# Examples of mathematics motivated by technological considerations

I would like examples of technological advances that were made possible only by the creation of new mathematics. I'm talking about technology that was desired in some period of history but for which the mathematics of the time were not enough.

It would be great if I could have an example in which the technology needed was associated with an industrial problem, and it would be even better if the technology developed had a significant impact in society.

• I had initially voted this down, but now I think that with some rewording, the question is fixable. Nov 23, 2009 at 2:35
• You might be interested in this question: mathoverflow.net/questions/2556/… . Nov 23, 2009 at 2:45
• I really appreciate this question! (And, even more so, the answers, of course) I'm really happy to see that there are more interesting answers than just cryptography. Mar 9, 2010 at 20:48
• noticing many refs from TCS below. so see also eg core algorithms deployed tcs.se
– vzn
May 7, 2014 at 20:26

One of the reasons we don't all speak German today is the combination of elementary group theory, and universal Turing machines.

Per a request below, I'll elaborate: The British, French and Poles had a joint intelligence effort against Germany during the 30s (at that time breaking cyphers was considered a liberal-arts high classes hobby). The French, using traditional cloak and dagger techniques, managed to get their hands on the German cypher machine (Enigma) plans - but did not know what to do with it. The Polish cypher bureau however had an amazing idea - they'll let math students play with it. From a group theoretic point of view, the machine worked by taking products in S_{letters}. However, the machine designers did not notice that the products were all of the form A^{-1} R A - (where A is constantly changing but R is not) which enabled the poles to discover the cycle structure of R. You can look here for more details.

When the Germans invaded Poland, the Poles let the French and the British on the secret, and the British took over the operation. The Poles used machines (bombes) from the very start to break the codes, but as the Germans improved the coding machines, the need arose the reconfigure the breaking machines as they run. Luckily for everyone involved, the head cryptanalysis on the British side was Turing, who built such a machine: Colossus.

Finally, a funny bit which I hear in Bletchley park last year: "everybody" knows that at the end of the war, Menzies and Churchill ordered the Collosi destroyed and the plans burnt, and yet they were rebuilt a few years after they were declassified - how come ? The easy part is that some of the engineers saw it coming and took backup plans home. However, it is much funnier to discover that the Royal post office continued to manufacture the parts until the 70s, and they were still in stock.

• While this is probably not the best place or medium to get into this kind of debate, I'm really uncomfortable with that first para. Personally, it would be Japanese that I'd be speaking in your counterfactual. Anyway, always good to hear about the Poles and Bletchley Nov 23, 2009 at 8:32
• @Yemon: Figuratively - I'm Jewish, so I would not be here to speak. Nov 23, 2009 at 8:45
• I suppose, it is worth to mention here the name of polish mathematician which has idea of using tools from group/permutation theory in order to analyze and break Enigma cipher, and who manage it to success with his colleagues. Marian Rejewski: en.wikipedia.org/wiki/Marian_Rejewski Mar 9, 2010 at 19:10
• The first paragraph is rude. Mar 10, 2010 at 3:27
• @Manuel: If you'll read previous comments you'll see that kakaz is annoyed by me not mentioning the original contributers by name, and Yemon by me not mentioning in the first paragraph the - how should I put it politely - "eastern" aspects of the war. You'd have a hard time convincing a Jew, a Pole, and a Korean that there is anything rude in my first paragraph. This is a question about the connection between math and real life. WWII was real enough. Mar 10, 2010 at 13:00

One example of mathematical achievement that has application to the "industry" of medicine is the Radon transform, that enables to produce x-ray tomographies.

The Radon transform of a plane section of your body is the set of information that you obtain after shooting x-rays through all the possible lines contained in that plane, and recording the intensity of the ray that comes out at the other side. More formally, its an integral transform: it's the result of integrating the density of the body tissue along each of the lines.

Johann Radon defined his transform in 1917, and calculated a formula for its inverse. That is, he deviced a way to recover from the transformed data the density of tissue in each point of the plane section. The inversion formula involved a lot of calculations that couldn't be handled at that time.

Cormack and Newbold implemented the idea with the aid of computers to handle the calculations. They got a Nobel Price in 1979 because of this.

• Lawrence Shepp, a mathematician and statistician at Rutgers, has done a lot of important work related to implementing Radon's work for medical and other applications.He has written many papers related to tomography and discrete tomography. His papers related to these matters are cited here: stat.rutgers.edu/~shepp/papers.html Mar 12, 2010 at 22:21
• I could have this wrong, but I believe that the great shame for mathematicians is that the engineers Cormack and Newbold rediscovered the mathematics independently, making it seem that mathematicians are not really needed. To my surprise, I have heard engineers claim (more than once) that engineers never learn anything from mathematicians, and instead rediscover independently all of the mathematics they ever use. Nov 7, 2019 at 8:49

Public-key cryptography.

Also, compressed sensing.

• Great example! No one can deny that e-commerce has had a significant impact on society, nor that good cryptography has been a long-standing technological desire. Nov 23, 2009 at 3:10
• But it's hard to say that public key required the creation of new mathematics. The math behind RSA encryption was developed by Euler. Nov 23, 2009 at 12:40
• The mathematics in question here is not the basic number theory of Euler, but the formal idea that cryptography can be based on a computationally difficult problem. Probably the first widely known (non-classified) example of the idea is Merkle's puzzles. Since the content of mathematics in made of the precisely formulated ideas, not the formal deductions, it is certain that public-key cryptography was new mathematics at the time. Nov 23, 2009 at 23:48

The work of Oliver Heaviside in electrical engineering is a prime example.

He first of all invented an "Operational Calculus" to solve differential equation, and did non-rigorous stuff that were beyond the pale from the mathematical point of view. But they did work, and Heaviside faced much stiff opposition from the British mathematicians of the time. Later this work was justified formally with the development of Laplace and Fourier transforms.

The Dirac delta function, its derivatives and the paraphernalia were all conceived by Heaviside. The Delta function being for a sudden impulse of current such as in lightning or a spark, etc.. This was again later used by Dirac, and justified later in the theory of distributions.

I can't think of a better example. The problems of electrical distribution, network theory, and above all, all electrical communication systems, arose out of this theory. Even up to digital signal processing.

First it was built non-rigorously by the engineer Heaviside, and it was put on sound footing only much later.

Without the development of error correction codes and data compression codes most of the recent developments in communications technology (cell phones, high definition television, audio CD's, DVD's) would not be possible. Two big pioneers in this were Richard Hamming who developed one of the early practical error correcting codes (others were developed by Golay) and David Huffman whose pioneering work helped with the dramatic developments in data compression we see today.

Basic information about Hamming codes can be found here:

Basic information about Huffman codes is also available on en.wiki.

The invention of logarithms in 1544 and later "popularized" in the 1600's, simplified calculations that were practically unfeasible beforehand. The following paragraph is quoted from the Wikipedia entry on logarithms:

Their use contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, they were used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides the utility of the logarithm concept in computation, the natural logarithm presented a solution to the problem of quadrature of a hyperbolic sector at the hand of Gregoire de Saint-Vincent in 1647.

The invention of the Sierpinski carpet by Sierpinski in 1916. Who knew that cell phone antennas would later be based on this shape?

Basic example from operations research: efficient resource allocation is important in any society, so one can argue that civilizations of the past (and the present) have been wanting in part because of the absence of good allocation heuristics. Linear programming is a first major step toward sound quantitative analysis, and the fact that it was known on one side and not the other apparently contributed to an allied advantage during World War 2.

• The simplex algorithm, in particular. Nov 23, 2009 at 2:38
• I've never heard about any of this. What's the simplex algorithm? Do these algorithms really improve efficiency in a noticeable way, even before computers started processing huge data sets? What's the connection with WW2? Mar 9, 2010 at 20:44
• I'd suggest you first read the Wikipedia article about the simplex algorithm. How well the algorithms improve efficiency depends on how badly you were allocating resources before using them, and how good you are at taking accurate data about your resource needs. On the subject of WWII, efficient resource allocation is generally a huge factor in winning wars. If you want an army to function effectively, you need to bring food, fuel, and ammunition to the soldiers. The history of war is full of disasters in which supply chains ran long or were cut off and soldiers starved and died en masse. Mar 10, 2010 at 1:48
• I guess I should rephrase my question. Did these algorithms really improve efficiency during WW2? How heavily were they used (versus just using common sense & educated guesses as to how to make a schedule of what to send where)? How were they implemented without computers -- is there a way to make them work with feasible sizes of data sets? Mar 10, 2010 at 18:48
• I can't answer your first two questions with much confidence, since my only information source was textbooks and word of mouth. I have heard that Dantzig's simplex algorithm was viewed as a very important advance. For your third question, I believe the algorithms were implemented using a combination of professional human calculators and mechanical calculating devices (that were not fully programmable computers). Mar 10, 2010 at 23:30

Feedback control theory. In the 1920 when people at Bell labs tried to build amplifiers, they oscillated; and oscillators went unstable. Then the likes of Bode and Nyquist came up with the idea of using complex variables to develop stability criteria, and classical control theory was born. With it the use of electronics in telecommunications became possible.

• I hadn't noted it before, but this is tightly connected with Heaviside's operational calculus mentioned above.
– Pait
Apr 14, 2011 at 1:39
• This is not quite correct. It is actually Black who built the operational feedback amplifier and solved a very very important problem. Bode and others analyzed the solution and made it rigorous. They even published together. Feb 16, 2015 at 18:40
• Yes, the mathematical theory was created with the goal of solving a practical problem, understanding stability of electronic circuits such as the amplifier.
– Pait
Feb 18, 2015 at 0:55

There are many examples from computing but I guess you're after areas that have mathematical interest in their own right.

I think I good one is the Monte Carlo method. The idea has been around since Buffon's idea of estimating $$\pi$$. But it became important leading up to, and during the Manhattan Project, when Fermi and Ulam both came up with the method as a way to make impossible seeming integrals over high dimensional spaces tractable. For example, those involved in neutron transport. Today it's in use everywhere from finance to 3D graphics (with Pixar holding the patent on its application to ray-tracing).

There have been all kinds of interesting variations such as Markov chain Monte Carlo and various 'stratification' strategies to increase reliability through 'even' sampling, and Las Vegas algorithms that guarantee correct results. There have been some interesting recent 'pure' applications to mathematics, eg. allowing 'exact' samplings of spaces such as the domino tilings of diamonds. And randomised algorithms are routine in areas like number theory - eg. primality testing.

With hindsight it seems like an obvious idea but I think that at the time the idea of using random numbers to compute a non-random quantity was a pretty big shift of mindset.

The Manhattan Project probably "had a significant impact in society".

• I thought the Monte Carlo method was invented by Teller (the first author of the paper is Metropolis who apparently didn't do too much) in 1953. So after the war and after Manhattan Project.
– lcv
Nov 7, 2019 at 10:48

Information Theory: the original 1948 paper by Shannon is clearly focused in putting the analysis of how to deal with information on sound mathematical basis.

Ocean navigation is greatly helped by our ability to integrate the secant.

Contemporary work in biology has lots of examples. With the wealth of sequencing data coming out of the machines at ever lower costs there is a huge need for new methods and models to analyze and understand the data.

Google is another good example. Better (or even acceptable) internet search was desired and it took a mathematical advance to move the technology forward.

• There was no mathematical advance --- the eigenvectors were known long before Google. Nov 23, 2009 at 12:27
• Well it depends on what you mean by mathematical advance. If your metric is the creation of an entire new field, such as graph theory itself in this case, then I guess not.
– marc
Nov 23, 2009 at 15:06
• No, I mean a creation of a formal mathematical idea. The idea here is that a position in generic time of an appropriate random walk on pages along the keywords should correlate with relevancy. This idea is not mathematical, but once this idea is there, one can ask how one can actually compute that position, which is where the old mathematics comes in. Nov 24, 2009 at 0:12

Methods for estimating risk (financial or otherwise) are important for decision-making, i.e., one often wants to anticipate the consequences of ones actions. Some modern statistical techniques have been motivated by actuarial questions in e.g., the insurance industry. Similarly, in the financial sector, people have developed quantitative tools for choosing hedging strategies to manage financial risk. (Recent events suggest that these tools should be used with some caution.)

I suppose that this is an example that doesn't pack the historical weight of what many others have said, but topology has played an interesting (and up-and-coming) role in computer graphics and network sensing. This site details some of these applications and how they are currently changing various technologies. For example, there are a few examples of persistent homology being using to detect network coverage (i.e. Cell Tower Coverage) a la Ghrist and de Silva (2007) and for feature detection in MRI. Similarly, the ALICE project has used various Laplace-Beltrami operators for Computer Graphics applications as well as the use of Cell Complexes in surface reconstruction. On the other hand, there are examples of homology being used to characterize inviscid flows and in Computational Statistics / Statistical Mechanics (Chapter 5).

Again, these probably don't hold the weight of some of the other answers and at the moment, we don't know how these mathematical innovations will push technology forward; however, they represent the potential future for mathematics as a driving force for technology.

Extremal problems in ODE's and Pontryagin's maximum (minimum) principle. I think there should be many other examples from Soviet school, where mathematicians were sometimes forced to work on problems of "people's interest" (like optimal control in this case).

Artillery cannot be developed without proper notion of reference frame, for example Cartesian one, which is needed in order to properly define notion of a curve given by equation. So before Cartesian reference frame was introduced, and notion of relation between equation of notion and drawn curve was introduced, artillery, and any knowledge working with trajectories was blind and unable to form useful domain of engineering beside some pure empirical rules. In fact it remain in this state long ago after even if mathematics was possible to use equation of motion to develop proper description of trajectory.

Statistical process control: developed because of the needs of the factory process to control variation of all the parts that make up the final product. It contributed greatly to the rise the Japanese economy, post WWII.

The specific technologies listed in the Compressed Sensing Hardware page ( http://sites.google.com/site/igorcarron2/compressedsensinghardware ) could not have existed or have a justifiable theoretical background before 2003-2004 the date of the first papers on compressed sensing by Donoho, Romberg, Tao and Candes.

Only time will tell on how much of an impact they will have.

• Most (if not all) of those technologies seem to be from academia. Are there genuine private enterprises using compressed sensing? I recall something related to Netflix but I'm not sure if they went for compressed sensing in the end.
– lcv
Nov 7, 2019 at 10:57

Hydraulic press, mill, simple mechanics, gearwheel, clocks in order to be made has to be projected, and very important conception here is ability to count mass or volume of parts for this mechanisms. Hellenic mathematician Eudoxus was one of the first who correctly define integral. It allows to precise describe relation between mass and volume. Unfortunately his works was forgotten or misunderstand, and was not intellectually accessible during Middle Ages, so large parts of technology was impossible reconstruct although was present before!

It is example of technology and mathematical knowledge which was invented, and forgotten and then invented again, whit straight ties to technology!

This, and many more examples of such things may be found in excellent book of Luccio Russo "The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn"

I would be interested in what the new mathematics is that enabled compressed sensing.

PS: Can't post this as a comment to that answer. If someone can fix that please do.

• You need to have a certain (small) amount of reputation to comment. Nov 24, 2009 at 2:55
• Anyway you might want to try Terence Tao's description: terrytao.wordpress.com/2007/04/13/… Nov 24, 2009 at 2:56
• It looks like you should be able to comment now. Mar 9, 2010 at 20:50

When steam engine was constructed in fact there was no theory of thermodynamics. Heat engines theory was developed by Carnot which was 10 years old when steam engine was constructed by James Watt. So as technology here was present and works well without any mathematics, there was no means of theory for such mechanisms, and then no other than empirical by trial and error way to develop better solutions.

Statistical design of experiments: This was first developed by Fisher in connection with agricultural experiments, later the development continued by George Box and others in connection with industrial experiments. This led to ideas such as factorial designs and fractional factorial designs. Later Taguchi developed what is now called https://en.wikipedia.org/wiki/Taguchi_methods in connection to control and bettering of quality.

Magnetic Resonance Imaging (MRI) associates certain spatial changes in composition of materials with a visible image which may be represented by some real signal (vector) $$m\in\mathbb{R}^N$$. The vector $$m$$ often can be represented by a small number of coefficients in some basis $$\Psi\in \mathbb{R}^{N\times N}$$ (through a wavelet transform for example). However, the MRI signal $$m$$ is not measured directly but through some number $$M$$ of approximate samples of the Fourier transform

$$\hat{f}(\xi)=\int f(x)e^{2\pi i x \cdot \xi} dx,$$

these being denoted by $$\mathcal{F}_{u}m$$, where $$\mathcal{F}_{u}:\mathbb{R}^{N}\rightarrow \mathbb{R}^{M}$$. In one dimension these Fourier measurements might correspond for example to a sub-sampled DFT matrix with entries powers of $$\omega=e^{\frac{2\pi i}{N}}$$, so that

$$(\mathcal{F}_{u}m)_k=\sum\limits_{n=0}^{N-1} m_{n}\omega^{kn},\ k\in S\subset [0,N],\ |S|=M.$$

Under appropriate conditions on $$\Psi\mathcal{F}_{u}^{*}\mathcal{F}_{u}\Psi^{*}$$, with Fourier measurements $$b$$ observed, and with coefficients $$\Psi m$$ $$s$$-sparse ($$\Psi m$$ has at most $$s$$ non-zero entries), a primary method of retrieval is by solving the optimization problem $$\min\limits_{m} \|\Psi m\|_1$$ subject to $$\|\mathcal{F}_{u}m-b\|<\epsilon.$$

Research in compressed sensing explains under what conditions this recovery is possible, and now accounts for a ten times speed up in imaging times from about ten years ago before systematic studies of this optimization problem.

• Nice. Are you saying that modern MRI machines use compressed sensing?
– lcv
Nov 7, 2019 at 10:51
• @Icv Compressive MRI machines do; see statistics.stanford.edu/sites/g/files/sbiybj6031/f/2017-09.pdf. Nov 7, 2019 at 12:52
• Cool. Although, again, this is an academic document.
– lcv
Nov 7, 2019 at 13:00
• @Icv Sure, but in it the machines made by GE are mentioned: gehealthcare.com/-/media/…. Nov 7, 2019 at 13:05
• Oh this is cool. So compressed sensing allows for faster MRI scans. I should remember to ask for it next time I get an MRI of the brain (as of now you have to stay in a crazy sounding tube for an hour)
– lcv
Nov 7, 2019 at 13:36

Markov chains have applications in speech recognition (see A tutorial on hidden Markov models and selected applications in speech recognition by Rabiner), analysis of DNA sequences (see Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids by Durbin, Eddy, Krogh, Mitchison), in Google PageRank engine etc, see Wikipedia article for a longer list of applications.

One of the most important tools used in Computer Science is the Fast Fourier Transform, which basically made much of image processing possible - described by Gilbert Strang as "the most important numerical algorithm of our lifetime". Note however that unknown to the inventors James Cooley and John Tukey it seems that Gauss actually had already discovered the algorithm (see this paper for more details) but not published it!

Much of modern medical imaging is only possible due to mathematical advances - for a flavour here are a few example projects from the University of Eindhoven:

1. Lie Group Theory & Differential Geometry for Medical Image Analysis, Remco Duits
2. Riemann-Finsler Geometry for Brain Connectivity and Tractography, Luc Florack & Andrea Fuster
3. Differential Geometry in Complex Medical Imaging & Relativity Theory, Andrea Fuster