.

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

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

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

## Related polytopes

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