Elongated pentagonal gyrobirotunda

The elongated pentagonal gyrobirotunda (OBSA: epgybro) is one of the 92 Johnson solids (J₄₃). 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 icosidodecahedron, seen as a pentagonal gyrobirotunda.

Elongated pentagonal gyrobirotunda
(3D model)
Rank	3
Type	CRF
Notation
Bowers style acronym	Epgybro
Coxeter diagram	xoxxfo5ofxxox&#xt
Elements
Faces	10+10 triangles 10 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	${\frac {45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}}{6}}\approx 21.52973$
Dihedral angles	3–4: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	5–4: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	4–4: 144°
	3–5: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
Central density	1
Number of external pieces	42
Level of complexity	16
Related polytopes
Army	Epgybro
Regiment	Epgybro
Dual	Dual of elongated pentagonal gyrobirotunda
Conjugate	Elongated pentagrammic gyrobirotunda
Abstract & topological properties
Flag count	320
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(10)×A1)/2, order 20
Flag orbits	16
Convex	Yes
Nature	Tame

Vertex coordinates

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

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

External links

• Klitzing, Richard. "epgybro".
• Quickfur. "The Elongated Pentagonal Gyrobirotunda".

• Wikipedia contributors. "Elongated pentagonal gyrobirotunda".
• McCooey, David. "Elongated Pentagonal Gyrobirotunda"