Pentachoric prism

The pentachoric prism or penp is a prismatic uniform polyteron that consists of 2 pentachora and 5 tetrahedral prisms as facets. Each vertex joins 1 pentachoron and 4 tetrahedral prisms. As the name suggests, it is a prism based on the pentachoron, which also makes it a convex segmentoteron.

Vertex coordinates
The vertices of a pentachoric prism of edge length 1 are given by:
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,±\frac12\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt6}{4},\,-\frac{\sqrt{10}}{20},\,±\frac12\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,-\frac{\sqrt6}{12},\,-\frac{\sqrt{10}}{20},\,±\frac12\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,-\frac{\sqrt6}{12},\,-\frac{\sqrt{10}}{20},\,±\frac12\right).$$


 * (0, 0, 0, $\sqrt{2}$/5, ±1/2),
 * (0, 0, $\sqrt{10}$/4, –$\sqrt{6}$/20, ±1/2),
 * (0, $\sqrt{10}$/3, –$\sqrt{3}$/12, –$\sqrt{6}$/20, ±1/2),
 * (±1/2, –$\sqrt{10}$/6, –$\sqrt{3}$/12, –$\sqrt{6}$/20, ±1/2).

Representations
A pentachoric prism has the following Coxeter diagrams:


 * x x3o3o3o (full symmetry)
 * xx3oo3oo3oo&#x (bases seen separately)
 * xx ox3oo3oo&#x (A3×A1 axial, tetrahedral pyramidal prism)
 * xx xo ox3oo&#x (A2×A1×A1 axial, trigonal scalene prism)
 * xxx oxo oox&#x (A1×A1×A1 axial, disphenoid pyramidal prism)
 * xxx oox3ooo&#x (A2×A1 axial, trigonal pyramidal pyramidl prism)
 * xxxx ooox&#x (A1×A1 axial, bases bilateral symmetric)
 * xxxxx&#x (bases irregular)
 * xxoo3oooo3oooo&#xr (A3 symmetry, vertex-fisrt)
 * xxoo3oooo ooxx&#xr (A2×A1 axial, triangle-first)