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When Math(s) Turns Out To Be Useful

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David Parkins's artThe 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.)

E8Among 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.

About the Author: Davide Castelvecchi is a freelance science writer based in Rome and a contributing editor for Scientific American magazine. Follow on Twitter @dcastelvecchi.

The views expressed are those of the author and are not necessarily those of Scientific American.

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  1. 1. Uncle.Al 4:00 pm 07/19/2011

    “Although to be fair to Einstein, the geometry of spacetime goes beyond Riemann’s” General relativity (GR) (Einstein, 1916) is a subset of teleparallel gravitation (Einstein, et al., 1931). Teleparallism contains testable failures of GR arising from angular momentum – physical spin, quantum spin, spin-orbit coupling, and quantitative chirality through moments of inertia (reduced to practice within chemistry not physics).

    Fundamental physical theory arises from mirror symmetries chosen for mathematical ease not empirical validity. Consider BRST invariance and Calabi-Yau manifolds in quantum gravitation (string theory), and Standard Model supersymmetry. Empirically defective theory is “corrected” with manually inserted symmetry breakings. Physics demands isotropic mirror-symmetric vacuum given zero photon vacuum refraction, dispersion, dichroism, and gyrotropy (arxiv:0912.5057, 0905.1929, 0706.2031, 1106.1068). Photons are massless bosons not massed fermions. Perhaps the universe is exactly as it appears to be, chiral toward massed fermions (matter) all the way down. Chiral weak interactions are then fundamental and diagnostic; strong interactions are racemic blurs.

    GR in pseudo-Riemannian spacetime is fundamentally mirror-symmetric, with gravitation as geometric curvature. Teleparallelism in Weitzenböck spacetime is fundamentally chiral, with gravitation as spacetime torsion, a chiral force like Lorentz force. The difference is testable in existing apparatus. Left and right shoes will vacuum free fall non-identically if there is a chiral vacuum background (vacuum left foot), spacetime torsion not spacetime curvature.

    Chemically and macroscopically identical, opposite geometric parity atomic mass distributions (left shoes versus right shoes on a vacuum left foot) will falsify the Equivalence Principle in existing bench top apparatus, a geometric Eotvos experiment, in a 90-day experiment. Required are single crystal test masses in which one set has all its atoms contained in parallel homochiral left-handed helices, and the other set has all its atoms contained in parallel homochiral right-handed helices. The two controls are each against a chemically identical amorphous solid or achiral single crystal.

    The universe wants to be discovered. Such single crystal test masses are respectively enantiomorphic space groups
    P3(2)21 and P3(1)21 alpha-quartz, with fused silica control. Space groups P3(2) and P3(1) gamma-glycine are another set, with achiral space group P2(1)/n alpha-glycine control.

    The worst it can do is succeed: MOND vs. dark matter is resolved because trace anisotropic vacuum toward fermionic mass means angular momentum is not conserved, for Noether’s theorems do not act on absolute discontinuous symmetry parity (chirality in all directions). Matter vs. antimatter abundance and neutrino-antineutrino reaction channel divergence are a vacuum left foot fitted by left and right shoes – of course they diverge. Chiral beta-, positron- (Na-22), and electron capture-decay (Co-57) rates are vacuum anisotropy sensitive; achiral alpha-decay is inert. Biological homochirality is the universal default. SUSY and quantum gravitations arising from vacuum mirror symmetries are defective at the founding postulate level.

    No prior observation would be contradicted. Physics would no longer be embarrassed by 10^500 string theory acceptable vacua that each and all together predict nothing testable, or a universe filled with postulated SUSY partners but zero being observed. The greatest obstacle to understanding reality is not ignorance but the illusion of knowledge. The proper test of spacetime geometry is atomic scale test mass geometry. Somebody should look.

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  2. 2. fnori 12:47 pm 07/22/2011

    Interesting post. A more general issue, sometimes not appreciated by some funding agencies, is the following: “When Fundamental Research Turns Out To Be Useful”. Following your example on curved spaces: years ago, general relativity was considered to be extremely useful. The advent of the GPS changed that view. Many other examples exist.

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  3. 3. @dcastelvecchi 1:53 pm 07/22/2011

    Excellent example, thank you for pointing that out. My friend Julie Rehmeyer describes the math of GPS in the July issue of Wired

    Link to this
  4. 4. fnori 9:26 pm 07/28/2011

    Correction to my post above: It should have used the word “useless” instead of “useful”. Sorry for the “typo”.

    The corrected text would be:

    Interesting post. A more general issue, sometimes not appreciated by some funding agencies, is the following: “When Fundamental Research Turns Out To Be Useful”. Following your example on curved spaces:
    years ago, general relativity was considered to be extremely *useless*. The advent of the GPS changed that view. Many other examples exist.

    Link to this
  5. 5. Mike Butler Snr 12:37 am 11/19/2012

    As a VERY newcomer to the Scientific American community, I am puzzled as to why I — as a well read and well educated Australian — struggle to understand these amazing mathematical concepts, yet can thrive on concepts that involve the use of the English language! According to tests conducted 45 years ago when I was at college, my IQ is definitely in the Mensa range, yet mathematics leave me cold. I know this is “off topic” on this post, but forgive me as I am a “newbie”, but if there is a neuro-scientist reading this who can tell me why we have such compartmentalised brains — even though our basic intelligence is sky-high — I would really love to know! I turn 67 in December and for at least 40 years, this has puzzled me!

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