Triangular orthobicupola


In geometry, the triangular orthobicupola is one of the Johnson solids . 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. 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, 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 ; 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, which is constructed by elongating this solid, has a special relationship with the rhombicuboctahedron.
The dual of the triangular orthobicupola is the trapezo-rhombic dodecahedron. It has 6 rhombic and 6 trapezoidal faces, and is similar to the rhombic dodecahedron.

Formulae

The following formulae for volume, surface area, and circumradius can be used if all faces are regular, with edge length a:
The circumradius of a triangular orthobicupola is the same as the edge length.

Related polyhedra and honeycombs

The rectified cubic honeycomb can be dissected and rebuilt as a space-filling lattice of
triangular orthobicupolae and square pyramids.