Pentagonal rotunda

Pentagonal rotunda
Rank3
TypeCRF
Notation
Bowers style acronymPero
Coxeter diagramofx5xox&#xt
Stewart notationR5
Elements
Faces
Edges5+5+5+10+10
Vertices5+5+10
Vertex figures5+5 rectangles, edge lengths 1 and 1+5)/2
10 scalene triangles, edge lengths 1, (1+5)/2, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Volume${\displaystyle {\frac {45+17{\sqrt {5}}}{12}}\approx 6.91776}$
Dihedral angles3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
3–10: ${\displaystyle \arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 79.18768^{\circ }}$
5–10: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density1
Number of external pieces17
Level of complexity14
Related polytopes
ArmyPero
RegimentPero
DualSemibisected pentagonal rhombitrapezohedron
ConjugatePentagrammic rotunda
Abstract & topological properties
Flag count140
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
Flag orbits14
ConvexYes
NatureTame

The pentagonal rotunda (OBSA: pero) is one of the 92 Johnson solids (J6). It consists of 5+5 triangles, 1+5 pentagons, and 1 decagon. It is a rotunda based on a pentagon, and the only rotunda that results in a Johnson solid.

It can be constructed by cutting an icosidodecahedron in half along one of its decagonal circles of edges. This produces two pentagonal rotundas with the bases in opposite orientation, so the icosidodecahedron can be thought of as the pentagonal gyrobirotunda.

Vertex coordinates

Coordinates for a pentagonal rotunda of edge length 1 has vertices are given by the following:

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

These coordinates create a pentagonal rotunda with a decagonal base on the xy plane.

An alternate set of coordinates can be obtained as a subset of the vertices of the icosidodecahedron:

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

Related polyhedra

Two pentagonal rotundas can be attached at their decagonal bases in the same orientation to form a pentagonal orthobirotunda. If the second rotunda is rotated by 36° the result is the pentagonal gyrobirotunda, better known as the icosidodecahedron. If a pentagonal cupola is attached, the result is either a pentagonal orthocupolarotunda (if the base pentagons are in the same orientation) or a pentagonal gyrocupolarotunda (if the base pentagons are rotated 36°).

A decagonal prism can be attached to the pentagonal rotunda's decagonal base to form the elongated pentagonal rotunda. If a decagonal antiprism is attached instead, the result is the gyroelongated pentagonal rotunda.