Pentagonal orthocupolarotunda

Pentagonal orthocupolarotunda
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Pocuro
Coxeter diagram	xoxx5ofxo&#xt
Elements
Faces	5+5+5 triangles, 5 squares, 1+1+5 pentagons
Edges	5+5+5+5+10+10+10
Vertices	5+5+5+10
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5}/2, √2
	10 rectangles, edge lengths 1 and (1+√5)/2
	10 irregular tetragons, edge lengths 1, √2, 1, (1+√5)/2
Measures (edge length 1)
Volume	$5\frac{11+5\sqrt5}{12} ≈ 9.24181$
Dihedral angles	3–4 cupolaic: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	3–5 rotundaic: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	3–4 join: $\arccos\left(-\frac{\sqrt{15}-\sqrt3}{6}\right) ≈ 110.95106°$
	3–5 join: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81237°$
Central density	1
Related polytopes
Army	Pocuro
Regiment	Pocuro
Dual	Pentadeltodecadeltopentadelto-pentarhombic icosipentahedron
Conjugate	Retrograde pentagrammic orthocupolarotunda
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The pentagonal orthocupolarotunda is one of the 92 Johnson solids (J₃₂). It consists of 5+5+5 triangles, 5 squares, and 1+1+5 pentagons. It can be constructed by attaching a pentagonal cupola and a pentagonal rotunda at their decagonal bases, such that the two pentagonal bases are in the same orientation.

If the cupola and rotunda are joined such that the bases are rotated 36º, the result is the pentagonal gyrocupolarotunda.

Vertex coordinates

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

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

Related polyhedra

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

The pentagonal orthocupolarotunda also has a connection with the regular icosahedron, being a partial Stott expansion of a pentagonal-symmetric faceting of the icosahedron.

External links

• Klitzing, Richard. "pocuro".

• Quickfur. "The Pentagonal Orthocupolarotunda".

• Wikipedia Contributors. "Pentagonal orthocupolarotunda".
• Hi.gher.Space Wiki Contributors. "Pentagonal orthocupolarotunda".

• McCooey, David. "Pentagonal Orthocupolarotunda"