The ancient Prussian city of Königsberg contained seven bridges. According to a tradition among the residents of Königsberg, it was impossible to walk over all seven bridges without crossing one of them at least twice. That problem came to the attention of the Swiss mathematician Leonhard Euler in the mid-eighteenth century, and he decided to develop a theory that would explain the puzzle. The solution Euler devised was eventually to form the basis of an entirely new branch of mathematics known as topology.

Topology is a field of geometry that examines changes that occur in an object when it is stretched, bent, twisted, or otherwise deformed in some way. It is different from other forms of mathematics in that it does not deal with metrics (distances and angles, for example), but with shapes. It is thus primarily qualitative rather than quantitative in nature. Two...