Pentagonal scalene

{{Infobox polytope 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.
 * type=Segmentotope
 * dim = 4
 * obsa = Pesc
 * cells = 5 tetrahedra, 2 pentagonal pyramids
 * faces = 5+10 triangles, 1 pentagon
 * edges = 1+5+10
 * vertices = 2+5
 * verf=2 pentagonal pyramids, edge length 1
 * verf2=5 digonal disphenoids, edge lengths (1+$\sqrt{5}$)/2 (1 base) and 1 (remaining edges)
 * coxeter = xo ox5oo&#x
 * army=Pesc
 * reg=Pesc
 * symmetry=H{{sub|2}}×A{{sub|1}}×I, order 20
 * circum = $$\frac{1+\sqrt5}{2} ≈ 1.61803$$
 * height = Point atop peppy: $$\frac{\sqrt5-1}{4} ≈ 0.30902$$
 * height2 = Dyad atop perp peg: $$\sqrt{\frac{5-2\sqrt5]{20}} ≈ 0.16246$$
 * hypervolume = $$\frac{\sqrt5}{96} ≈ 0.023292$$
 * dich= Tet–3–tet: $$\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751°$$
 * dich2= Peppy–5–peppy: 144°
 * dich3= Tet–3–peppy: $$\arccos\left(\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 22.23876°$$
 * dual=Pentagonal scalene
 * conjugate=Pentagrammic scalene
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

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)