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.

K4.107: Octahedron
atop Rhombicuboctahedron

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).


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

Parallel projection of
K4.107, showing octahedral cell

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:

Parallel projection of
K4.107, showing 8 triangular prisms

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

Parallel projection of
K4.107, showing 12 more triangular prisms

The remaining gaps are filled by 6 square pyramids:

Parallel projection of
K4.107, showing 12 more triangular prisms

Finally, the last cell is the antipodal rhombicuboctahedron:

Parallel projection of
K4.107, showing antipodal rhombicuboctahedron

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


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

Last updated 18 Jun 2019.

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