# Octahedron atop Rhombicuboctahedron

Octahedron atop rhombicuboctahedron (octahedron || rhombicuboctahedron), or K4.107 among Dr. Richard Klitzing's segmentochora, is a 4D CRF polytope that consists of an octahedron and a rhombicuboctahedron in parallel hyperplanes, connected to each other by 20 triangular prisms and 6 square pyramids, for a total of 28 cells, 82 polygons (40 triangles, 42 squares), 84 edges, and 30 vertices.

Eight copies of K4.107 can be attached to the runcitruncated 16-cell to form the octa-augmented runcitruncated 16-cell, in which the square pyramids of K4.107 and the cubes of the runcitruncated 16-cell lie in coincident hyperplanes and thus merge into elongated square bipyramids (J15).

## Structure

The structure of K4.107 is quite simple. We shall explore it using its parallel projections into 3D:

The above image shows the octahedral cell of K4.107. It lies closest to this 4D viewpoint.

The next image shows 8 of the triangular prisms that are attached to this octahedron:

Between these triangular prisms are more triangular prisms, another 12 of them:

The remaining gaps are filled by 6 square pyramids:

Finally, the last cell is the antipodal rhombicuboctahedron:

For clarity, we have omitted the other cells that have already been shown.

## Coordinates

The Cartesian coordinates of K4.107 with edge length 2 are:

• (0, 0, ±√2, 1)
• (0, ±√2, 0, 1)
• (±√2, 0, 0, 1)
• (±1, ±1, ±(1+√2), 0)
• (±1, ±(1+√2), ±1, 0)
• (±(1+√2), ±1, ±1, 0)

Last updated 18 Jun 2019.