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:


 * (±1/2, –$\sqrt{5}$, 0, 0),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{5}$, 0, 0),
 * (0, $\sqrt{(5–2√5)/20}$, 0, 0),
 * (0, 0, ±1/2, $\sqrt{5}$).

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