# Square gyrobicupola

Square gyrobicupola
Rank3
TypeCRF
Notation
Bowers style acronymSquigybcu
Coxeter diagramxxo4oxx&#xt
Elements
Faces8 triangles, 2+8 squares
Edges8+8+16
Vertices8+8
Vertex figure8 isosceles trapezoids, edge lengths 1, 2, 2, 2; 8 rectangles, edge lengths 1 and 2
Measures (edge length 1)
Volume${\displaystyle {\frac {6+4{\sqrt {2}}}{3}}\approx 3.88562}$
Dihedral angles3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 135°
3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {3-2{\sqrt {2}}}{6}}}\right)\approx 99.73561^{\circ }}$
Central density1
Number of external pieces18
Level of complexity8
Related polytopes
ArmySquigybcu
RegimentSquigybcu
DualJoined square antiprism
Abstract & topological properties
Flag count128
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(8)×A1)/2, order 16
ConvexYes
NatureTame

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:

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

## Related polyhedra

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