Elongated square gyrobicupola

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.

Elongated square gyrobicupola
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymEsquigybcu
Coxeter diagramoxxx4xxxo&#xt
Elements
Faces8 triangles, 2+8+8 squares
Edges8+8+8+8+16
Vertices8+16
Vertex figure8+16 isosceles trapezoids, edge lengths 1, 2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{5+2\sqrt2}}{2} ≈ 1.39897}$
Volume${\displaystyle 2\frac{6+5\sqrt2}{3} ≈ 8.71404}$
Dihedral angles3–4: ${\displaystyle \arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°}$
4–4: 135°
Central density1
Related polytopes
ArmyEsquigybcu
RegimentEsquigybcu
DualGyrodeltoidal icositetrahedron
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(8)×A1)/2, order 16
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

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:

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