The equilateral elongated square dipyramid, illustrated above together with its net, is Johnson solid .

A version of the elongated square dipyramid that is "squashed" along the 4-fold symmetry axis so that the pyramid edge lengths are in ratio of to the central cube edge lengths is a space-filling
polyhedron and stereohedron.

These and the canonical variant of the elongated square dipyramid are illustrated above. They will are implemented in the Wolfram
Language as PolyhedronData["EquilateralElongatedSquareDipyramid"],
PolyhedronData["StereohedralElongatedSquareDipyramid"],
and PolyhedronData["CanonicalElongatedSquareDipyramid"],
respectively.

An elongated square dipyramid (having a central ribbon composed of rectangles instead of squares) appears in the top center as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).