Erlanger program-an article written by the great mathematician Felix Klein in 1872, in which he outlined his vision of the direction in which the study of geometry should proceed in the years that followed. Klein's key idea was that geometric structures on spaces should be understood in terms of the group of transformations that preserve the structures. Klein's insight, which seemed very abstract to his contemporaries, was a source of inspiration to the mathematicians that followed him, notably Henri Poincaré, and has become fundamental to our current understanding of geometry.

In the years preceding Klein's Erlanger program, the world of geometry had been changed by the discovery of non-Euclidean geometries, in which Klein was instrumental. Mathematicians realized that the same space could admit more than one geometric structure, and it became important to make a precise definition of a geometric structure on a space. Two complementary...