Pentagonal scalene

The pentagonal scalene, pentagonal pyramidal pyramid, or pesc, is a CRF segmentochoron (designated K-4.86 on Richard Klitzing's list). It consists of 2 pentagonal pyramids and 5 tetrahedra. It can be thought of as a pyramid based on the pentagonal pyramid.

Apart from being a point atop pentagonal pyramid, it has an alternate segmentochoron representation as a dyad atop perpendicular pentagon.

Being a pyramid of a pentagonal pyramid, it is a part of an icosahedral pyramid, and also forms the edge-first cap of the hexacosichoron.

Vertex coordinates
The vertices of a pentagonal scalene with unit edge length are given by:


 * $$\left(±\frac{1}{2},\,–\sqrt{\frac{5+2\sqrt{5}}{20}},\,0,\,0\right),$$
 * $$\left(±\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{5-\sqrt{5}}{40}},\,0,\,0\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt{5}}{10}},\,0,\,0\right),$$
 * $$\left(0,\,0,\,±\frac{1}{2},\,\sqrt{\frac{5-2\sqrt{5}}{20}}\right).$$

Representations
The pentagonal scalene has the following Coxeter diagrams:
 * xo ox5oo&#x (full symmetry, dyad atop fully orthogonal pentagon)
 * oox5ooo&#x (H2 symmetry, pentagonal pyramidal pyramid)