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elongated square gyrobicupola
In geometry, the elongated square gyrobicupola or pseudorhombicuboctahedron is one of the Johnson solids (''J''37). It is sometimes considered to be an Archimedean solid, because its faces consist of regular polygons that meet in the same pattern at each of its vertices. However, unlike the rest of the Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex.
This shape 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 (allegedly by mistake while attempting to construct a model of the rhombicuboctahedron) and again by V. G. Ashkinuse in 1957 .
== Construction and relation to the rhombicuboctahedron ==
As the name suggests, it can be constructed by elongating a square gyrobicupola (''J''29) and inserting an octagonal prism between its two halves.
The solid can also be seen as the result of twisting one of the square cupolae (''J''4) on a rhombicuboctahedron (one of the Archimedean solids; a.k.a. the elongated square orthobicupola) by 45 degrees. Its similarity to the rhombicuboctahedron gives it the alternative name pseudorhombicuboctahedron. It has occasionally been referred to as "the fourteenth Archimedean solid".
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