Gyroelongated square bicupola

Gyroelongated square bicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Gyesquibcu
Coxeter diagram	soxo8oxos&#xt
Elements
Faces	3×8 triangles 2+8 squares
Edges	7×8
Vertices	3×8
Vertex figures	8 isosceles trapezoids, edge lengths 1, √2, √2, √2
	16 irregular pentagons, edge lengths 1, 1, 1, 1, √2
Measures (edge length 1)
Volume	$2{\frac {3+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}+{\sqrt {146+103{\sqrt {2}}}}}}}{3}}\approx 8.15357$
Dihedral angles	3–3 antiprismatic: $\arccos \left({\frac {1-2{\sqrt {2+{\sqrt {2}}}}}{3}}\right)\approx 153.96238^{\circ }$
	3–3 join: $\arccos \left({\frac {\sqrt {3}}{3}}\right)+\arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 151.33013^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	3–4 join: $\arccos \left({\frac {\sqrt {2}}{2}}\right)+\arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 141.59451^{\circ }$
	4–4: 135°
Central density	1
Number of external pieces	34
Level of complexity	28
Related polytopes
Army	Gyesquibcu
Regiment	Gyesquibcu
Dual	Tetradeltoctapentagonal icositetrahedron
Conjugate	Gyroelongated retrograde square bicupola
Abstract & topological properties
Flag count	224
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(B2×A1)+, order 8
Flag orbits	28
Convex	Yes
Nature	Tame

The gyroelongated square bicupola (OBSA: gyesquibcu) is one of the 92 Johnson solids (J₄₅). It consists of 8+8+8 triangles and 2+8 squares. It can be constructed by attaching square cupolas to the bases of the octagonal antiprism.

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

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {{\sqrt {2}}+2H}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,H\right)$,
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,H\right)$,
• $\left(0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,-H\right)$,
• $\left(\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,0,\,-H\right)$,
• $\left(\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,-H\right)$,
• $\left({\sqrt {\frac {2+{\sqrt {2}}}{8}}},\,{\sqrt {\frac {2-{\sqrt {2}}}{8}}},\,-{\frac {{\sqrt {2}}+2H}{2}}\right)$,
• $\left(-{\sqrt {\frac {2+{\sqrt {2}}}{8}}},\,-{\sqrt {\frac {2-{\sqrt {2}}}{8}}},\,-{\frac {{\sqrt {2}}+2H}{2}}\right)$,
• $\left(-{\sqrt {\frac {2-{\sqrt {2}}}{8}}},\,{\sqrt {\frac {2+{\sqrt {2}}}{8}}},\,-{\frac {{\sqrt {2}}+2H}{2}}\right)$,
• $\left({\sqrt {\frac {2-{\sqrt {2}}}{8}}},\,-{\sqrt {\frac {2+{\sqrt {2}}}{8}}},\,-{\frac {{\sqrt {2}}+2H}{2}}\right)$.

where $H={\sqrt {\frac {-2-2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{8}}}$ is the distance between the octagonal antiprism's center and the center of one of its bases.

External links

• Klitzing, Richard. "gyesquibcu".
• Quickfur. "The Gyroelongated Square Bicupola".

• Wikipedia contributors. "Gyroelongated square bicupola".
• McCooey, David. "Gyroelongated Square Bicupola"

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