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I would like to know some examples of interesting (to the layman or young student), easy-to-describe examples of mathematics that has had profound unanticipated useful applications in the real world. For my own purposes, the longer the gap between the theory and the application, the better.

My purpose is to explain to the people I know why studying theoretical mathematics isn't a waste of time, and more importantly to motivate students. The reason I would like to have a longer time gap is that I want it to be clear that the mathematicians could not possibly have had the future applications of their work in mind.

A good example of this is Robert Lang's TED talk about origami, in which he describes how origami artists applied circle packing to their own work to construct designs, and how later engineers used origami to construct devices that can be transported compactly and then unfold to fill a larger space. His premier example is that of transporting large telescope lenses into space; since they are made of glass, they have to be carefully folded, and not just stuffed into a canister.

Other examples are how number theory was developed and later used in cryptography, and how polynomials were studied and later found to be useful in all sorts of applications. These have drawbacks, however. Cryptography and its uses in computer science are basically still math, and it's quite complicated. It's also not so clear that people studying polynomials weren't aware of their many potential applications.

Are there other good examples that fit my criteria? I think the latter two examples I mentioned could also be good examples, if presented correctly, but I'm not sure how best to go about that (and I am most interested in hearing of other connections).

Edit: If my motivation (or its wording) bothers you, please just ignore it and instead note that surprising later applications and connections make for interesting and engaging talks.

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    $\begingroup$ Gamma function and String theory tops the list for me $\endgroup$
    – Mr. Math
    Sep 7, 2013 at 21:24
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    $\begingroup$ Shouldn't this be community wiki? $\endgroup$
    – mrf
    Sep 7, 2013 at 22:53
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    $\begingroup$ Studying theoretical mathematics is for the most part a waste of time. People should waste their time on things they enjoy. $\endgroup$
    – emory
    Sep 8, 2013 at 2:59
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    $\begingroup$ I thought mathematicians do math because they enjoy it. $\endgroup$
    – achille hui
    Sep 8, 2013 at 5:04
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    $\begingroup$ @achillehui Mathematicians wouldn't do math if they didn't enjoy it... $\endgroup$
    – zerosofthezeta
    Sep 8, 2013 at 17:18

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A classic example is conic sections, which were studied as pure math in ancient Greece and turned out to describe planetary orbits in Newtonian physics (about 2000 years later).

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    $\begingroup$ I believe this is the example! $\endgroup$
    – egreg
    Jul 30, 2014 at 13:10
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    $\begingroup$ It might have had some applications back then, for optics and construction of sundials. $\endgroup$
    – MaudPieTheRocktorate
    May 21, 2019 at 1:58
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Radon Transform is an obscure piece of mathematics which study the integral transform consisting of the integral of a function over straight lines. This was studied as early as 1917.

In the second half of $20^{th}$-century, this piece of mathematics find its uses in medical imaging when computer becomes available. It is now widely used in all sort of tomography, to reconstruct the inner image of a patient using scattering data of penetration waves from multiple directions.

Next time when you or your family need to go to a doctor and takes a CT, MRI or PET. You are being benefited by this piece of mathematics.

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David Hilbert said: "I developed my theory of infinitely many variables from purely mathematical interests and even called it 'spectral analysis' without any pressentiment that it would later find an application to the actual spectrum of physics."

C. Reid. Hilbert–Courant. Springer-Verlag, New York, 1986.

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    $\begingroup$ This isn't a bad example, but it's a little hard to describe to someone without a math background already. $\endgroup$
    – Eric Tressler
    Sep 7, 2013 at 23:44
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"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years."

Written by G.H. Hardy in A Mathematician's Apology in 1940. (Chapter 28, page 140)

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    $\begingroup$ Just to flesh this out a little: among many other applications, number theory is now the basis for RSA and related encryption software, used in secure electronic communications from Amazon to Wikileaks; and relativity is crucial to the workings of GPS, used not only by tourists in New York but also soldiers in Afghanistan. $\endgroup$
    – Peter LeFanu Lumsdaine
    Sep 8, 2013 at 0:39
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    $\begingroup$ Yes, number theory is also behind the encryption behind every cell phone call or credit card transaction, etc. And I'm aware of the fact that GPS depends upon relativity, but I've had a hard time impressing that fact on people, because it's so intangible. $\endgroup$
    – Eric Tressler
    Sep 8, 2013 at 0:48
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    $\begingroup$ @PeterLeFanuLumsdaine: relativity is crucial to the fine-calibration of GPS, but that doesn't make GPS a "warlike purpose" of relativity IMO. You could design a satellite positioning system to work with Newtonian gravity & ether-wave electrodynamics just as well, if those had happened to be the right physical models. – Where SRT is important on a more fundamental level is for nuclear energy, so the Hiroshima and Nagasaki bombs are much closer to being such a warlike purpose (though again the connection, $E=mc^2$ and the enourmous devestation power of nuclear weapons, is often exaggerated). $\endgroup$
    – leftaroundabout
    Sep 8, 2013 at 1:24
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    $\begingroup$ @leftaroundabout: Of course you could have built a satellite positioning system on any sufficiently precise theory. The point is that relativity is, Maxwell isn't, and Hardy failed to predict that. Seriously: the relativistic effects of GPS are about 2100 meters (7 microseconds) per day. $\endgroup$
    – MSalters
    Sep 8, 2013 at 16:18
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    $\begingroup$ @MSalters: sure those $2100\: \mathrm{m}$ are a very significant contribution. My point is, if GRT had still been unknown when GPS was launched, the effect might have scared the heck out of the designers, but they would probably have been able to add the corrections on a purely empirical basis – similar to how NASA corrected the images from the HST before its mirror was fixed. $\endgroup$
    – leftaroundabout
    Sep 8, 2013 at 16:58
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Haar wavelet, a series of "square shaped" functions that can be used, similar to Fourier transforms, to transform functions into an alternate representation. It had roots as far back as 1909, and at the time was regarded as a mathematical curiosity.

The key characteristics of the Haar wavelet are that it is extremely simple, and when used on image data, the most significant components of a transform also happen to be the most perceptually significant. This mean that it's very useful in image compression. It forms part of the JPEG2000 format, and perhaps more commonly, is used for content-based image searches, such as Google's Search by Image, or "reverse image search". For this purpose, the first usage comes from the paper Fast Multiresolution Image Querying, in 1995. That's a gap of at least 86 years.

There's a more in-depth and not-quite-lay explanation of Haar wavelets here.

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  • $\begingroup$ I thought JPEG was dependent on the discrete cosine transform? Edit: Nevermind, I followed your link, and I see that the DCT has the Haar wavelet as a predecessor. $\endgroup$
    – Eric Tressler
    Sep 8, 2013 at 0:49
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    $\begingroup$ JPEG2000 is not JPEG; yes JPEG does use 8x8 DCT but JPEG2000 uses a wavelet transform. $\endgroup$
    – congusbongus
    Sep 8, 2013 at 0:51
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It is strange that so far nobody mentioned quaternions (discovered in 1843) and their use in computer animation.

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    $\begingroup$ Perhaps it's because the quaternions had early uses in physics, so they feel less like a curiosity and more like a useful idea. $\endgroup$
    – Gyu Eun Lee
    Sep 10, 2013 at 2:54
  • $\begingroup$ In fact, the whole reason the quaternions were even "discovered" was to model rotations in three-dimensional space. But yeah, it seems that some of their special advantages over rotation matrices (such as circumventing gimbal lock) were probably not anticipated. $\endgroup$
    – Josse van Dobben de Bruyn
    Jul 23, 2017 at 12:59
  • $\begingroup$ @GyuEunLee I believe that the applications to Maxwell's equation, the cross product, and everything else including in algebra and beyond came years after the discovery of quaternions. Here's a good read about it jstor.org/stable/… but Hamilton was shooting for things like maintaining that modulus of the product is the product of the moduli - and not, apparently physics but an abstraction of rotations in space as Josse van Dobben de Bruyn said so it would seem. $\endgroup$
    – Pineapple Fish
    Jun 25, 2020 at 0:31
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Non-euclidean Geometries and relativity-theory. When the first concepts of non-Euclidean geometry were developed (like the famous parallel postulate and it's independence from the previous four ones) the majority of people thought that they lived in an Euclidean space let's say. And "exotic geometries" were treated like funny brain-teasers. 100 years laters the first experiments shows that in fact we live in a non-Euclidean space and so subjected to the rule\theorems which was developed hundred years before.

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    $\begingroup$ I still think that I live in Euclidean space (perhaps higher dimensional, bit still Euclidean). $\endgroup$
    – Martin Brandenburg
    Sep 8, 2013 at 16:22
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Group theory came out long before it (and representational theory) was used by chemists to model molecular and other structures. Also, computer science was really pure mathematics prior to the advent of the computer. (Yes, computer science existed BEFORE computers believe it or not).

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  • $\begingroup$ Do you mean pure instead of applied? $\endgroup$
    – RghtHndSd
    Sep 8, 2013 at 1:58
  • $\begingroup$ Either would really be acceptable, but I changed it to more closely apply to the question. $\endgroup$
    – Padawan
    Sep 8, 2013 at 2:00
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    $\begingroup$ @achillehui AFAIK Babbage never actually finished building his computer. Also, Ada Lovelace, daughter of Lord Byron, helped Babbage devising algorithms for his Analytical Engine. If you include algorithm design in computer science then this proves that computer science started before any computer was built. (actually we could include a lot of algorithms devised by the greeks also, making computer science about 2000 years older than computers). $\endgroup$
    – Bakuriu
    Sep 8, 2013 at 11:36
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    $\begingroup$ Let me clarify that... I was meaning the common definition of "computer" (mechanical/electrical) and not the technical/theoretical definition. $\endgroup$
    – Padawan
    Sep 8, 2013 at 16:33
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    $\begingroup$ As soon as man learnt to sort, computer science was invented. $\endgroup$
    – Kris
    Sep 10, 2013 at 22:30
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Imaginary numbers came out long before their use in electrical engineering became apparent.

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Boolean Algebra, or the notion that many mathematical concepts can be expressed with variables that only take the values true or false, was pretty applicationless until the advent of the digital computer.

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    $\begingroup$ Is that really true? I mean, boolean algebra already had applications simply for logical reasoning, and was used very early on, so I wouldn't say that was an unintended consequence at all. $\endgroup$
    – haylem
    Sep 9, 2013 at 11:45
  • $\begingroup$ I should clarify, applicationless in engineering. $\endgroup$
    – Dougvj
    Nov 16, 2013 at 20:48
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All of complex analysis was pure mathematics before it started to be applied to physics. Two examples are Maxwell equations and the Schrödinger equation.

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  • $\begingroup$ The Maxwell equations are purely real. $\endgroup$
    – Mark Viola
    Jun 8, 2018 at 23:51
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Fractal geometry is a wonderful example of a mathematical field that has found broad practical application. From fractal cell phone antennas to improvements in the diagnosis and treatment of liver disease, to the alien landscapes of Hollywood, fractal analysis and design have become an integral part of modern life. Here is a link to a page from Michael Frame's site at Yale on various applications:

https://users.math.yale.edu/public_html/People/frame/Fractals/Panorama/Welcome.html

Here are a couple of links regarding liver treatment:

http://www.ncbi.nlm.nih.gov/pubmed/12768879

http://www.fractal.org/Fractal-Research-and-Products/Fractal-properties.pdf

While Benoit Mandelbrot knew that fractal geometry held great promise in many fields, he pursued it primarily for the love of exploration, knowledge, and underlying truth. It's hard to believe now, but his early work introducing the concept of non-integer dimensions was met with derision in some quarters. One can appreciate the sincerity of his pursuit by considering that one of the last things he had been working on (with Frame) was a theory of negative dimension:

http://www.worldscientific.com/doi/abs/10.1142/S0218348X09004211

In short, given its relative accessibility, fractal geometry provides rich pathways for motivating math students of all ages:

https://users.math.yale.edu/public_html/People/frame/Fractals/BonniePictures/BonniePictures.html

If you'd like, I'd be happy to correspond further regarding the education-based work we did at the NSF-funded Fractal Geometry Workshops at Yale.

Finally, I should mention fractal invisibility cloaks (always a crowd-pleaser), though currently they only work in the in microwave region:

https://users.math.yale.edu/public_html/People/frame/Fractals/Panorama/ManuFractals/InvisibilityCloak/InvisibilityCloak.html

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    $\begingroup$ This is a great example, that I had forgotten about. Especially given the technical nature of the Mandelbrot set, the interesting visuals, and how very surprising some of the applications are, I should bring this up more often. $\endgroup$
    – Eric Tressler
    Sep 11, 2013 at 16:43
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I think a lot of Graph theory was made without thinking about how much applications there are.

One can thing of a city as a graph, where the edges are the streets and the vertices are the crossroads. Now it is interesting for postmen to find ways through the city where avoid walking through a street more than once.

Furthermore it is used for network analysis and here are even a lot more of applications mentioned

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    $\begingroup$ This might be true, but the bridges of Koenigsburg problem was the motivation behind creating graph theory, so that was kind of an application itself. Graph theorists knew there were applications from the outset (though of course, as with all math, some branches found much later application). $\endgroup$
    – Eric Tressler
    Sep 8, 2013 at 16:28
  • $\begingroup$ @EricTressler Well for me the bridges of Koenigsburg is not a real useful application. I think of it more than as a puzzle as a real application $\endgroup$
    – Dominic Michaelis
    Sep 8, 2013 at 16:32
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    $\begingroup$ Of course it is only a puzzle, but it does indicate that the useful, non-puzzle applications were anticipated immediately. $\endgroup$
    – Eric Tressler
    Sep 8, 2013 at 16:42
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    $\begingroup$ @EricTressler at least a lot of the applications surprised me $\endgroup$
    – Dominic Michaelis
    Sep 9, 2013 at 6:50
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Von Neumann algebras are special operator algebras, introduced by John Von Neumann who was motivated by problems in operator theory, representation theory, ergodic theory and quantum mechanic. Decades later they found applications in knot theory, statistical mechanics, quantum field theory, local quantum physics, free probability and noncommutative geometry.

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    $\begingroup$ I think this is kind of "application in mathematics". $\endgroup$
    – Ziyuan
    Sep 9, 2013 at 10:50
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    $\begingroup$ Mainly, yes, but also in applied mathematics and physics. $\endgroup$
    – Martin Brandenburg
    Sep 9, 2013 at 14:38
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    $\begingroup$ I'm sceptical. Von Neumann was most likely motivated by quantum mechanics for quantum mechanics. $\endgroup$
    – Nikolaj-K
    Sep 11, 2013 at 15:00
  • $\begingroup$ @NickKidman is correct. "Third, various aspects of the quantum mechanical formalism suggest strongly the elucidation of this subject [von Neumann algebras]." from the original paper "On Rings of Operators" $\endgroup$
    – minimalrho
    Sep 15, 2013 at 3:58
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An outstanding example is RSA public-key cryptography. It is based on number theory, and in particular on the difficulty of decomposing a natural number in its prime factors.

This application is noteworthy because of its ubiquity in Internet secure transactions. It's also interesting that G. H. Hardy, the great mathematician, took pride in the belief that number theory was useless (see "A Mathematician's Apology"), which it has become not to be.

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A great example is Pascal's Triangle. Despite being named after Pascal, this arrangement (or others) of the binomial coefficients have been known for at least two thousand years.

Pascal's triangle turns out to be necessary for the interpretation of NMR spectroscopy data. NMR spectroscopy is the technology underlying the MRI.

And, although I have never seen a biologist use it this way, the binomial distribution is useful for solving the problem that Punnett Squares were invented to solve in Genetics.

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    $\begingroup$ Pascal used it himself to solve the gambling problem of the Chevalier de Méré. $\endgroup$
    – Raskolnikov
    Sep 8, 2013 at 16:11
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Humans have been interested in knot-like designs for a long time, but knot theory as we know of it began in the 18-th and 19-th centuries. But knot theory wasn't particularly useful for anything for a while - aside from Lord Kelvin's conjecture that atoms are knots in the aether (false anyway), there really wasn't much purpose to studying knots (as far as I know).

Later on, knots turned out to be very important objects in low-dimensional topology; e.g. they often form boundaries of three-manifolds. One can think of the classification of three-manifolds as a difficult problem because it involves the classification of knots, which is challenging in and of itself.

Most surprising though are the ways knots have found themselves applied in recent years: DNA topology and chemistry (people have even synthesized a knotted molecule!) come to mind.

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Matrix theory, which is estimated to have been used theoretically as early as 300 BCE, is being used in making computer-animated movies in order to represent 3D objects on a 2D screen. In fact, I attended a lecture that pointed out that, when Monsters Inc. was being created, one of the biggest challenges the crew needed to overcome was making Sulley's hair move in a realistic fashion. Their inevitable answer involved using transformation matrices!

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This is an extension to ricardo's answer on non-euclidean geometry and general relativity.

I just want to add to his answer that the independence of the fifth Euclid's postulate was a well-known open problem for over 2000 years. In fact, the beginning of non-euclidean geometries were motivated by attempts to prove the independence of the fifth postulate by contradiction ("assume it doesn't hold..."); however, no contradiction was found.

It is also noteworthy that already Gauss (100 years before Einstein) tried to do physical experiments to test the fifth postulate by measuring angles and distances in many-kilometers-large triangles; he was probably one of the first who realized that the Pytagorean theorem (as well as the "sum of angles in a triangle=180 degrees") doesn't need to hold in the "real space". (He was not able to disprove it with his aparatus, just derived a very small upper bound on possible deviation from these theorems based on his measurements)

As for the applications, today the general relativity theory based on non-euclidean computations is not only used in space-missions but also in satelitte navigation: that means, in GPS systems, broadcasting, flight navigation... It is very likely that at some point, it is used to deliver this text to you.

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Arguably, it's hard to beat the Pythagorean theorem, both on the number of useful applications and how much later they were found.

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    $\begingroup$ Could you elaborate? The ancient Greeks knew about triangulation (Aristarchus calculated the distance to the Sun), but they use more trigonometry than just the Pythagorean theorem. That is, they also knew about the applications; are there any non-obvious ones? $\endgroup$
    – Eric Tressler
    Sep 8, 2013 at 16:40
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    $\begingroup$ As far as I know, Pytagorean theorem was first known to engineers and only much later proved theoretically. $\endgroup$
    – Peter Franek
    Jul 19, 2014 at 14:46

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