In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a birotunda is any member of a family of
dihedral-symmetric polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
, 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.
The pentagonal birotundas can be formed with regular faces, one a
Johnson solid, the other a
semiregular polyhedron
In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors.
Definitions
In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...
:
*
pentagonal orthobirotunda
In geometry, the pentagonal orthobirotunda is one of the Johnson solids (). It can be constructed by joining two pentagonal rotundae () along their decagonal faces, matching like faces.
Related polyhedra
The pentagonal orthobirotunda is als ...
,
* pentagonal gyrobirotunda, which is also called an
icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...
.
Other forms can be generated with
dihedral symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...
and distorted equilateral pentagons.
Examples
See also
*
Gyroelongated pentagonal birotunda
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
*
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.
* The first proof that there are only 92 Johnson solids.
{{polyhedron navigator
Johnson solids