Elongated pentagonal orthobicupola

Elongated pentagonal orthobicupola
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epobcu
Coxeter diagram	oxxo5xxxx&#xt
Elements
Faces	10 triangles, 5+5+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	Epobcu
Regiment	Epobcu
Dual	Orthodeltodeltoidal triacontahedron
Conjugate	Elongated retrograde pentagrammic orthobicupola
Abstract & topological properties
Flag count	240
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Convex	Yes
Nature	Tame

The elongated pentagonal orthobicupola is one of the 92 Johnson solids (J₃₈). It consists of 10 triangles, 5+5+10 squares, and 2 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal orthobicupola.

Vertex coordinates[edit | edit source]

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

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