Mathematics

Mathematical crystallography involves many branches of mathematics.  The relationship may be ancient: William Waterhouse proposed that the notion of a polyhedron may have been inspired by encounters with crystals, which would mean that humans have been puzzling over crystals for a very long time.

Historically, there were two mathematical approaches to crystals.  One is Waterhouse’s ancient one: what do crystals look like? As our pre-Renaissance literature consists of those hand-written books that survived, the lack of literary references to crystals means little.  Whatever the case, the literary references start soon after the Renaissance, when books started pouring off the brand new presses, on every subject anyone could think of, and one of those subjects was the taxonomy of crystals by their appearance, a subject that could only go so far on its own: Carl Linnaeus’s De Crystallorum Generatione sits in an intellectual cul-de-sac.

Crystallography took off once scientists dusted off the politically toxic Atomic Theory, and speculated about the nanoscopic structure of crystals.

• Johannes Kepler, best known for proposing that the planets move in elliptic orbits around the Sun, was very interested in polyhedra, and also proposed that ice crystals might have a periodic hexagonal structure.  People in this thread, like Nicolas Steno and Rene Hauy, were various kinds of geologists and chemists.
• But the atomic theory provided a model for mathematicians – both professional like Augustin Cauchy and amateur like Auguste Bravais – during the Nineteenth century.  Crystallography became a geometric problem, and during the Nineteenth century it became (through the motivation and development of the theory of symmetries, which became the basis for much of Felix Klein‘s Erlangen program) one of the motivators and sources for the development of Group Theory.

By the beginning of the Twentieth century, mathematical crystallography involved polyhedra, algebra (especially group theory and vector algebra), and, through the Fyodorov–Schoenflies-Bieberbach Theorem (see also this entry), Riemann manifolds. But that was also the era of X-rays, and the development of x-ray diffraction photography entailed the application of areas of functional analysis, especially Fourier analysis.

Since World War II, a growing interest in discrete structures (enabled by modern computers) encouraged modeling crystals by crystal nets. This involves discrete geometry, graph theory, and other areas of combinatorics. Many other areas are involved as well, even (via Wang tilings) my own field of mathematical logic.