 Rank3
TypePseudo-uniform
SpaceSpherical
Notation
Bowers style acronymGyquerco
Elements
Faces8 triangles, 2+8+8 squares
Edges8+8+8+8+16
Vertices8+16
Measures (edge length 1)
Circumradius$\frac{\sqrt{5-2\sqrt2}}{2} \approx 0.73681$ Volume$2\frac{5\sqrt2-6}{3} \approx 0.71404$ Dihedral angles4–4: 45°
4–3: $\arccos\left(\frac{\sqrt6}{3}\right) \approx 35.26439^\circ$ Related polytopes
ConjugateElongated square gyrobicupola
Abstract & topological properties
Flag count192
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(8)×A1)/2, order 16
ConvexNo
NatureTame

The elongated retrograde square gyrobicupola or pseudo great rhombicuboctahedron, gyrated quasirhombicuboctahedron, or gyquerco, is one of the two known pseudo-uniform polyhedra, the other being the elongated square gyrobicupola or pseudorhombicuboctahedron. All its faces are regular and the same configuration of faces meet at each vertex, but it is not isogonal. This is because the vertices divide into two sets, one of size 16 and the other of size 8, that are distinct according to the symmetries of the polyhedron.

The pseudo great rhombicuboctahedron is the result of starting with a quasirhombicuboctahedron and gyrating a retrograde square cupola by 45 degrees. This breaks up four of the quasirhombicuboctahedron's six octagrammic pseudo-faces. Just as the elongated square gyrobicupola is constructed as an octagonal prism sandwiched (via blending) between two square cupolae at 45-degree angles, the elongated retrograde square gyrobicupola is may be constructed as a blend of two retrograde square cupola at 45-degree angles and an octagrammic prism.

## Vertex coordinates

A elongated retrograde square gyrobicupola of edge length 1 has vertex coordinates given by all permutations of the first two coordinates of:

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