Elongated pentagonal gyrocupolarotunda

{{Infobox polytope The elongated pentagonal gyrocupolarotunda, or epgycuro, is one of the 92 Johnson solids (J40). It consists of 5+5+5 triangles, 5+5+5 squares, and 1+1+5 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal gyrocupolarotunda.
 * type=CRF
 * img=Elongated pentagonal gyrocupolarotunda 2.png
 * 3d=J41 elongated pentagonal gyrocupolarotunda.stl
 * dim = 3
 * obsa = Epgycuro
 * faces = 5+5+5 triangles, 5+5+5 squares, 1+1+5 pentagons
 * edges = 5+5+5+5+5+5+10+10+10+10
 * vertices = 5+5+5+10+10
 * verf = 5 trapezoids, edge lengths 1, $\sqrt{2}$, (1+$\sqrt{5}$}/2, $\sqrt{2}$
 * verf2 = 10 rectangles, edge lengths 1 and (1+$\sqrt{5}$)/2
 * verf3 = 10 trapezoids, edge lengths 1, $\sqrt{2}$, $\sqrt{2}$, $\sqrt{2}$
 * verf4 = 10 irregular tetragons, edge lengths 1, (1+$\sqrt{5}$)/2, $\sqrt{2}$, $\sqrt{2}$
 * coxeter = xoxxo5ofxxx&#xt
 * army=Epgycuro
 * reg=Epgycuro
 * symmetry = H2×I, order 10
 * volume = 5(11+5$\sqrt{5}$+6$\sqrt{5+2√5}$)/12 ≈ 16.93602
 * dih = 3–4 rotundaic join: acos($\sqrt{2(5+√5)/15}$) ≈ 169.18768º
 * dih2 = 3–4 cupolaic: acos(–($\sqrt{3}$+$\sqrt{15}$)/6) ≈ 159.09484º
 * dih3 = 4–5 join: acos(–2$\sqrt{5}$/5) ≈ 153.43495º
 * dih4 = 4–5 cupolaic: acos(–$\sqrt{(5+√5)/10}$) ≈ 148.28253º
 * dih5 = 4–4 prismatic: 144º
 * dih6 = 3–5: acos(–($\sqrt{5+2√5)/15}$) ≈ 142.62263º
 * dih7 = 3–4 cupolaic join: acos(–$\sqrt{(10–2√5)/15}$) ≈ 127.37737º
 * dih8 = 4–4 join: acos(–$\sqrt{(5–√5)/10}$) ≈ 121.71747º
 * smm = Yes
 * conjugate = Rlongated retrograde pentagrammic fgyrocupolarotunda
 * conv=Yes
 * orientable=Yes
 * nat=Tame}}

Vertex coordinates
An elongated pentagonal gyrocupolarotunda of edge length 1 has vertices given by the following coordinates:


 * (±1/2, ±$\sqrt{(5+2√5)}$/2, ±1/2),
 * (±(3+$\sqrt{5}$)/4, ±$\sqrt{(5+√5)/8}$, ±1/2),
 * (±(1+$\sqrt{5}$)/2, 0, ±1/2),
 * (±1/2, –$\sqrt{(5+2√5)/20}$, (1+2$\sqrt{(5+2√5)/5}$)/2),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{(5+√5)/40}$, (1+2$\sqrt{(5+2√5)/5}$)/2),
 * (0, $\sqrt{(5+√5)/10}$, (1+2$\sqrt{(5+2√5)/5}$)/2),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{(25+11√5)/40}$, (1+2$\sqrt{(5+√5)/10}$)/2),
 * (±(3+$\sqrt{5}$)/4, –$\sqrt{(5+√5)/40}$, (1+2$\sqrt{(5+√5)/10}$)/2),
 * (0, –$\sqrt{(5+2√5)/5}$, (1+2$\sqrt{(5+√5)/10}$)/2),
 * (±1/2, $\sqrt{(5+2√5)/20}$, –(1+2$\sqrt{(5–√5)/10}$)/2),
 * (±(1+$\sqrt{5}$)/4, –$\sqrt{(5+√5)/40}$, –(1+2$\sqrt{(5–√5)/10}$)/2),
 * (0, –$\sqrt{(5+√5)/10}$, –(1+2$\sqrt{(5–√5)/10}$)/2).