Elongated square gyrobicupola
|Elongated square gyrobicupola|
J36 - J37 - J38
|Dual polyhedron||Pseudo-deltoidal icositetrahedron|
|Properties||convex, singular vertex figure|
In geometry, the elongated square gyrobicupola or pseudorhombicuboctahedron is one of the Johnson solids (J37). 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 (Grünbaum 2009).
A Johnson solid is one of 92 strictly convex regular-faced polyhedra, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. They are named by Norman Johnson who first enumerated the set in 1966.
Construction and relation to the rhombicuboctahedron
Exploded sections of
The solid can also be seen as the result of twisting one of the square cupolae (J4) 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".
Symmetry and classification
The elongated square gyrobicupola possesses D4d symmetry. It is locally vertex-regular — the arrangement of the four faces incident on any vertex is the same for all vertices; this is unique among the Johnson solids. However, it is not vertex-transitive, and consequently not usually considered to be one of the Archimedean solids, as there are pairs of vertices such that there is no isometry of the solid which maps one into the other. Essentially, the two types of vertices can be distinguished by their "neighbors of neighbors." Another way to see that the polyhedron is not vertex-transitive is to note that there is exactly one belt of eight squares around its equator, which distinguishes vertices on the belt from vertices on either side. With faces colored by its D4d symmetry, it can look like this:
There are 8 (green) squares around its equator, 4 (red) triangles and 4 (yellow) squares above and below, and one (blue) square on each pole.
- Grünbaum, Branko (2009). "An enduring error". Elemente der Mathematik 64 (3): 89–101. doi:10.4171/EM/120. MR 2520469. Reprinted in Pitici, Mircea, ed. (2011). The Best Writing on Mathematics 2010. Princeton University Press. pp. 18–31..
- Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra, prisma and antiprisms, p. 25 Pseudo-rhombicuboctahedron
- Sommerville, D. M. Y. (1905). "Semi-regular networks of the plane in absolute geometry". Transactions of the Royal Society of Edinburgh 41: 725–747. doi:10.1017/s0080456800035560.. As cited by Grünbaum (2009).