In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces.

The same shape is also called hexakaidecadeltahedron[1], heccaidecadeltahedron,[2] or tetrakis square antiprism;[1] these last names mean a polyhedron with 16 triangular faces.

It is an example of deltahedron, and of a Johnson solid.

The gyroelongated square pyramid appears in chemistry as the basis for the bicapped square antiprismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem.

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.

[3][4] 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.

[5] 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

[6] One possible system of Cartesian coordinates for the vertices of a gyroelongated square bipyramid, giving it edge length 2, is:[1]

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

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:[4]

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:[7] The dual polyhedron of a gyroleongated square bipyramid is the square truncated trapezohedron.

[citation needed] It has eight pentagons and two squares.

[8] 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.

[9] It has 10 vertices and 24 edges, corresponding to the closo polyhedron with

An example is nickel carbonyl carbide anion Ni10C(CO)2−18, a 22 skeletal electron chemical compound with ten Ni(CO)2 vertices and the deficiency of two carbon monoxides.

[10] The Thomson problem concerning the minimum-energy configuration of

places the points at the vertices of a gyroelongated square bipyramid, inscribed in a sphere.