Triangular orthobicupola

In geometry, the triangular orthobicupola is the 27th Johnson solid. As the name suggests, it can be constructed by attaching two triangular cupolae 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,^[1] or disheptahedron.^[2][3]

The dual polyhedron of a triangular orthobicupola is a trapezo-rhombic dodecahedron. It is a plesiohedron, space-filling polyhedron defined by Voronoi diagram.

The triangular orthobicupola can be found in the coordination structure of crystals with hexagonal closed-packing spheres in chemistry.

The triangular orthobicupola is constructed by attaching two triangular cupolas to their bases. Similar to the cuboctahedron, which would be known as the triangular gyrobicupola, the difference is that the two triangular cupolas that 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.^[4] Hence, another name for the triangular orthobicupola is the anticuboctahedron.^[5] By such a construction, the triangular orthobicupola is a composite polyhedron.^[6]

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, ${\displaystyle J_{27}}$.^[7][8]

The surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ of a triangular orthobicupola are the same as those of a cuboctahedron. Its surface area is the sum of all of its polygonal faces, and its volume is obtained by slicing it off into two triangular cupolas and adding their volume. With edge length ${\displaystyle a}$, they are:^[7]$${\displaystyle A=\left(6+2{\sqrt {3}}\right)a^{2}\approx 9.464a^{2},\quad V={\frac {5{\sqrt {2}}}{3}}a^{3}\approx 2.357a^{3}.}$$

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 a twelve-faced rhombic dodecahedron of six rhombi and six trapezoidal shapes, a trapezo-rhombic dodecahedron;^[5] alternative names are rhombo-trapezoidal dodecahedron and trapezoidal dodecahedron.^[10] The trapezo-rhombic dodecahedron 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]

A trapezo-rhombic dodecahedron has two different edges lengths, ${\textstyle {\frac {2}{3}}a}$ and ${\textstyle {\frac {4}{3}}a}$. For its surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$, they can be formulated as^[11]$${\displaystyle A=8{\sqrt {2}}a^{2}\approx 11.314a^{2},\quad V={\frac {16}{9}}{\sqrt {3}}a^{3}\approx 3.079a^{3}.}$$

• Weisstein, Eric W., "Johnson solid" ("Triangular orthobicupola") at MathWorld.