Elongated square gyrobicupola

In geometry, the elongated square gyrobicupola or pseudorhombicuboctahedron is one of the Johnson solids (J₃₇). The 92 Johnson solids were named and described by Norman Johnson in 1966.

Relation to Rhombicuboctahedron

As the name suggests, it can be constructed by elongating a square gyrobicupola (J₂₉) and inserting an octagonal prism between its two halves. The resulting solid 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 truly vertex-transitive, and consequently not 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, two types of vertices can be distinguished by their "neighbors of neighbors." Another way to see that the polyhedron is not vertex-regular 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.

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. 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

With faces colored by its D_4d symmetry, it can look like this:

There are 8 (green) squares around its equator, 4 and 4 (yellow) squares above and below, and one (blue) square on each pole.

External links

• Eric W. Weisstein, Elongated square gyrobicupola (Johnson solid) at MathWorld.
• George Hart: pseudo-rhombicuboctahedra