# Blended cube

Blended cube
Rank3
Dimension4
Notation
Schläfli symbol${\displaystyle \{4,3\}\#\{\}}$
Elements
Faces6 skew squares
Edges12
Vertices8
Petrie polygons4 skew hexagons
Related polytopes
Petrie dualBlended Petrial cube
HalvingTetrahedron
Convex hullTetrahedral antiprism
Abstract & topological properties
Euler characteristic2
Schläfli type{4,3}
OrientableYes
Genus0
Properties
Symmetry(A3×2×A1)/2, order 48
ConvexNo
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 convex 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.

The blended cube is the orthoplex realization of the abstract polyhedron {4,3}.

## 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.