Gyroelongated square cupola

Gyroelongated square cupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Gyescu
Coxeter diagram	oxo8sox&#xt
Elements
Faces	4+4+4+8 triangles, 1+4 squares, 1 octagon
Edges	4+4+4+8+8+8+8
Vertices	4+4+4+8
Vertex figures	4 isosceles trapezoids, edge lengths 1, √2, √2, √2
	8 irregular pentagons, edge lengths 1, 1, 1, 1, √2
	8 isosceles trapezoids, edge lengths 1, 1, 1, √2+√2
Measures (edge length 1)
Volume	${\frac {3+2{\sqrt {2}}+2{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}}{3}}\approx 6.21077$
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°
	3–8: $\arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 96.59451^{\circ }$
Central density	1
Number of external pieces	26
Level of complexity	22
Related polytopes
Army	Gyescu
Regiment	Gyescu
Dual	Tetradeltoctapentagonal hemitrapezohedron
Conjugate	Gyroelongated retrograde square cupola
Abstract & topological properties
Flag count	176
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Convex	Yes
Nature	Tame

The gyroelongated square cupola is one of the 92 Johnson solids (J₂₃). It consists of 4+4+4+8 triangles, 1+4 squares, and 1 octagon. It can be constructed by attaching an octagonal antiprism to the octagonal base of the square cupola.

If a second cupola is attached to the other octagonal base of the antiprism, the result is the gyroelongated square bicupola.

Vertex coordinates

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

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

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. "gyescu".
• Quickfur. "The Gyroelongated Square Cupola".

• Wikipedia contributors. "Gyroelongated square cupola".
• McCooey, David. "Gyroelongated Square Cupola"