The Euler characteristic is a number that can be associated to a polyhedron that gives an indication of its topological qualities (qualities of an object that don't change when the object is stretched or distorted, without tearing). The characteristic number was first studied by the great mathematician Leonhard Euler, who wrote his famous formula for a polyhedron in a letter in 1750 to Christian Goldbach: v-e+f=2, where v is the number of vertices (corners) of the polyhedron, e is the number of edges, and f is the number of faces (flat sides). Thus, for example, a cube has 8 vertices, 12 edges, and 6 faces, and 8-12+6=2. Although polyhedra had been studied quite intensely before the time of Euler by such great mathematicians as Archimedes and Descartes, historians believe that none of them noticed this simple relationship. Euler's work was something of a departure from the usual study of...