# Elongated pentagonal orthobirotunda

Elongated pentagonal orthobirotunda
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymEpobro
Coxeter diagramxoxxox5ofxxfo&#xt
Stewart notationR5P10R5
J42
Elements
Faces10+10 triangles, 5+5 squares, 2+10 pentagons
Edges10+10+10+10+20+20
Vertices10+10+20
Vertex figures20 rectangles, edge lengths 1 and (1+5)/2
20 irregular tetragons, edge lengths 1, (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle \frac{45+17\sqrt5+15\sqrt{5+2\sqrt5}}{6} ≈ 21.52973}$
Dihedral angles3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768^\circ}$
5–4: ${\displaystyle \arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495^\circ}$
4–4: 144°
3–5: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263^\circ}$
Central density1
Number of external pieces42
Level of complexity16
Related polytopes
ArmyEpobro
RegimentEpobro
DualPentarhombideltorthodecatetragonal tetracontahedron
ConjugateElongated pentagrammic orthobirotunda
Abstract & topological properties
Flag count320
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
ConvexYes
NatureTame

The elongated pentagonal orthobirotunda is one of the 92 Johnson solids (J42). It consists of 10+10 triangles, 5+5 squares, and 2+10 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal orthobirotunda.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right),}$
• ${\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),}$
• ${\displaystyle \left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,±\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,±\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,±\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right).}$