Cubic tegum

The cubic tegum or cute, also called the cubic bipyramid, is a CRF polychoron with 12 square pyramids as cells. As the name suggests, it is a tegum based on the cube, formed by attaching 2 cubic pyramids together at their common base. It is unique among CRF tegums as the height from the top to the bottom vertex is identical to the cube's edge length.

A regular icositetrachoron can be decomposed into 8 CRF cubic tegums sharing one common vertex.

Vertex coordinates
Coordinates for the vertices of a cubic tegum of edge length 1 are given by:
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,0\right),$$
 * $$\left(0,\,0,\,0,\,±\frac12\right).$$

Representations
A cubic tegum has the following Coxeter diagrams:


 * oxo4ooo3ooo&#xt (full symmetry)
 * oxo oxo4ooo&#xt (square prismatic tegum)
 * oxo oxo oxo&#xt (cuboid tegum)
 * xo ox4oo3oo&#zx
 * oxo xox4ooo&#xt (square-first)
 * oxo xox xox&#xt (as above with rectangular symmetry)

Variations
The cubic tegum can have the heights of its pyramids varied while maintaining its full symmetry. These variants generally have 12 non-CRF square pyramids as cells.

One notable variation can be obtained as the dual of the uniform octahedral prism, which can be represented by m2o4o3m. In this variation the height between the top and bottom vertices of the tegum is $$\sqrt2 ≈ 1.41421$$ times the length of the edges of the base cube, and all the dichoral angles are $$\arccos\left(-\frac13\right) ≈ 109.47122°$$.