Gyroelongated square bipyramid

Gyroelongated square bipyramid
Type	Gyroelongated bipyramid, Deltahedron, Johnson J16 – J17 – J18
Faces	16 triangles
Edges	24
Vertices	10
Vertex configuration	${\displaystyle 2\times (3^{4})+8\times (3^{5})}$
Symmetry group	${\displaystyle D_{4d}}$
Dual polyhedron	Truncated square trapezohedron
Properties	convex
Net

In

, and every convex deltahedron is a Johnson solid. The gyroelongated square bipyramid is numbered among the Johnson solids as ${\displaystyle J_{17}}$.

One possible system of

$${\displaystyle {\begin{aligned}\left(\pm 1,\pm 1,2^{-1/4}\right),\qquad &\left(\pm {\sqrt {2}},0,-2^{-1/4}\right),\\\left(0,\pm {\sqrt {2}},-2^{-1/4}\right),\qquad &\left(0,0,\pm \left(2^{-1/4}+{\sqrt {2}}\right)\right).\end{aligned}}}$$

Properties

The surface area of a gyroelongated square bipyramid is 16 times the area of an equilateral triangle, that is:[4]

$${\displaystyle 4{\sqrt {3}}a^{2}\approx 6.928a^{2},}$$

$${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}}{3}}a^{3}\approx 1.428a^{3}.}$$

It has the same three-dimensional symmetry group as the square antiprism, the dihedral group of ${\displaystyle D_{4d}}$ 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: the dihedral angle of two adjacent triangles in an equilateral square pyramid is ${\displaystyle 109.47^{\circ }}$, and that of square and triangle in the same pyramid is ${\displaystyle 54.74^{\circ }}$. The dihedral angle of a square antiprism between the two adjacent triangles is ${\displaystyle 127.55^{\circ }}$, and that between square and triangle is ${\displaystyle 103.83^{\circ }}$. The dihedral angle of the square and triangle, on the edge where the square pyramid attaches the antiprism, is ${\displaystyle 103.83^{\circ }+54.74^{\circ }=158.57^{\circ }}$.^[7]

Dual polyhedron

The dual polyhedron of a gyroleongated square bipyramid is the

with ${\displaystyle 2n+2}$ skeletal electrons. An example is nickel carbonyl carbide anion ${\displaystyle {\ce {Ni_{10}C(CO)_{18}^{2-}}}}$, a 22 skeletal electron chemical compound with ten ${\displaystyle {\ce {Ni(CO)_2}}}$ vertices and the deficiency of two

The Thomson problem concerning the minimum-energy configuration of ${\displaystyle n}$ charged particles on a sphere. The minimum solution known for ${\displaystyle n=10}$ places the points at the vertices of a gyroelongated square bipyramid, inscribed in a sphere.^[1]

References