View source for Elongated pentagonal gyrobirotunda

{{Infobox polytope
|type = [[CRF]]
|img = Elongated pentagonal gyrobirotunda 2.png
|3d = J43 elongated pentagonal gyrobirotunda.stl
|rank = 3
|obsa = {{BSAhelp
  |E|elongated
  |p|pentagonal
  |gy|gyro-
  |b|bi-
  |ro|rotunda
  }}
|faces = {{hlist
  |10+10 [[triangle]]s
  |10 [[square]]s
  |2+10 [[pentagon]]s
  }}
|edges = 10+10+10+10+20+20
|vertices = 10+10+20
|verf = 20 [[rectangle]]s, edge lengths 1 and (1+{{radic|5}})/2
|verf2 = 20 [[irregular tetragon]]s, edge lengths 1, (1+{{radic|5}})/2, {{radic|2}}, {{radic|2}}
|coxeter = xoxxfo5ofxxox&#xt
|army = '''Epgybro'''
|reg = '''Epgybro'''
|symmetry = [[Pentagonal antiprismatic symmetry|(I{{sub|2}}(10)×A{{sub|1}})/2]], order 20
|orbits = 16
|volume = <math>\frac{45+17\sqrt5+15\sqrt{5+2\sqrt5}}{6} \approx 21.52973</math>
|dih = 3–4: <math>\arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) \approx 169.18768^\circ</math>
|dih2 = 5–4: <math>\arccos\left(-\frac{2\sqrt5}{5}\right) \approx 153.43495^\circ</math>
|dih3 = 4–4: 144°
|dih4 = 3–5: <math>\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) \approx 142.62263^\circ</math>
|smm = Yes
|pieces = 42
|flags = 320
|loc = 16
|dual = [[Dual of elongated pentagonal gyrobirotunda]]
|conjugate = [[Elongated pentagrammic gyrobirotunda]]
|conv = Yes
|orientable = Yes
|nat = Tame
}}
The '''{{lc:{{PAGENAME}}}}''' ([[OBSA]]: '''{{BSAhelp|e|elongated|p|pentagonal|gy|gyro-|b|bi-|ro|rotunda}}''') is one of the 92 [[Johnson solids]] (J{{sub|43}}). It consists of 10+10 [[triangle]]s,  5+5 [[square]]s, and 2+10 [[pentagon]]s. It can be constructed by inserting a [[decagonal prism]] between the two halves of the [[icosidodecahedron]], seen as a pentagonal gyrobirotunda.

==Vertex coordinates==

An {{lc:{{PAGENAME}}}} of edge length 1 has the following vertices:
* <math>\left(\pm\frac12,\,\pm\frac{\sqrt{5+2\sqrt5}}{2},\,\pm\frac12\right)</math>,
* <math>\left(\pm\frac{3+\sqrt5}{4},\,\pm\sqrt{\frac{5+\sqrt5}{8}},\,\pm\frac12\right)</math>,
* <math>\left(\pm\frac{1+\sqrt5}{2},\,0,\,\pm\frac12\right)</math>,
* <math>\pm\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right)</math>,
* <math>\pm\left(\pm\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right)</math>,
* <math>\pm\left(\pm\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right)</math>,
* <math>\pm\left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right)</math>,
* <math>\pm\left(\pm\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right)</math>,
* <math>\pm\left(\pm\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right)</math>.

==External links==

* {{Incmats|epgybro}}
* {{Eusebeia|J43|The Elongated Pentagonal Gyrobirotunda}}
* {{Wikipedia|Elongated pentagonal gyrobirotunda}}
* {{Dmccooey|ElongatedPentagonalGyrobirotunda|Elongated Pentagonal Gyrobirotunda}}

[[Category:Johnson solids|43]]
[[Category:Epgybro army]]
[[Category:(I2(10)×A1)/2 symmetry]]
[[Category:Polyhedra with 42 faces]]
{{acra|5-3-5-3|5-4-4-3}}