Leonhard Euler"s polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller"s buildings and giant all-carbon molecules. Yet Euler"s formula is so simple it can be explained to a child. Euler"s Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today"s cutting-edge research, Euler"s Gem celebrates the discovery of Euler"s beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula"s scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians...