Pentagonal orthobicupola

{{Infobox polytope The pentagonal orthobicupola is one of the 92 Johnson solids (J30). 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 in the same orientation.
 * type=CRF
 * img=Pentagonal orthobicupola 2.png
 * 3d=J30 pentagonal orthobicupola.stl
 * off=Pentagonal orthobicupola.off
 * dim = 3
 * obsa = Pobcu
 * faces = 10 triangles, 10 squares, 2 pentagons
 * edges = 5+5+10+20
 * vertices = 10+10
 * verf = 10 isosceles trapezoids, edge lengths 1, $\sqrt{2}$, (1+$\sqrt{5}$}0/2, $\sqrt{2}$
 * verf2 = 10 kites, edge lengths 1 and $\sqrt{2}$
 * coxeter = xxx5oxo&#xt
 * army=Pobcu
 * reg=Pobcu
 * symmetry = H2×A1, order 20
 * volume = $$\frac{5+4\sqrt5}{3} \approx 4.64809$$
 * dih = 3–4: $$\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) \approx 159.09484°$$
 * dih2 = 4–5: $$\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) \approx 148.28253°$$
 * dih3 = 3–3: $$\arccos\left(\frac{4\sqrt5-5}{15}\right) \approx 74.75474°$$
 * dih4 = 4–4: $$\arccos\left(\frac{\sqrt5}{5}\right) \approx 63.43495°$$
 * height = $$\sqrt{\frac{10-2\sqrt{5}}{5}}\approx 1.05146$$
 * smm = Yes
 * dual = Deltotrapezohedral icosahedron
 * conjugate = Retrograde pentagrammic orthobicupola
 * conv=Yes
 * orientable=Yes
 * nat=Tame}}

If the cupolas are joined such that the bases are rotated 36º, the result is the pentagonal gyrobicupola.

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


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

Related polyhedra

 * A decagonal prism can be inserted between the two halves of the pentagonal orthobicupola to produce the elongated pentagonal orthobicupola.
 * A slightly non-uniform pentagonal prism can be excavated to form the excavated pentagonal orthobicupola, a near-miss Stewart toroid.