Elongated Square Gyrobicupola


The elongated square gyrobicupola nonuniform polyhedron obtained by rotating the bottom third of a small rhombicuboctahedron (Ball and Coxeter 1987, p. 137). It is also called Miller's solid, the Miller-aškinuze solid, or the pseudorhombicuboctahedron, and is Johnson solid J_(37).

Although some writers have suggested that the elongated square gyrobicupola should be considered a fourteenth Archimedean solid, its twist allows vertices "near the equator" and those "in the polar regions" to be distinguished. Therefore, it is not a true Archimedean like the small rhombicuboctahedron, whose vertices cannot be distinguished (Cromwell 1997, pp. 91-92).

The elongated square gyrobicupola has volume


and Dehn invariant


where the first expression uses the basis of Conway et al. (1999). It can be dissected into the small rhombicuboctahedron, from which it differs only by relative rotation of the top and bottom cupolas.

See also

Archimedean Solid, Johnson Solid, Small Rhombicuboctahedron

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Aškinuze, V. G. "O čisle polupravil'nyh mnogogrannikov." Math. Prosvešč. 1, 107-118, 1957.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 137-138, 1987.Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Coxeter, H. S. M. "The Polytopes with Regular-Prismatic Vertex Figures." Phil. Trans. Roy. Soc. 229, 330-425, 1930.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 91-92, 1997.Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.Miller, J. C. P. "Polyhedron." Encyclopædia Britannica, 11th ed.

Cite this as:

Weisstein, Eric W. "Elongated Square Gyrobicupola." From MathWorld--A Wolfram Web Resource.

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