Triangular orthobicupola

In geometry, the triangular orthobicupola is one of the Johnson solids (J_27). As the name suggests, it can be constructed by attaching two triangular cupolas (J_3) 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. 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 (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 (J35), which is constructed by elongating this solid, has a (different) 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. 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 (C = a). The rectified cubic honeycomb can be dissected and rebuilt as a space-filling lattice of triangular orthobicupolae and square pyramids.