Pentagonal scalene

Pentagonal scalene
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Pesc
Coxeter diagram	xo ox5oo&#x
Elements
Cells	5 tetrahedra, 2 pentagonal pyramids
Faces	5+10 triangles, 1 pentagon
Edges	1+5+10
Vertices	2+5
Vertex figures	2 pentagonal pyramids, edge length 1
	5 digonal disphenoids, edge lengths (1+√5)/2 (1 base) and 1 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {1+{\sqrt {5}}}{2}}\approx 1.61803$
Hypervolume	${\frac {\sqrt {5}}{96}}\approx 0.023292$
Dichoral angles	Tet–3–tet: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Peppy–5–peppy: 144°
	Tet–3–peppy: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
Heights	Point atop peppy: ${\frac {{\sqrt {5}}-1}{4}}\approx 0.30902$
	Dyad atop perp peg: ${\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\approx 0.16246$
Central density	1
Related polytopes
Army	Pesc
Regiment	Pesc
Dual	Pentagonal scalene
Conjugate	Pentagrammic scalene
Abstract & topological properties
Flag count	200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×A1×I, order 20
Flag orbits	10
Convex	Yes
Nature	Tame

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:

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

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)

External links

• Klitzing, Richard. "pesc".