In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a birotunda is the name for several different polyhedra, formed from two rotunda

Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids . It can be seen as half an icosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966.-Formulae:...

s adjoined through their common regular-decagon

Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...

 faces. In an orthobirotunda, 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. In addition, a rotunda may be elongated in various ways by a ring of faces that separates one rotunda from the other.

There are five birotundas among the Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s:

• the pentagonal orthobirotunda
  Pentagonal orthobirotunda
  In geometry, the pentagonal orthobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by joining two pentagonal rotundae along their decagonal faces, matching like faces...
  
  ,
• the icosidodecahedron
  Icosidodecahedron
  In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...
  
  , which is known as the pentagonal gyrobirotunda
• the elongated pentagonal orthobirotunda
  Elongated pentagonal orthobirotunda
  In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda by inserting a decagonal prism between its congruent halves...
  
  , in which a ring of ten square faces separates two mirrored rotundas,
• the elongated pentagonal gyrobirotunda
  Elongated pentagonal gyrobirotunda
  In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron , by inserting a decagonal prism between its congruent halves...
  
  , and
• the gyroelongated pentagonal birotunda
  Gyroelongated pentagonal birotunda
  In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda by inserting a decagonal antiprism between its two halves.The gyroelongated pentagonal birotunda is one of five Johnson solids...
  
  , in which a ring of twenty triangular faces separates the two rotundas.

Another Johnson solid, the bilunabirotunda

Bilunabirotunda

In geometry, the bilunabirotunda is one of the Johnson solids . It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....

, is not a birotunda despite having that word as part of its name.