# Elongated pentagonal gyrobicupola

Elongated pentagonal gyrobicupola
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymEpigybcu
Coxeter diagramoxxx5xxxo&#xt
Elements
Faces10 triangles, 10+10 squares, 2 pentagons
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${\displaystyle \frac{10+8\sqrt5+15\sqrt{5+2\sqrt5}}{6} ≈ 12.34230}$
Dihedral angles3–4 cupolaic: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484^\circ}$
4–5: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253^\circ}$
4–4 prismatic: 144°
3–4 join: ${\displaystyle \arccos\left(-\sqrt{\frac{10-2\sqrt5}{15}}\right) ≈ 127.37737^\circ}$
4–4 join: ${\displaystyle \arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747^\circ}$
Central density1
Number of external pieces32
Level of complexity12
Related polytopes
ArmyEpigybcu
RegimentEpigybcu
DualGyrodeltodeltoidal triacontahedron
Abstract & topological properties
Flag count240
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(10)×A1)/2, order 20
ConvexYes
NatureTame

The elongated pentagonal gyrobicupola is one of the 92 Johnson solids (J39). It consists of 10 triangles, 10+10 squares, and 2 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal gyrobicupola.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle ±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5+\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle ±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),}$