Pentagonal gyrobicupola

Pentagonal gyrobicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Pegybcu
Coxeter diagram	xxo5oxx&#xt
Elements
Faces	10 triangles, 10 squares, 2 pentagons
Edges	10+10+20
Vertices	10+10
Vertex figures	10 isosceles trapezoids, edge lengths 1, √2, (1+√5}0/2, √2
	10 rectangles, edge lengths 1 and √2
Measures (edge length 1)
Volume	${\frac {5+4{\sqrt {5}}}{3}}\approx 4.64809$
Dihedral angles	3–4 cupolaic: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	3–4 join: $\arccos \left({\frac {{\sqrt {15}}-{\sqrt {3}}}{6}}\right)\approx 69.09484^{\circ }$
Height	${\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\approx 1.05146$
Central density	1
Number of external pieces	22
Level of complexity	8
Related polytopes
Army	Pegybcu
Regiment	Pegybcu
Dual	Joined pentagonal antiprism
Conjugate	Retrograde pentagrammic gyrobicupola
Abstract & topological properties
Flag count	160
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(10)×A1)/2, order 20
Convex	Yes
Nature	Tame

The pentagonal gyrobicupola is one of the 92 Johnson solids (J₃₁). It consists of 10 triangles, 10 squares, and 2 pentagons. It can be constructed by attaching two pentagonal cupolas at their decagonal bases, such that the two pentagonal bases are rotated 36° with respect to each other.

It is topologically equivalent to the rectified pentagonal antiprism.

If the cupolas are joined in the same orientation, the result is the pentagonal orthobicupola.

Vertex coordinates

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

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

Related polyhedra

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

External links

• Klitzing, Richard. "pegybcu".
• Quickfur. "The Pentagonal Gyrobicupola".

• Wikipedia contributors. "Pentagonal gyrobicupola".
• McCooey, David. "Pentagonal Gyrobicupola"