Blended cube

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 where the total number of negative coordinates is odd.
 * $$\left(\pm\frac{\sqrt{1-x^2}}{2},\pm\frac{\sqrt{1-x^2}}{2},\pm\frac{\sqrt{1-x^2}}{2},\pm \frac{x}{2}\right)$$,

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: These correspond to the vertices of a hexadecachoron with unit edge length.
 * $$\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$$,
 * $$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$$.

Related polytopes
The has the same faces and vertex figure as the blended petrial tetrahedron, and the two are abstractly equivalent.