Triangular orthobicupola

The triangular orthobicupola is constructed by attaching two triangular cupolae at their bases. This construction also applies to the cuboctahedron, which could also be called the triangular gyrobicupola; the difference is that the two triangular cupolas that make up the triangular orthobicupola are joined so that the triangular bases have the same orientation (hence, "ortho"); the cuboctahedron is joined so that the triangular bases have opposite orientations. Given a triangular orthobicupola, a 60° rotation of one cupola before the joining yields a cuboctahedron.^[4] Hence, another name for the triangular orthobicupola is the anticuboctahedron.^[5]

The triangular orthobicupola is a convex polyhedron with regular polygonal faces (eight equilateral triangles and six squares) and is therefore a Johnson solid, named after American mathematician Norman W. Johnson, who enumerated the 92 such polyhedra (excluding the uniform polyhedra). It is enumerated as the twenty-seventh Johnson solid, ${\displaystyle J_{27}}$.^[6][7] It is a composite polyhedron, because it can be sliced into two triangular cupolae.^[8]

The surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ of a triangular orthobicupola are the same as those of a cuboctahedron. The surface is the sum of all of its polygonal faces: six squares and eight equilateral triangles. For edge length ${\displaystyle a}$, the areas of these of polygons are given by ${\displaystyle a^{2}}$ and ${\textstyle {\frac {\sqrt {3}}{4}}a^{2}}$, respectively. Therefore,^[6] $${\displaystyle A=6(a^{2})+8\left({\frac {\sqrt {3}}{4}}a^{2}\right)=\left(6+2{\sqrt {3}}\right)a^{2}\approx 9.464a^{2}.}$$ Its volume is twice the volume of a triangular cupola:^[6]$${\displaystyle V={\frac {5{\sqrt {2}}}{3}}a^{3}\approx 2.357a^{3}.}$$

A triangular orthobicupola has the same symmetry as a triangular prism, the dihedral group ${\displaystyle \mathrm {D} _{3\mathrm {h} }}$ of order six, which contains one three-fold axis and three two-fold axes.^[1]

A triangular orthobicupola has three different dihedral angles (angles between two polygonal faces):^[9]

• An angle between two adjacent triangles is around 141°. This is obtained by adding the two triangular cupolae's triangle-to-hexagon angles.
• An angle between two adjacent squares is around 109.4°. This is obtained by adding the two triangular cupolae's square-to-hexagon angles.
• A square-to-triangle angle is the same angle as in the triangular cupola, around 125.3°.

The packing of congruent spheres can be arranged densely into a triangular orthobicupola.^[2] The twelve vertices represent the spheres, or ligancy. Such an arrangement is called the hexagonal close-packing, and is found in crystal coordination.^[3] Magnesium and helium are chemical elements with such a structure.^[2]

The dual polyhedron of a triangular orthobicupola is the trapezo-rhombic dodecahedron^[5] or trapezoidal dodecahedron,^[10] a twelve-faced solid whose faces are six rhombi and six trapezoids. It is obtained by cutting a rhombic dodecahedron in half through the hexagonal cross-section and rotating the halves 60° with respect to each other.^[11] It is a plesiohedron, a special kind of polyhedron that can tile a space with its copy, defined as the Voronoi cell of a symmetric Delone set, generated by the hexagonal close-packing.^[10][1]