# Birotunda

In geometry, a **birotunda** is any member of a family of dihedral-symmetric polyhedra, formed from two rotunda adjoined through the largest face. They are similar to a bicupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. There are two forms, ortho- and gyro-: an *orthobirotunda* has one of the two rotundas is placed as the mirror reflection of the other, while in a *gyrobirotunda* one rotunda is twisted relative to the other.

Set of birotundas | |
---|---|

(Example Ortho/gyro pentagonal forms) | |

Faces | 2 n-gons 2 n pentagons4 n triangles |

Edges | 12n |

Vertices | 6n |

Symmetry group | Ortho: D_{nh}, [n,2], (*n22), order 4nGyro: D |

Rotation group | D_{n}, [n,2]^{+}, (n22), order 2n |

Properties | convex |

The pentagonal birotundas can be formed with regular faces, one a Johnson solid, the other a semiregular polyhedron:

- pentagonal orthobirotunda,
- pentagonal gyrobirotunda, which is also called an icosidodecahedron.

Other forms can be generated with dihedral symmetry and distorted equilateral pentagons.

## See also

## References

- Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics,
**18**, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. - Victor A. Zalgaller (1969).
*Convex Polyhedra with Regular Faces*. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.

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