# Cuboctahedron atop Truncated Cube

Cuboctahedron atop Truncated Cube (cuboctahedron || cube), or **K4.129**
among Dr. Richard Klitzing's *segmentochora*, is a 4D CRF
polytope that consists of a cuboctahedron and a
truncated cube in parallel hyperplanes, connected to
each other by 8 triangular prisms and 6
square cupolae. It has a total of 16 cells, 64 polygons (28
triangles, 30 squares, 6 octagons), 84 edges, and 36 vertices.

Besides the defining construction of taking the convex hull of a
cuboctahedron and a truncated cube placed in parallel hyperplanes, K4.129 can
also be constructed as a Stott expansion of the cubical
pyramid (a *cantellation*).

Eight copies of K4.129 can be attached to the truncated tesseract to form the octa-augmented truncated tesseract, in which adjacent square cupolae lie in coincident hyperplanes and thus merge into square orthobicupolae (J28).

## Structure

The structure of K4.129 is quite simple. We shall explore it using its parallel projections into 3D:

The above image shows the cuboctahedral cell. Its triangular faces are joined to 8 prisms, shown next:

The square faces of the cuboctahedron are joined to 6 square cupolae:

Finally, closing up the entire shape is the truncated cube:

For clarity, we show only the truncated cube and omit the cells previously seen.

## Coordinates

The Cartesian coordinates of K4.129 with edge length 2 are:

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