Pentagonal gyrocupolarotunda

Pentagonal gyrocupolarotunda
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Pegycuro
Coxeter diagram	xoxo5ofxx&#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}0/2, √2
	10 rectangles, edge lengths 1 and (1+√5)/2
	10 irregular tetragons, edge lengths 1, 1, √2, (1+√5)/2
Measures (edge length 1)
Volume	$5\frac{11+5\sqrt5}{12} ≈ 9.24181$
Dihedral angles	3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	4–5 cupolaic: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	3–5: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	3–3: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	4–5 join: $\arccos\left(-\sqrt{\frac{25-11\sqrt5}{50}}\right) ≈ 95.15242°$
Central density	1
Related polytopes
Army	Pegycuro
Regiment	Pegycuro
Dual	Pentadeltodecatrapezopentadelto-pentarhombic icosipentahedron
Conjugate	Retrograde pentagrammic gyrocupolarotunda
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The pentagonal gyrocupolarotunda is one of the 92 Johnson solids (J₃₃). 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 rotated 36° with respect to each other.

If the cupola and rotunda are joined such that the bases are in the same orientation, the result is the pentagonal orthocupolarotunda.

Vertex coordinates

A pentagonal gyrocupolarotunda 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 gyrocupolarotunda to produce the elongated pentagonal gyrocupolarotunda.

External links

• Klitzing, Richard. "pegycuro".

• Quickfur. "The Pentagonal Gyrocupolarotunda".

• Wikipedia Contributors. "Pentagonal gyrocupolarotunda".
• McCooey, David. "Pentagonal Gyrocupolarotunda"