# Pentagonal cupofastegium

Jump to navigation Jump to search
Pentagonal cupofastegium
Rank4
TypeSegmentotope
Notation
Bowers style acronymPecuf
Coxeter diagramox xx5xo&#x
Elements
Cells5 tetrahedra, 5 triangular prisms, 1 pentagonal prism, 2 pentagonal cupolas
Faces10+10 triangles, 5+10 squares, 2 pentagons, 1 decagon
Edges5+5+5+10+20
Vertices10+10
Vertex figures10 isosceles trapezoidal pyramids, base edge lengths 1, 2, (1+5)/2, 2, side edge lengths 1, 1, 2. 2
10 sphenoids, edge lengths 1 (3), 2 (2), and (5+5)/2 (1)
Measures (edge length 1)
Circumradius${\displaystyle 3+{\sqrt {5}}\approx 5.23607}$
Hypervolume${\displaystyle 3{\frac {8+3{\sqrt {5}}}{32}}\approx 1.37889}$
Dichoral anglesTet–3–trip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }}$
Pip–4–trip: ${\displaystyle \arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }}$
Pecu–10–pecu: 144°
Pecu–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }}$
Pecu–4–trip: ${\displaystyle \arccos \left({\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }}$
Pip–5–pecu: 18°
HeightsPeg atop pecu: ${\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}$
Pip atop dec: ${\displaystyle {\sqrt {\frac {5-2{\sqrt {5}}}{20}}}=0.16246}$
Central density1
Related polytopes
ArmyPecuf
RegimentPecuf
DualPentagonal cupolanotch
ConjugateRetrograde pentagrammic cupofastegium
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2×A1×I, order 20
ConvexYes
NatureTame

The pentagonal cupofastegium, or pecuf, also sometimes called the pentagonal orthobicupolic ring, is a CRF segmentochoron (designated K-4.154 on Richard Klitzing's list). It consists of 1 pentagonal prism, 5 tetrahedra, 5 triangular prisms, and 2 pentagonal cupolas.

It has two representations as a segmentochoron: pentagon atop pentagonal cupola or pentagonal prism atop decagon.

The pentagonal cupofastegium can be obtained as a cap of the small disprismatohexacosihecatonicosachoron in pentagonal prism first orientation.

## Vertex coordinates

The vertices of a pentagonal cupofastegium with edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(0,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,0\right).}$

## Representations

A pentagonal cupofastegium has the following Coxeter diagrams:

• ox xx5xo&#x (full symmetry)
• xxx5oxo&#x (H2 symmetry only, seen with pentagon atop pentagonal cupola)