elongated square gyrobicupola

It is the dual of the pseudorhombicuboctahedron (also known as the elongated square gyrobicupola).

Related to the square gyrobicupola is the elongated square gyrobicupola.

It is also a Johnson solid (J) and can also be called the elongated square gyrobicupola.

Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".

The elongated square gyrobicupola possesses D symmetry.

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids.

Elongated square gyrobicupola forms space-filling honeycombs with Tetrahedron, cubes and Cuboctahedron.

These pieces can be reassembled to give a new solid called the elongated square gyrobicupola or pseudorhombicuboctahedron, with the symmetry of a square antiprism.

There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images of two enantiomorphs, see below, are counted separately).

All but the elongated square gyrobicupola can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

Elongated square gyrobicupola forms space-filling honeycombs with Tetrahedron, Square pyramid and either or combination of (cube, Elongated square pyramid, Elongated square bipyramid).

Of the Johnson solids, the elongated square gyrobicupola ('J'37) is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 triangle.

It is argued whether or not the elongated square gyrobicupola is an Archimedean solid because, although it meets every other standard necessary to be an Archimedean solid, it is not highly symmetric.

Based on its existence, has suggested a terminological distinction in which an Archimedean solid is defined as having the same vertex figure at each vertex (including the elongated square gyrobicupola) while a uniform polyhedron is defined as having each vertex symmetric to each other vertex (excluding the gyrobicupola).

This difference in definitions controls whether the elongated square gyrobicupola (pseudo-rhombicuboctahedron) is considered an Archimedean solid or a Johnson solid: it is the unique convex polyhedron that has regular polygons meeting in the same way at each vertex, but that does not have a global symmetry taking every vertex to every other vertex.