Square gyrobicupola

{{Infobox polytope The square gyrobicupola, or squigybcu, 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 45º from each other.
 * type=CRF
 * img=Square gyrobicupola 2.png
 * 3d=J29 square gyrobicupola.stl
 * off=Square gyrobicupola.off
 * dim = 3
 * obsa = Squigybcu
 * faces = 8 triangles, 2+8 squares
 * edges = 8+8+16
 * vertices = 8+8
 * verf = 8 isosceles trapezoids, edge lengths 1, $\sqrt{2}$, $\sqrt{2}$, $\sqrt{2}$; 8 rectangles, edge lengths 1 and $\sqrt{2}$
 * coxeter = xxo4oxx&#xt
 * army=Squigybcu
 * reg=Squigybcu
 * symmetry = I2(8)×A1+, order 16
 * volume = $$2\frac{3+2\sqrt2}{3} ≈ 3.88562$$
 * dih = 3–4 cupolaic: $$\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$$
 * dih2 = 4–4: 135°
 * dih3 = 3–4 join: $$\arccos\left(-\sqrt{\frac{3-2\sqrt2}{6}\right) ≈ 99.73561°$$
 * dual = Joined square antiprism
 * conjugate = Retrograde square gyrobicupola
 * conv=Yes
 * orientable=Yes
 * nat=Tame
 * smm = Yes}}

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.