Gyroelongated square bipyramid

Like other gyroelongated bipyramids, the gyroelongated square bipyramid can be constructed by attaching two equilateral square pyramids onto the square faces of a square antiprism; this process is known as gyroelongation. These pyramids cover each square, replacing it with four equilateral triangles, so that the resulting polyhedron has 16 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the gyroelongated square bipyramid. More generally, the convex polyhedron in which all faces are regular is the Johnson solid, and every convex deltahedron is a Johnson solid. The gyroelongated square bipyramid is numbered among the Johnson solids as J_{17} . One possible system of Cartesian coordinates for the vertices of a gyroelongated square bipyramid, giving it edge length 2, is: \begin{align} \left(\pm 1, \pm 1, 2^{-1/4} \right),\qquad &\left(\pm \sqrt{2}, 0, -2^{-1/4} \right), \\ \left(0, \pm \sqrt{2}, -2^{-1/4} \right),\qquad &\left(0, 0, \pm \left(2^{-1/4} + \sqrt{2}\right)\right). \end{align} == Properties ==

The surface area of a gyroelongated square bipyramid is 16 times the area of an equilateral triangle, that is: 4\sqrt{3}a^2 \approx 6.928a^2, and the volume of a gyroelongated square bipyramid is obtained by slicing it into two equilateral square pyramids and one square antiprism, and then adding their volume: \frac{\sqrt{2} + \sqrt{4 + 3\sqrt{2}}}{3}a^3 \approx 1.428a^3. It has the same three-dimensional symmetry group as the square antiprism, the dihedral group of D_{4d} of order 8. Its dihedral angle is similar to the gyroelongated square pyramid, by calculating the sum of the equilateral square pyramid and the square antiprism's angle in the following: • the dihedral angle of an equilateral square pyramid between two adjacent triangles, approximately 109.47^\circ • the dihedral angle of a square antiprism between two adjacent triangles, approximately 127.55^\circ • the dihedral angle between two adjacent triangles, on the edge where an equilateral square pyramid is attached to a square antiprism, is 158.57^\circ, which is obtained by adding the dihedral angles between a square and a triangle of both the pyramid and the antiprism. The dual polyhedron of a gyroelongated square bipyramid is the square truncated trapezohedron. As suggested by the name, it is the family of truncated trapezohedron by means of truncating two axial vertices of a square trapezohedron. The resulting polyhedron has eight pentagons and two squares. == Application ==

Gyroelongated square bipyramid can be visualized in the geometry of chemical compounds as the atom cluster surrounding a central atom as a polyhedron, and the compound of such cluster is the bicapped square antiprismatic molecular geometry. It has 10 vertices and 24 edges, corresponding to the closo polyhedron with 2n + 2 skeletal electrons. An example is nickel carbonyl carbide anion , a 22 skeletal electron chemical compound with ten vertices and the deficiency of two carbon monoxides. The Thomson problem concerns the minimum-energy configuration of n charged particles on a sphere. The minimum solution known for n = 10 places the points at the vertices of a gyroelongated square bipyramid, inscribed in a sphere. == References ==