Pentachoric tegum

The pentachoric tegum, also called the pentachoric bipyramid, is a Blind polytope and CRF polyteron with 10 identical regular pentachora as tera. As the name suggests, it is a tegum based on the pentachoron, formed by attaching two regular hexatera at a common facet.

It is part of an infinite family of Blind polytopes known as the simplicial bipyramids. It is one of two non-uniform Blind polytopes in five dimensions, the other being the hexadecachoric pyramid.

Vertex coordinates
The vertices of a pentachoric tegum of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,0\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,0\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,0\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,0\right),$$
 * $$\left(0,\,0,\,0,\,0,\,±\frac{\sqrt{15}}{5}\right).$$

Variations
The pentachoric tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 10 non-CRF tetrahedral pyramids as tera.

One notable variation can be obtained as the dual of the uniform pentachoric prism, which can be represented by m2m3o3o3o.