Square gyrobicupola

The square gyrobicupola is one of the 92 Johnson solids (J29). It consists of 8 triangles and 2+8 squares. It can be constructed by attaching two square cupolas at their octagonal bases, such that the two square bases are rotated 45° from each other.

It is topologically equivalent to the rectified square antiprism.

If the cupolas are joined such that the bases are in the same orientation, the result is the square orthobicupola.

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


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

Related polyhedra
An octagonal prism can be inserted between the two halves of the square gyrobicupola to produce the elongated square gyrobicupola.