Elongated square gyrobicupola

Elongated square gyrobicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Esquigybcu
Coxeter diagram	oxxx4xxxo&#xt
Elements
Faces	8 triangles, 2+8+8 squares
Edges	8+8+8+8+16
Vertices	8+16
Vertex figure	8+16 isosceles trapezoids, edge lengths 1, √2, √2, √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{5+2\sqrt2}}{2} ≈ 1.39897$
Volume	$2\frac{6+5\sqrt2}{3} ≈ 8.71404$
Dihedral angles	3–4: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	4–4: 135°
Central density	1
Related polytopes
Army	Esquigybcu
Regiment	Esquigybcu
Dual	Gyrodeltoidal icositetrahedron
Conjugate	Elongated retrograde square gyrobicupola
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(8)×A1)/2, order 16
Convex	Yes
Nature	Tame

The elongated square gyrobicupola, also called the pseudo-rhombicuboctahedron, is one of the 92 Johnson solids (J₃₇). It consists of 8 triangles and 2+8+8 squares. It can be constructed by inserting an octagonal prism between the two halves of the square gyrobicupola. It can also be constructed from the small rhombicuboctahedron by rotating one of its square cupola segments 45°, and could be called the gyrate rhombicuboctahedron.

It is notable because it has the same set of faces around each vertex (one triangle and three squares), but is not a uniform polyhedron, as it is not actually vertex-transitive. It shares this property with its conjugate, which can be constructed as a gyration of the quasirhombicuboctahedron. The elongated square gyrobicupola is the only convex polyhedron with this property, but it is not known if there are any other nonconvex polyhedra with the property.

Vertex coordinates

An elongated square gyrobicupola of edge length 1 has the following vertices:

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

External links

• Klitzing, Richard. "esquigybcu".

• Quickfur. "The Elongated Square Gyrobicupola".

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