# Pentagonal scalene

Pentagonal scalene
Rank4
TypeSegmentotope
Notation
Bowers style acronymPesc
Coxeter diagramxo ox5oo&#x
Elements
Cells5 tetrahedra, 2 pentagonal pyramids
Faces5+10 triangles, 1 pentagon
Edges1+5+10
Vertices2+5
Vertex figures2 pentagonal pyramids, edge length 1
5 digonal disphenoids, edge lengths (1+5)/2 (1 base) and 1 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Hypervolume${\displaystyle {\frac {\sqrt {5}}{96}}\approx 0.023292}$
Dichoral anglesTet–3–tet: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Peppy–5–peppy: 144°
Tet–3–peppy: ${\displaystyle \arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }}$
HeightsPoint atop peppy: ${\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}$
Dyad atop perp peg: ${\displaystyle {\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\approx 0.16246}$
Central density1
Related polytopes
ArmyPesc
RegimentPesc
DualPentagonal scalene
ConjugatePentagrammic scalene
Abstract & topological properties
Flag count200
Euler characteristic0
OrientableYes
Properties
SymmetryH2×A1×I, order 20
Flag orbits10
ConvexYes
NatureTame

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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,0\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,0\right)}$,
• ${\displaystyle \left(0,\,0,\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right)}$.

## 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)