Elongated pentagonal gyrocupolarotunda

Elongated pentagonal gyrocupolarotunda
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epgycuro
Coxeter diagram	xoxxo5ofxxx&#xt
Elements
Faces	5+5+5 triangles, 5+5+5 squares, 1+1+5 pentagons
Edges	5+5+5+5+5+5+10+10+10+10
Vertices	5+5+5+10+10
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5}/2, √2
	10 rectangles, edge lengths 1 and (1+√5)/2
	10 trapezoids, edge lengths 1, √2, √2, √2
	10 irregular tetragons, edge lengths 1, (1+√5)/2, √2, √2
Measures (edge length 1)
Volume	$5\frac{11+5\sqrt5+6\sqrt{5+2\sqrt5}}{12} ≈ 16.93602$
Dihedral angles	3–4 rotundaic join: $\arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768°$
	3–4 cupolaic: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	4–5 join: $\arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°$
	4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	4–4 prismatic: 144°
	3–5: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	3–4 join: $\arccos\left(-\sqrt{\frac{10-2\sqrt5}{15}}\right) ≈ 127.37737°$
	4–4 join: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°$
Central density	1
Related polytopes
Army	Epgycuro
Regiment	Epgycuro
Dual	Pentadeltodecadeltogyrodecatetragopentarhombirhombic triacontapentahedron
Conjugate	Rlongated retrograde pentagrammic fgyrocupolarotunda
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The elongated pentagonal gyrocupolarotunda is one of the 92 Johnson solids (J₄₁). It consists of 5+5+5 triangles, 5+5+5 squares, and 1+1+5 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal gyrocupolarotunda.

Vertex coordinates

An elongated pentagonal gyrocupolarotunda of edge length 1 has vertices given by the following coordinates:

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

External links

• Klitzing, Richard. "epgycuro".

• Quickfur. "The Elongated Pentagonal Gyrocupolarotunda".

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