Pentagonal orthobirotunda

Pentagonal orthobirotunda
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Pobro
Coxeter diagram	xoxox5ofxfo&#xt
Elements
Faces	10+10 triangles, 2+10 pentagons
Edges	5+5+10+20+20
Vertices	10+10+10
Vertex figures	10+10 rectangles, edge lengths 1 and (1+√5)/2
	10 kites, edge lengths 1 and (1+√5)/2
Measures (edge length 1)
Circumradius	$\frac{1+\sqrt5}{2} ≈ 1.61803$
Volume	$\frac{45+17\sqrt5}{6} ≈ 13.83552$
Dihedral angles	3–3: $\arccos\left(-\frac{5+4\sqrt5}{15}\right) ≈ 158.37537°$
	3–5: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	5–5: $\arccos\left(-\frac35\right) ≈ 126.86990°$
Central density	1
Related polytopes
Army	Pobro
Regiment	Pobro
Dual	Rhombitrapezohedral triacontahedron
Conjugate	Pentagrammic orthobirotunda
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Convex	Yes
Nature	Tame

The pentagonal orthobirotunda is one of the 92 Johnson solids (J₃₄). It consists of 10+10 triangles and 2+10 pentagons. It can be constructed by attaching two pentagonal rotundas at their decagonal bases, such that the two pentagonal bases are in the same orientation.

If the rotundas are joined such that the bases are rotated 36°, the result is the pentagonal gyrobirotunda, better known as the uniform icosidodecahedron. Conversely, the pentagonal orthobirotunda can be seen as a gyrate icosidodecahedron, since it is an icosidodecahedron with one half rotated.

Vertex coordinates

A pentagonal orthobirotunda of edge length 1 has vertices given by the following coordinates:

• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),$
• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),$
• $\left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right),$
• $\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0\right),$
• $\left(±\frac{1+\sqrt5}{2},\,0,\,0\right).$

Related polyhedra

A decagonal prism can be inserted between the two halves of the pentagonal orthobirotunda to produce the elongated pentagonal orthobirotunda.

External links

• Klitzing, Richard. "pobro".

• Quickfur. "The Pentagonal Orthobirotunda".

• Wikipedia Contributors. "Pentagonal orthobirotunda".
• McCooey, David. "Pentagonal Orthobirotunda"