Tetrahedron atop cuboctahedron

Tetrahedron atop cuboctahedron, or tetaco, is a CRF segmentochoron (designated K-4.23 on Richard Klitzing's list). As the name suggests, it consists of a tetrahedron and a cuboctahedron as bases, connected by 4 further tetrahedra and 4+6 triangular prisms.

It is also sometimes referred to as a tetrahedral cupola, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

Two tetrahedron atop cuboctahedron segmentochora can be attached at their cuboctahedral bases, such that the tetrahedral bases are in dual positions, to form the small prismatodecachoron.

Vertex coordinates
The vertices of a tetrahedron atop cuboctahedron segmentochoron of edge length 1 are given by:
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt{10}}{4}\right)$$ and all even sign changes of first three coordinates
 * $$\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0\right)$$ and all permuations of first three coordinates

Alternative coordinates can be obtained from those of the small prismatodecachoron by removing the vertices of one of its tetrahedral cells:


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

Related polychora
This segmentochoron can be split into a triangular cupofastegium and the segmentochoron tetrahedron atop triangular cupola, joining at a common triangular cupola cell.