Poincaré and his polarization sphere

In a text (1892) on light, Jules Henri Poincaré introduced a geometrical device for tracking the polarization of state of light interacting with matter. Poincaré first mapped all polarization ellipses onto the surface of sphere and changes of state were represented as rotations from one point to another about prescribed axes depending on linear optical properties of the medium. Here, we consider how Poincaré, the mathematician, invented his sphere. Poincaré's professional activities in the service of geodesy appear at first glance to provide a borrowed geometry for his one-to-one mapping of polarization ellipses to global lines of latitude and longitude. However, this association falls apart in the face of a close reading of Poincaré's biography and his influences, especially the research interests of his teachers from the École Polytechnique and the Écoles de Mines. The work of Tissot and Mallard on distortion ellipses in cartography and the etiology of optical activity in crystals, respectively, together with Poincaré's own study of the qualitative theory of differential equations, provide the iconography of, and motivation for, the sphere. Whether Poincaré's mentors were unwitting partners in the invention of the sphere-it is impossible to be sure-Poincaré's otherworldly geometric sensibilities carried him through isometries (rotations) on hyperbolic planes and beyond. The apparent ingenuity behind Poincaré's sphere is diminished in comparison to his fulsome achievements. Moreover, Poincaré's considerations of the psychology of invention further emphasize that sometimes great ideas arrive to those fortunate to receive them by mechanisms that resist interpretation.