Square cupola

The square cupola is one of the 92 Johnson solids (J4). It consists of 4 triangles, 1+4 squares, and 1 octagon. It is a cupola based on the square, and is one of three Johnson solid cupolas, the other two being the triangular cupola and the pentagonal cupola.

It can be obtained as a segment of the small rhombicuboctahedron, when considered as an elongated square orthobicupola.

Vertex coordinates
A square cupola of edge length 1 has vertices given by the following coordinates:


 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt2}{2}\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,0\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,0\right).$$

These can be obtained from placing a square and octagon in parallel planes.

Representations
A square cupola has the following Coxeter diagrams:


 * ox4xx&#x
 * so8ox&#x
 * oqxw qowx&#xr (bases in digonal symmetry)

Related polyhedra
Two square cupolas can be attached at their octagonal bases in the same orientation to form a square orthobicupola. If the second cupola is rotated by 45º the result is the square gyrobicupola.

An octagonal prism can be attached to the square cupola's octagonal base to form the elongated square cupola. If an octagonal antiprism is attached instead, the result is the gyroelongated square cupola.