Созылған шаршылы гиробикупола

37th Johnson solid

In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It is also known as pseudo rhombicuboctahedron because many mathematicians mistakenly constructed a rhombicuboctahedron. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron.

The elongated square gyrobicupola can be constructed similarly to the rhombicuboctahedron, by attaching two regular square cupolas onto the bases of octagonal prism, a process known as elongation. The difference between these two polyhedrons is that one of two square cupolas of the elongated square gyrobicupola is twisted by 45 degrees, a process known as gyration, making the triangular faces staggered vertically. The resulting polyhedron has 8 equilateral triangles and 18 squares. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid, and the elongated square gyrobicupola is among them, enumerated as the 37th Johnson solid
The elongated square gyrobicupola may have been discovered by Johannes Kepler in his enumeration of the Archimedean solids, but its first clear appearance in print appears to be the work of Duncan Sommerville in 1905. It was independently rediscovered by J. C. P. Miller in 1930 by mistake while attempting to construct a model of the rhombicuboctahedron, which is known as pseudo rhombicuboctahedron or sometimes Miller's solid. This solid was discovered again by V. G. Ashkinuse in 1957.

The elongated square gyrobicupola can form a space filling honeycomb with the regular tetrahedron, cube, and cuboctahedron. It can also form another honeycomb with the tetrahedron, square pyramid and various combinations of cubes, elongated square pyramids, and elongated square bipyramids. The pseudo great rhombicuboctahedron is a nonconvex analog of the pseudo rhombicuboctahedron, constructed in a similar way from the nonconvex great rhombicuboctahedron.

The polyvanadate ion [V18O42]12− has a pseudo rhombicuboctahedral structure, where each square face acts as the base of a VO5 pyramid.