Pentagonal orthobicupola

Pentagonal orthobicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Pobcu
Coxeter diagram	xxx5oxo&#xt
Elements
Faces	10 triangles 10 squares 2 pentagons
Edges	5+5+10+20
Vertices	10+10
Vertex figures	10 isosceles trapezoids, edge lengths 1, √2, (1+√5}/2, √2
	10 kites, edge lengths 1 and √2
Measures (edge length 1)
Volume	${\frac {5+4{\sqrt {5}}}{3}}\approx 4.64809$
Dihedral angles	3–4: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	3–3: $\arccos \left({\frac {4{\sqrt {5}}-5}{15}}\right)\approx 74.75474^{\circ }$
	4–4: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$
Height	${\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\approx 1.05146$
Central density	1
Number of external pieces	22
Level of complexity	8
Related polytopes
Army	Pobcu
Regiment	Pobcu
Dual	Deltotrapezohedral icosahedron
Conjugate	Retrograde pentagrammic orthobicupola
Abstract & topological properties
Flag count	160
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Flag orbits	8
Convex	Yes
Nature	Tame

The pentagonal orthobicupola (OBSA: pobcu) 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.

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:

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

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.

External links

• Klitzing, Richard. "pobcu".
• Quickfur. "The Pentagonal Orthobicupola".

• Wikipedia contributors. "Pentagonal orthobicupola".
• McCooey, David. "Pentagonal Orthobicupola"

Bibliography

• Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686-119 36-3.