The current issue of Nature has a great feature about how mathematical inventions and discoveries often find unexpected applications, sometimes decades after their first appearance.

Mathematicians usually pursue their theories out of curiosity, aesthetical interest, ambition, or that intangible quality of being mathematically deep (a notion that's worth a whole blog post of its own). They do so with a loftiness that frustrates just about everybody else, but especially scientists—and funding agencies.

But being able to solve abstract, general questions often means that when a scientist or an engineer shows up at their door with a specific problem, mathematicians have an answer ready for her.

It is a pattern that repeats time and again, and that is worth analyzing, although to me what’s even more interesting is that math and science have more of a complicated two-way relationship: mathematicians create whole new theories inspired by real-life problems. This interplay has been especially spectacular in recent decades, with some of the most sophisticated mathematical advances being inspired by physics, and by string theory in particular. But I digress.

For the feature, Peter Rowlett, a math blogger and podcaster who teaches at the University of Birmingham and belongs to the British Society for the History of Mathematics, invited seven of his fellow society members (including himself) to write brief articles describing their favorite examples of such historical twists.

The list includes some predictable examples--though told with freshness—but also some that I had not heard of and that I found very intriguing. I’ll just mention a couple.

In the predictable category, computer scientists Mark McCartney and Tony Mann talk about Irish Mathematician William Rowan Hamilton’s discovery of the algebra of quaternions, which, incidentally, was discussed at length in a recent Scientific American article by John Baez and John Huerta. Quaternions are a four-dimensional generalization of ordinary numbers and they show up in innumerable places in both math and physics. (The Baez-Huerta article was mostly about an even weirder number system called the octonions, as my colleague Michael Moyer recounts.)

McCartney and Mann point out how quaternions have turned out to be convenient tools in robotics, computer graphics, and the gaming industry in particular, because they simplify calculations of rotations in three-dimensional space.

Graham Hoare, an editor at Mathematics Today, discusses what is perhaps the most celebrated case of mathematicians’ prescience: Georg Bernhard Riemann, who invented modern geometry and who is one of my personal heroes. His theory of “Riemannian manifolds,” Hoare points out, was the basis for Albert Einstein’s general theory of relativity, and in particular for the notion that spacetime is curved.

Although to be fair to Einstein, the geometry of spacetime goes beyond Riemann’s: it is much more subtle and less easy to visualize, as I plan to discuss in a future post. (In jargon: spacetime is not a metric space, and its geodesics are not the shortest paths between two points.)

Spanish physicist Juan Parrondo, writing with University of Greenwich mathematician Noel-Ann Bradshaw, describes a paradox Parrondo himself proposed in 1996 that gives a game-theory interpretation of the intriguing concept of Brownian ratchet, a device that appears to extract free energy from a fluid. (Equally fascinating to me is how pervasive Brownian ratchets seem to be in biology, where they power many of life’s nanomotors.)

Among the more surprising (to me) examples is the one narrated by Arkasas mathematician Edmund Harriss. He tells the story of Johannes Kepler, who conjectured in 1611 that the way oranges are stacked in grocery stores is the most efficient way of packing spherical objects in space. So far so good, but what I didn’t know was that these “sphere packings” have applications to telecommunications, and in particular to the efficient transmission of information over noisy channels.

I also had no idea that the notorious “E8” (Garrett Lisi’s favorite means of attempting to build a theory of everything – see the November Scientific American article by Lisi and James Owen Weatherall) was involved, as Harriss tells.

Rowlett’s introductory remarks make no secret of the fact that the issue here is funding—and the article may be intended for skeptical scientists (the bulk of Nature’s readership) who grumble when mathematics receives money that, in their view, would be better spent on vials and microscopes.

As much as I enjoyed the article, it must be said that picking some of the successful examples does not satisfactorily answer the broader question of whether the bulk of mathematical research is a "waste of time," in the sense that it will never find applications anywhere. It is a legitimate question, and one that I am not qualified to answer.

On the other hand, mathematicians are cheap. They just need a small office, some chalk, a computer and, once in a while, a ticket to a conference. They make you smile by wearing nerdy T-shirts. They are good to have around on university campuses in case you are a scientist who happens to have calculus (or Riemannian geometry) questions. Oh, and they teach math to students. Lots of students.

So a similar remark might apply to mathematics as a whole as what Malcolm Gladwell said in his recent New Yorker article about Xerox PARC (the legendary research center in Silicon Valley) and about whether pure research is a good investment for technology companies.

The laser printer, Gladwell writes, came out of a maverick engineer’s obsession. It was one of many high-risk projects, most of which never led anywhere. But in the end it made Xerox billions of dollars, and “it paid for every other single project at Xerox PARC, many times over.”

Scientific American is part of Nature Publishing Group.

Illustration credit: David Parkins/Nature.