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 cupolae
Example: pentagonal orthobirotunda
Faces	2 n-gons 2n pentagons 4n triangles
Edges	12n
Vertices	6n
Symmetry group	Ortho: Dnh, [n,2], (*n22), order 4n Gyro: Dnd, [2n,2+ ], (2*n), order 4n
Rotation group	Dn, [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.

Examples

Birotundas

4	5	6	7	8
square orthobirotunda	pentagonal orthobirotunda	hexagonal orthobirotunda	heptagonal orthobirotunda	octagonal orthobirotunda
square gyrobirotunda	pentagonal gyrobirotunda (icosidodecahedron)	hexagonal gyrobirotunda	heptagonal gyrobirotunda	octagonal gyrobirotunda

See also

• Gyroelongated pentagonal birotunda
• Elongated pentagonal orthobirotunda
• Elongated pentagonal gyrobirotunda

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.