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 c ...
, the triangular orthobicupola is one of the
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...
s (). As the name suggests, it can be constructed by attaching two
triangular cupolas () along their bases. It has an equal number of squares and triangles at each vertex; however, it is not
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...
. It is also called an ''anticuboctahedron'', ''twisted cuboctahedron'' or ''disheptahedron''. It is also a
canonical polyhedron.
The ''triangular orthobicupola'' is the first in an infinite set of
orthobicupolae.
Relation to cuboctahedra
The ''triangular orthobicupola'' has a superficial resemblance to the
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, which would be known as the ''triangular gyrobicupola'' in the nomenclature of Johnson solids — the difference is that the two triangular cupolas which make up the triangular orthobicupola are joined so that pairs of matching sides abut (hence, "ortho"); the cuboctahedron is joined so that triangles abut squares and vice versa. Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron. Hence, another name for the triangular orthobicupola is the ''anticuboctahedron''.
The
elongated triangular orthobicupola (''J''
35), which is constructed by elongating this solid, has a (different) special relationship with the