# Elongated pentagonal orthobicupola

Elongated pentagonal orthobicupola
Rank3
TypeCRF
Notation
Bowers style acronymEpobcu
Coxeter diagramoxxo5xxxx&#xt
Elements
Faces
Edges10+10+10+10+20
Vertices10+20
Vertex figures10 isosceles trapezoids, edge lengths 1, 2, (1+5)/2, 2
20 trapezoids, edge lengths 1, 2, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}}{6}}\approx 12.34230}$
Dihedral angles3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
4–4 prismatic: 144°
3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }}$
4–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
Central density1
Number of external pieces32
Level of complexity12
Related polytopes
ArmyEpobcu
RegimentEpobcu
DualOrthodeltodeltoidal triacontahedron
Abstract & topological properties
Flag count240
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
Flag orbits12
ConvexYes
NatureTame

The elongated pentagonal orthobicupola (OBSA: epobcu) is one of the 92 Johnson solids (J38). It consists of 10 triangles, 5+5+10 squares, and 2 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal orthobicupola.

## Vertex coordinates

An elongated pentagonal orthobicupola of edge length 1 has the following vertices:

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