# Pentagonal orthobirotunda

Pentagonal orthobirotunda
Rank3
TypeCRF
Notation
Bowers style acronymPobro
Coxeter diagramxoxox5ofxfo&#xt
Elements
Faces10+10 triangles, 2+10 pentagons
Edges5+5+10+20+20
Vertices10+10+10
Vertex figures10+10 rectangles, edge lengths 1 and (1+5)/2
10 kites, edge lengths 1 and (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Volume${\displaystyle {\frac {45+17{\sqrt {5}}}{6}}\approx 13.83552}$
Dihedral angles3–3: ${\displaystyle \arccos \left(-{\frac {5+4{\sqrt {5}}}{15}}\right)\approx 158.37537^{\circ }}$
3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
5–5: ${\displaystyle \arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }}$
Central density1
Number of external pieces32
Level of complexity12
Related polytopes
ArmyPobro
RegimentPobro
DualRhombitrapezohedral triacontahedron
ConjugatePentagrammic orthobirotunda
Abstract & topological properties
Flag count240
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
ConvexYes
NatureTame

The pentagonal orthobirotunda is one of the 92 Johnson solids (J34). It consists of 10+10 triangles and 2+10 pentagons. It can be constructed by attaching two pentagonal rotundas at their decagonal bases, such that the two pentagonal bases are in the same orientation.

If the rotundas are joined such that the bases are rotated 36°, the result is the pentagonal gyrobirotunda, better known as the uniform icosidodecahedron. Conversely, the pentagonal orthobirotunda can be seen as a gyrate icosidodecahedron, since it is an icosidodecahedron with one half rotated.

## Vertex coordinates

A pentagonal orthobirotunda of edge length 1 has vertices given by the following coordinates:

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

## Related polyhedra

A decagonal prism can be inserted between the two halves of the pentagonal orthobirotunda to produce the elongated pentagonal orthobirotunda.