Pentagonal gyrobicupola

{{Infobox polytope The pentagonal gyrobicupola is one of the 92 Johnson solids (J31). It consists of 10 triangles, 10 squares, and 2 pentagons. It can be constructed by attaching two pentagonal cupolas at their decagonal bases, such that the two pentagonal bases are rotated 36° with respect to each other.
 * type=CRF
 * img=Pentagonal gyrobicupola 2.png
 * 3d=J31 pentagonal gyrobicupola.stl
 * off=Pentagonal gyrobicupola.off
 * dim = 3
 * obsa = Pegybcu
 * faces = 10 triangles, 10 squares, 2 pentagons
 * edges = 10+10+20
 * vertices = 10+10
 * verf = 10 isosceles trapezoids, edge lengths 1, $\sqrt{2}$, (1+$\sqrt{5}$}0/2, $\sqrt{2}$
 * verf2 = 10 rectangles, edge lengths 1 and $\sqrt{2}$
 * coxeter = xxo5oxx&#xt
 * army=Pegybcu
 * reg=Pegybcu
 * symmetry = I2(10)×A1/2, order 20
 * volume = $$\frac{\sqrt{5+4\sqrt5}}{3} \approx 4.64809$$
 * dih = 3–4 cupolaic: $$\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) \approx 159.09484°$$
 * dih2 = 4–5: $$\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) \approx 148.28253°$$
 * dih3 = 3–4 join: $$\arccos\left(\frac{\sqrt{15}-\sqrt3}{6}\right) \approx 69.09484°$$
 * height = $$\sqrt{\frac{10-2\sqrt{5}}{5}}\approx 1.05146$$
 * smm = Yes
 * dual = Joined pentagonal antiprism
 * conjugate = Retrograde pentagrammic gyrobicupola
 * conv=Yes
 * orientable=Yes
 * nat=Tame}}

It is topologically equivalent to the rectified pentagonal antiprism.

If the cupolas are joined in the same orientation, the result is the pentagonal orthobicupola.

Vertex coordinates
A pentagonal gyrobicupola of edge length 1 has vertices given by the following coordinates:


 * $$±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$$
 * $$±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0\right).$$

Related polyhedra
A decagonal prism can be inserted between the two halves of the pentagonal orthobicupola to produce the elongated pentagonal gyrobicupola..