Octahedron atop truncated tetrahedron

Octahedron atop truncated tetrahedron, or octatut, is a CRF segmentochoron (designated K-4.52 on Richard Klitzing's list). As the name suggests, it consists of an octahedron and a truncated tetrahedron as bases, connected by 4 triangular prisms and 4 triangular cupolas.

It can be obtained as a segment of the small rhombated pentachoron, which can be constructed by joining this segmentochoron to a cuboctahedron atop truncated tetrahedron segmentochoron at their common truncated tetrahedral base.

Vertex coordinates
The vertices of an octahedron atop truncated tetrahedronsegmentochoron of edge length 1 are given by:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,\frac{\sqrt{10}}{4}\right)$$ and all permutations of first three coordinates
 * $$\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,0\right)$$ and all permutatoins and even sign changes of first three coordinates

Alternative coordinates can be obtained from those of the small rhombated pentachoron by removing the vertices of one of its cuboctahedral cells:


 * $$\left(-\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}6,\,-\frac{\sqrt3}3,\,0\right),$$
 * $$\left(-\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}6,\,\frac{\sqrt3}3,\,0\right),$$
 * $$\left(-\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,-\frac{2\sqrt3}3,\,0\right),$$
 * $$\left(-\frac{\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,-\frac{\sqrt3}3,\,0\right),$$
 * $$\left(-\frac{\sqrt{10}}{20},\,\frac{\sqrt6}4,\,0,\,±1\right),$$
 * $$\left(-\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,\frac{\sqrt3}3,\,±1\right),$$
 * $$\left(-\frac{\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,\frac{\sqrt3}6,\,±\frac12\right),$$
 * $$\left(-\frac{\sqrt{10}}{20},\,\frac{\sqrt6}4,\,±\frac{\sqrt3}2,\,±\frac12\right),$$
 * $$\left(-\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}6,\,\frac{\sqrt3}6,\,±\frac12\right),$$
 * $$\left(-\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}6,\,-\frac{\sqrt3}6,\,±\frac12\right).$$