Blended cube

The 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.

Vertex coordinates
For a skew cube with edge length 1 and skew distance $$x<1$$ it's 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 even permutations of the following: These correspond to half of the vertices of a tesseract with edge length $$\frac{\sqrt{2}}{2}$$.
 * $$\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$$,
 * $$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$$.