Elongated pentagonal orthobirotunda

Elongated pentagonal orthobirotunda
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epobro
Coxeter diagram	xoxxox5ofxxfo&#xt
Stewart notation	R5P10R5 J42
Elements
Faces	10+10 triangles, 5+5 squares, 2+10 pentagons
Edges	10+10+10+10+20+20
Vertices	10+10+20
Vertex figures	20 rectangles, edge lengths 1 and (1+√5)/2
	20 irregular tetragons, edge lengths 1, (1+√5)/2, √2, √2
Measures (edge length 1)
Volume	$\frac{45+17\sqrt5+15\sqrt{5+2\sqrt5}}{6} ≈ 21.52973$
Dihedral angles	3–4: $\arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768^\circ$
	5–4: $\arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495^\circ$
	4–4: 144°
	3–5: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263^\circ$
Central density	1
Number of external pieces	42
Level of complexity	16
Related polytopes
Army	Epobro
Regiment	Epobro
Dual	Pentarhombideltorthodecatetragonal tetracontahedron
Conjugate	Elongated pentagrammic orthobirotunda
Abstract & topological properties
Flag count	320
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Convex	Yes
Nature	Tame

The elongated pentagonal orthobirotunda is one of the 92 Johnson solids (J₄₂). It consists of 10+10 triangles, 5+5 squares, and 2+10 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal orthobirotunda.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "epobro".

• Quickfur. "The Elongated Pentagonal Orthobirotunda".

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