Icosahedral tegum

The icosahedral tegum or ite, also called the icosahedral bipyramid, is a CRF polychoron with 40 identical regular tetrahedra as cells. As such it is also a Blind polytope. As the name suggests, it is a tegum based on the icosahedron, formed by attaching two icosahedral pyramids at their common base.

Vertex coordinates
The vertices of an icosahedral tegum of edge length 1 are given by:
 * $$\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,0\right)$$ and all even permutations of the first 3 coordinates,
 * $$\left(0,\,0,\,0,\,±\frac{\sqrt5-1}{4}\right).$$

Representations
An icosahedral tegum has the following Coxeter diagrams:


 * ooo5ooo3oxo&#xt
 * vo oo5oo3ox&#zx

Variations
The icosahedral tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 40 non-CRF triangular pyramids as cells.

One notable variation can be obtained as the dual of the uniform dodecahedral prism, which can be represented by m2m5o3o. in this variation the height between the top and bottom vertices of the tegum is $$2+\sqrt5 ≈ 4.23607$$ times the length of the edges of the base icosahedron, and all the dichoral angles are $$\arccos\left(-\frac{8+3\sqrt5}{19}\right) ≈ 140.72495°$$.