Elongated retrograde square gyrobicupola

Elongated retrograde square gyrobicupola
Rank	3
Type	Pseudo-uniform
Space	Spherical
Notation
Bowers style acronym	Gyquerco
Elements
Faces	8 triangles, 2+8+8 squares
Edges	8+8+8+8+16
Vertices	8+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 angles	4–4: 45°
	4–3: $\arccos\left(\frac{\sqrt6}{3}\right) \approx 35.26439^\circ$
Related polytopes
Conjugate	Elongated square gyrobicupola
Abstract & topological properties
Flag count	192
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(8)×A1)/2, order 16
Convex	No
Nature	Tame

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:

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

Gallery

External links

• Klitzing, Richard. "gyquerco".