Question: In what year was the mathematician in whose enumeration of the Archimedean solids the elongated square gyrobiculopa may have been discovered born?

Context paragraphs:
Elongated pentagonal gyrobirotunda: In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids ("J").  As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron (one of the Archimedean solids), by inserting a decagonal prism between its congruent halves.  Rotating one of the pentagonal rotundae ("J") through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda ("J").

Elongated square pyramid: In geometry, the elongated square pyramid is one of the Johnson solids ("J").  As the name suggests, it can be constructed by elongating a square pyramid ("J") by attaching a cube to its square base.  Like any elongated pyramid, it is topologically (but not geometrically) self-dual.

Elongated triangular orthobicupola: In geometry, the elongated triangular orthobicupola is one of the Johnson solids ("J").  As the name suggests, it can be constructed by elongating a triangular orthobicupola ("J") by inserting a hexagonal prism between its two halves.  The resulting solid is superficially similar to the rhombicuboctahedron (one of the Archimedean solids), with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.

Archimedean graph: In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids.  There are 13 Archimedean graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vertex-connected planar graphs), and also Hamiltonian graphs.

List of small polyhedra by vertex count: In geometry, a polyhedron is a solid in three dimensions with flat faces and straight edges.  Every edge has exactly two faces, and every vertex is surrounded by alternating faces and edges.  The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices.  Named polyhedra primarily come from the families of platonic solids, Archimedean solids, Catalan solids, and Johnson solids, as well as dihedral symmetry families including the pyramids, bipyramids, prisms, antiprisms, and trapezohedrons.

Elongated square gyrobicupola: 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 by 1930 (by mistake while attempting to construct a model of the rhombicuboctahedron ) and again by V. G. Ashkinuse in 1957.

Elongated square bipyramid: In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids ("J").  As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves.

Elongated square cupola: In geometry, the elongated square cupola is one of the Johnson solids ("J").  As the name suggests, it can be constructed by elongating a square cupola ("J") by attaching an octagonal prism to its base.  The solid can be seen as a rhombicuboctahedron with its "lid" (another square cupola) removed.

Johannes Kepler: Johannes Kepler ( ; ] ; December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

Jovo: JOVO Click 'N Construct is a brand name of a construction toy system invented by Josef Volgger and produced by JOVO International of Helsinge, Denmark.  The toy consists of equilateral triangle, square and regular pentagon plastic pieces that click together on their edges, and the edges then act as hinges to allow them to be folded up into three dimensional shapes.  The pieces were designed to model regular-faced polyhedra, such as the Platonic solids, Archimedean solids, and Johnson solids.

Supporting paragraphs: Elongated square gyrobicupola Johannes Kepler
Answer: 1571