Square gyrobicupola

Square gyrobicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Squigybcu
Coxeter diagram	xxo4oxx&#xt
Elements
Faces	8 triangles, 2+8 squares
Edges	8+8+16
Vertices	8+8
Vertex figure	8 isosceles trapezoids, edge lengths 1, √2, √2, √2; 8 rectangles, edge lengths 1 and √2
Measures (edge length 1)
Volume	$2\frac{3+2\sqrt2}{3} ≈ 3.88562$
Dihedral angles	3–4 cupolaic: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	4–4: 135°
	3–4 join: $\arccos\left(-\sqrt{\frac{3-2\sqrt2}{6}}\right) ≈ 99.73561°$
Central density	1
Related polytopes
Army	Squigybcu
Regiment	Squigybcu
Dual	Joined square antiprism
Conjugate	Retrograde square gyrobicupola
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(8)×A1)/2, order 16
Convex	Yes
Nature	Tame
Discovered by	{{{discoverer}}}

The square gyrobicupola is one of the 92 Johnson solids (J₂₉). It consists of 8 triangles and 2+8 squares. It can be constructed by attaching two square cupolas at their octagonal bases, such that the two square bases are rotated 45° from each other.

It is topologically equivalent to the rectified square antiprism.

If the cupolas are joined such that the bases are in the same orientation, the result is the square orthobicupola.

Vertex coordinates

A square gyroobicupola of edge length 1 has vertices given by the following coordinates:

• $\left(±\frac12,\,±\frac12,\,\frac{\sqrt2}{2}\right),$
• $\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,0\right),$
• $\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,0\right),$
• $\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt2}{2}\right),$
• $\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt2}{2}\right).$

Related polyhedra

An octagonal prism can be inserted between the two halves of the square gyrobicupola to produce the elongated square gyrobicupola.

External links

• Klitzing, Richard. "squigybcu".

• Quickfur. "The Square gyrobicupola".

• Wikipedia Contributors. "Square gyrobicupola".
• McCooey, David. "Square Gyrobicupola"