Elongated pentagonal gyrobicupola

Elongated pentagonal gyrobicupola
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epigybcu
Coxeter diagram	oxxx5xxxo&#xt
Elements
Faces	10 triangles, 10+10 squares, 2 pentagons
Edges	10+10+10+10+20
Vertices	10+20
Vertex figures	10 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2
	20 trapezoids, edge lengths 1, √2, √2, √2
Measures (edge length 1)
Volume
Dihedral angles	3–4 cupolaic:
	4–5:
	4–4 prismatic: 144°
	3–4 join:
	4–4 join:
Central density	1
Number of external pieces	32
Level of complexity	12
Related polytopes
Army	Epigybcu
Regiment	Epigybcu
Dual	Gyrodeltodeltoidal triacontahedron
Conjugate	Elongated retrograde pentagrammic gyrobicupola
Abstract & topological properties
Flag count	240
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(10)×A1)/2, order 20
Convex	Yes
Nature	Tame

The elongated pentagonal gyrobicupola is one of the 92 Johnson solids (J₃₉). It consists of 10 triangles, 10+10 squares, and 2 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal gyrobicupola.

Vertex coordinates[edit | edit source]

An elongated pentagonal gyrobicupola of edge length 1 has the following vertices:

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