Gyroelongated pentagonal bicupola

Gyroelongated pentagonal bicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Gyepibcu
Coxeter diagram	soxo10oxos&#xt
Elements
Faces	10+10+10 triangles, 10 squares, 2 pentagons
Edges	10+10+10+10+10+10+10
Vertices	10+10+10
Vertex figures	10 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2
	20 irregular pentagons, edge lengths 1, 1, 1, 1, √2
Measures (edge length 1)
Volume	${\frac {10+8{\sqrt {5}}+5{\sqrt {2-2{\sqrt {5}}+2{\sqrt {650+290{\sqrt {5}}}}}}}{6}}\approx 11.39738$
Dihedral angles	3–3 antiprismatic: $\arccos \left({\frac {1-{\sqrt {10+2{\sqrt {5}}}}}{3}}\right)\approx 159.18651^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	3–3 join: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)+\arccos \left(-{\sqrt {\frac {11+4{\sqrt {5}}-2{\sqrt {50+22{\sqrt {5}}}}}{3}}}\right)\approx 132.62401^{\circ }$
	3–4 join: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)+\arccos \left(-{\sqrt {\frac {11+4{\sqrt {5}}-2{\sqrt {50+22{\sqrt {5}}}}}{3}}}\right)\approx 126.96412^{\circ }$
Central density	1
Number of external pieces	42
Level of complexity	28
Related polytopes
Army	Gyepibcu
Regiment	Gyepibcu
Dual	Pentadeltodecapentagonal triacontahedron
Conjugate	Gyroelongated retrograde pentagrammic bicupola
Abstract & topological properties
Flag count	280
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(H2×A1)+, order 10
Convex	Yes
Nature	Tame

The gyroelongated pentagonal bicupola is one of the 92 Johnson solids (J₄₆). It consists of 10+10+10 triangles, 10 squares, and 2 pentagons. It can be constructed by attaching pentagonal cupolas to the bases of the decagonal antiprism.

It is one of the five Johnson solids to be chiral.

Vertex coordinates

A gyroelongated pentagonal bicupola of edge length 1 has the following vertices:

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

where H = ${\sqrt {\frac {-4-2{\sqrt {5}}+{\sqrt {50+22{\sqrt {5}}}}}{8}}}$is the distance between the decagonal antiprism's center and the center of one of its bases.

External links

• Klitzing, Richard. "gyepibcu".
• Quickfur. "The Gyroelongated Pentagonal Bicupola".

• Wikipedia contributors. "Gyroelongated pentagonal bicupola".
• McCooey, David. "Gyroelongated Pentagonal Bicupola"