Cube atop Cuboctahedron


Cube atop Cuboctahedron (cube || cuboctahedron), or K4.35, is one of Richard Klitzing's convex segmentochora, 4D CRF polytopes whose vertices lie on two parallel hyperplanes and which are orbiform (can be inscribed into a 4D sphere). It is constructed by placing a cube and a cuboctahedron in two appropriately-spaced parallel hyperplanes and taking their convex hull. Its surface consists of a cube, a cuboctahedron, 6 square antiprisms, and 8 tetrahedra.

K4.35

This polychoron is notable for having square antiprisms as cells.

Structure

We shall explore the structure of K4.35 using its parallel projections into 3D.

Parallel projection of
K4.35, showing cubical cell

The above image shows the cubical cell lying on the top hyperplane of the polychoron. Its square faces are joined to 6 square antiprisms:

Parallel projection of
K4.35, showing 2/6 square antiprisms

Parallel projection of
K4.35, showing 4/6 square antiprisms

Parallel projection of
K4.35, showing 6/6 square antiprisms

These square antiprisms appear quite flattened, because they lie at an angle to the 4D viewpoint. In 4D, they are perfectly uniform square antiprisms. Their exposed square faces are joined to the antipodal cuboctahedron:

Parallel projection of
K4.35, showing cuboctahedron

For clarity's sake, we have omitted the cells previously seen.

The triangular faces of this cuboctahedron are joined to 8 tetrahedra:

Parallel projection of
K4.35, showing 8 tetrahedra

These tetrahedra are foreshortened because they lie at an angle to the 4D viewpoint; in 4D they are perfectly regular tetrahedra.

These are all the cells of K4.35. The following table summarizes the cell counts:

Region Cube Square antiprisms Tetrahedra Cuboctahedra
Near side 1 6 0 0
Far side 0 0 8 1
Grand total 1 6 8 1
16 cells

Coordinates

The Cartesian coordinates of K4.35, with edge length 2, are:


Last updated 09 Mar 2018.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid HTML 5! Proud to be Microsoft-free