Tetrahedral tegum

The tetrahedral tegum or tete, also called the tetrahedral bipyramid, is a CRF polychoron with 8 identical regular tetrahedra as cells. As such it is also a Blind polytope. As the name suggests, it is a tegum based on the tetrahedron, formed by attaching two regular pentachora at a common cell.

Vertex coordinates
The vertices of a tetrahedral tegum of edge length 1 are given by:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,0\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,0\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,0\right),$$
 * $$\left(0,\,0,\,0,\,±\frac{\sqrt{10}}{4}\right).$$

Representations
A tetrahedral tegum has the following Coxeter diagrams:


 * oxo3ooo3ooo&#xt
 * yo ox3oo3oo&#xt (y = $$\frac{\sqrt{10}}{2}$$, as full tegum)
 * oyo oox3ooo&#xt (as triangular pyramidal tegum)

Variations
The tetrahedral tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 8 non-CRF triangular pyramids as cells.

One notable variation can be obtained as the dual of the uniform tetrahedral prism, which can be represented by m2m3o3o. In this variation the height between the top and bottom vertices of the tegum is exactly $$\frac12 = 0.5$$ times the length of the edges of the base tetrahedron, and all the dichoral angles are $$\arccos\left(-\frac15\right) ≈ 101.53696°$$.