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I would like to ask about examples where experimentation by computers have led to major mathematical advances.

Motivation

I am aware about a few such cases and I think it will be useful to gather such examples together. I am partially motivated by the recent polymath5 which, at this stage, have become an interesting experimental mathematics project. So I am especially interested in examples of successful "mathematical data mining"; and cases which are close in spirit to the experimental nature of polymath5. My experience is that it can be, at times, very difficult to draw useful insights from computer data.

Meta Question: classification of experimental mathematics

I asked: Is there a useful way to classify the various types of experimental mathematics?

Let me try to propose, based on the answers so far, a tentative answer describing five categories of experimental mathematics:

1) Mathematical conjectures that were arrived at by examining experimental data

2) General purpose programs which interactively or automatically lead to posing mathematical conjectures.

3) Large mathematical databases

4) Computer-assisted proofs of mathematical theorems

5) Various computer programs which allow proving automatically theorems or generating automatically proofs in a specialized field.

Bounty:

There were many excellent answers so let's give the bounty to Gauss...

Related question: Where have you used computer programming in your career as an (applied/pure) mathematician?, What could be some potentially useful mathematical databases?

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en.wikipedia.org/wiki/… –  Steve Huntsman Jan 17 '10 at 8:35
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You might want to email Doron Zeilberger at Rutgers. –  Charles Siegel Jan 17 '10 at 15:39
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Indeed, Doron Zeilberger has even taught classes called "experimental mathematics": math.rutgers.edu/~zeilberg/teaching.html . –  Qiaochu Yuan Jan 17 '10 at 15:50
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@Mark: URLs are automatically turned into links in comments; no need for html markup. Corrected version of the link: emis.de/journals/EM –  Anton Geraschenko Jan 20 '10 at 18:53
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Doron has recently posted a "videoed lecture" about experimental mathematics: math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/future.html –  Douglas S. Stones Apr 3 '10 at 0:48

33 Answers 33

up vote 29 down vote accepted

The Prime Number Theorem was conjectured by Gauss from looking (very hard, one can presume...) at a table of the primes $\leq10^6$. It is not with too much effort that one can read his Disquisitiones as a set of tricks to determine primality with as little work as possible, and one can understand the motivation: he was his own computer, in a way :P

(I don't know where Legendre got the statement from, but he must surely have had tables of primes too!)

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This may be apocryphal, but I remember being told that Gauss also conjectured and proved quadratic reciprocity after expanding 1/p in base q for various pairs of primes p, q. –  Qiaochu Yuan Jan 18 '10 at 15:58
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Considering Euler died when Gauss was seven or eight, I'd think that in some sense it's very likely Euler predated Gauss. (I don't know if Euler published his conjectured statement, though.) –  Harrison Brown Jan 18 '10 at 16:29
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You get 1/p as a/(q^n-1) so p is a factor of q^n-1, and the multiplicative order of q is n mod p. If (p-1)/n is even, q is a quadratic residue. –  Douglas Zare Jan 18 '10 at 21:28
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I would like to mention two "computers" of the 18th century. The first one is Anton Felkel who produced a table of primes in 1776. In scs.uiuc.edu/~mainzv/exhibitmath/exhibit/felkel.htm one can read that the Austrian artillery used his output in battles : just the paper, not numbers! In 1849 Gauss mentions in his 4 page letter to his student JF Encke that he used Vega's tables to confirm his estimate. Jurij Vega published his table of primes in 1797. Who knows if Felkel's tables were used for cartridges in 1789 when Vega commanded a battery of mortars at the battle for Belgrade. –  Tomaž Pisanski Mar 11 '10 at 17:09

Computer experiments (in the early 1960s!) led Birch and Swinnerton-Dyer to the formulation of their conjecture, which stimulated the development of much of arithmetic geometry.

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A lot of Rich Schwartz's work begins with computer experiments.

Perhaps most notably is his proof that a triangle whose largest angle is less than 100 degrees has a periodic billiard trajectory.

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I think Rich's work is also a good example of how to present computer-assisted mathematics: lots of discussion of his motivation for performing various experiments, lots of friendly-to-use Java applets... –  David Hansen Mar 5 '10 at 15:02

Another thing you might like to check out is Herb Wilf's very nice article in the Princeton Companion to Mathematics, in which he talks about the interplay between theory and experiment, and the many forms it can take. To give one of his examples (or rather classes of examples), if you generate a sequence of integers, you can plug it into Sloane's database and you may find that it is a known sequence, but generated in a completely different way. In that case, you have an instant interesting conjecture -- that there is a connection. But he discusses several other kinds of example. Another one that deserves to be mentioned is the Bailey-Borwein-Plouffe amazing formula for pi.

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I have one personal experience with experimental mathematics. It started with a computer assisted simple proof of a result that every five dimensional polytope contains a 2-simensional face which is a triangle or a quadrangle. It turns out that the negation of this theorem implies a certain number of inequalities for the so called flag numbers which together with known inequalities lead to a contradiction. At the end the proof can be checked easily by hand.

The next step was with Gunter Meisinger and Peter Kleinschmidt (see this paper), and we made a computer program called FLAGTOOLS that was able to prove automatically theorem of a similar type. At a later stage we tried to let the program test systematically for such theorems. Among the theorems we proved is the following: that there is a finite list of 3-polytopes so that every 9-polytope has a face from that list. Some similar work for quasisimplicial polytopes which had additional features of automations were carried out by Shahar Lovett.

Overall, it was difficult to use the automatic systems to obtain unanticipated "meaningful" or "interesting" results, and this had become harder the higher the level of automataion was.

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A quotient of a polytope P corresponds to an interval in the face lattice: all the faces that contain a face G and are contained in a face H. The set of these faces are the set of faces of a polytope Q of dmension dim G - dim H - 1. Intervals of the form [emptyset, F] simply correspond to the faces of F. Polyope duality allow you to think easily about intervals of the form [F,P]. –  Gil Kalai Jan 26 '10 at 22:30

The proof of the four color theorem, the proof of the nonexistence of a projective plane of order 10, the proof of the Kepler conjecture, the search for Mersenne primes, the discovery of the Lorenz attractor and the Feigenbaum constant among others are examples. These are in the wikipedia article on experimental mathematics with other examples. There is also a journal of experimental mathematics. Also there are some videos here from June and July of 2008 about experimental mathematics. Many of them are about the Landen transformation.

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A fairly classical (and pretty dated at that) example is the study of solitons which was to a large extent triggered by sloving numerically the so-called Fermi--Pasta--Ulam chain and then of its continuous limit, the Korteweg--de Vries equation. It would not be much of exaggeration to say that the whole modern theory of integrable systems grew out of this. For more details see e.g. the Wikipedia entry on solitons and the first chapter of the book Solitons in Mathematics and Physics by Alan Newell.

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You asked for a classification of types of experimental mathematics. I would like to mention just one class that interests me greatly (which is the class that predominates in polymath5). There are many examples of mathematical statements that become easier to prove when you strengthen them, because you have much more of a clue of what sort of proof is appropriate. To give one example, consider the following two statements.

  1. For every k there exists n such that every real sequence of length n has a monotonic subsequence of length k.

  2. Every real sequence of length mn+1 has an increasing subsequence of length m+1 or a decreasing subsequence of length n+1.

Even though 2 is a stronger statement, it is much easier to come up with the correct proof, because the nature of the bound gives you a big clue (particularly after you check easily that this bound is best possible).

In research mathematics, we often find ourselves with problems like 1, and we want to convert them into problems like 2. And a fantastic way of doing that, which was not available to our mathematical forebears to anything like the same extent, is to program a computer to solve the problem by brute force (or better still, with the help of clever algorithms) in small, but not too small, cases. Right now in polymath5 we are trying to find a proof using semidefinite programming, which involves coming up with a quadratic form with certain properties. Both finding the form and proving that it has the required properties seem to be difficult, but the first task is much easier than it might have been, because for pretty large n one can use a computer to find the best possible quadratic form, and one can then find out a lot from that about which forms are likely to be good and which less good.

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I'm surprised that no one has mentioned the Mandelbrot set yet, arguably the most famous new mathematical object of the last 30 years, at least among the general public. Benoit Mandelbrot discovered it in 1979 as a result of computer experiments. He says that when he first saw it he was so surprised by its appearance that he thought it must be the result of a computer malfunction. In his book Fractals and Chaos, Mandelbrot argues that his discovery of the Mandelbrot set contributed to the revival of experimental mathematics and led to a general change in the attitude of mathematicians to experiments in mathematics. On page 25 he writes:

"The culture of mathematics during the 1960s and 1970s

Within that culture the Mandelbrot set could not have been discovered. Hence its discovery marked a historical departure. Today -- but not yesterday -- only a minority among mathematicians would agree with the opinion due to someone who did not discover that set, that the study of M reflects "a rather infantile and somewhat dull mathematical sensibility" (Brooks 1989)."

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My understanding is that John Hubbard discovered the set and described it to Mandelbrot, who had access to much better computers. –  Victor Protsak Jun 6 '10 at 19:29
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The Mandelbrot set was first discovered in 1978 by Robert Brooks and Peter Matelski (The dynamics of 2-generator subgroups of PSL(2,C), in "Riemann Surfaces and Related Topics", ed. Kra and Maskit, Ann. Math. Stud. 97, 65–71, ISBN 0-691-08264-2) They drew a crude but recognizable picture of it in their paper. So the statement that it could not have been discovered in the mathematical culture of the 1970s is simply false: it WAS discovered in the mathematical culture of the 1970s! –  Richard Borcherds Aug 15 '10 at 14:12
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Thanks for the correction. The paper by Brooks and Matelski is available online. The picture of the Mandelbrot set is on the second last page. math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf –  Marko Amnell Aug 15 '10 at 18:10

Also V.I. Arnol'd is quite enthusiastic about the experimental mathematics. In particular, he wrote two books on the subject in Russian, Experimental mathematics, Fazis, Moscow, 2005, and Experimental observation of mathematical facts, MCCME, Moscow, 2006, where, I guess, one could find a number of examples of the kind you ask for. Unfortunately, to the best of my knowledge, neither of these books was translated into English so far.

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Arnol'd gave some lectures at Berkeley a couple years ago. I remember he mentioned something about generating random numbers by standing at the side of a road and taking note of license plate numbers of cars that went by. –  Kevin H. Lin Jan 18 '10 at 18:35

The Monstrous Moonshine relationships between the Monster group (the largest sporadic finite simple group) and modular functions were investigated after observations of several numerical coincidences:

Several of the lowest coefficients of the Fourier expansion of the $j$ modular function are small sums of the smallest dimensions of irreducible representations of the Monster group. For example, the linear term is $196884q$, and the two smallest irreducible representations of the Monster group have dimensions $1$ and $196883$. The quadratic coefficient is the sum of the dimensions of the three smallest irreducible representations. This led to the conjecture of the existence and then the construction of an infinite graded vertex algebra on which the Monster acts whose pieces have dimensions equal to the Fourier coefficients of $j$.

I'm not sure whether computers were involved historically, but this was a shocking pattern of numerical coincidences of a type which is much easier to find now with tools like Sloane's online encyclopedia of integer sequences.

Also, it was noted that the primes $p$ such that $\mathbb H^2/\Gamma_0(p)^*$ has genus zero are precisely the primes dividing the order of the Monster group.

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+1, but a couple notes: The j function observation was due to McKay, the genus zero observation was due to Ogg, and the moonshine module is a vertex algebra, but not an algebra in the usual sense of the word. –  S. Carnahan Jan 19 '10 at 20:56

Donald Knuth comments in one of his papers on the key role of the use of computers in making his paper possible.

D. E. Knuth, A class of projective planes. Transactions of the American Mathematical Society 115 (1965) 541–549.

Basically what happened was that Knuth used a computer to study an example and once he saw how that example worked he was able to see what was going on in general.

This is a common way that "experimental" mathematics can work. Doing experiments can lead to examples that provide the proof of how some general construction can be implemented.

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I am familiar with a few more examples of various types. I think that in order to be useful we can think about various different types of "computer aided experimental mathematics".

A) Conjectures obtained from computer experimentations.

1) A famous example is Feigenbaum's discovery: (Quote from the wikipedia article) "Some mathematical mappings involving a single linear parameter exhibit the apparently random behavior known as chaos when the parameter lies within certain ranges. As the parameter is increased towards this region, the mapping undergoes bifurcations at precise values of the parameter. At first there is one stable point, then bifurcating to an oscillation between two values, then bifurcating again to oscillate between four values and so on. In 1975, Dr. Feigenbaum, using the small HP-65 calculator he had been issued, discovered that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692... He was then able to provide a mathematical proof of that fact, and he then showed that the same behavior, with the same mathematical constant, would occur within a wide class of mathematical functions, prior to the onset of chaos. For the first time, this universal result enabled mathematicians to take their first steps to unravelling the apparently intractable "random" behavior of chaotic systems. This "ratio of convergence" is now known as the first Feigenbaum constant."

2) There are various mathematical packages with more "general-purpose" experimentation in various fields and some of those have led to interesting conjectures.

B) Computerized proof

1) Also here there are several examples of different nature. In some cases (like Hales proof of the density packing conjecture) verifying the proof requires full replication of the experiment. In some other cases, like the proof of Cohn and Kumar for the densest lattice in 24 dimensions, checking an automated proof can be carried out (again usually using computers) with less effort compared to the effort in finding the proof.

2) There are computer packages which allow more "general-purpose" forming-conjectures-and-proving-them in various (still quite restricted) areas.

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I don't think any precise theorems have fallen out yet, but the computations done by the 'Atlas of Lie groups' software of Adams, Vogan, du Cloux and many others has "completely changed the way [they] look at the field"

Edit: Jeff Adams agreed to give a lecture this semester where he would outline what surprising facts have been discovered, what conjectures have been made, what statements have been proved, etc. by looking at Atlas data. If this happens I'll post some kind of summary here.

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When I was an undergrad, Jon Borwein showed me the inverse symbolic calculator. You type in a number, and the inverse calculator tells you what it 'thinks' your number is, for example $\pi^2$. I thought it was a useful tool for experimenting. For example, if you have a series or integral which you can't evaluate exactly, you can evaluate it numerically and plug it into the inverse calculator to see if it spits out a 'meaningful' answer. You then have something that you can actually try to prove.

This isn't a very deep application, but there is more information on this page for experimental mathematics.

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One should surely also mention the Riemann hypothesis since Riemann computed a few zeros of the zeta function by hand and observed that they were on the critical line. I guess that this contributed to his statement, together with the functional equation, of course. (I hope that I did not overlook this response in the many previous answers, sorry otherwise).

Another experimental result with a happy ending is the birth of random matrix theory by Wigner who guessed the circular law after computer experimennts if I am not mistaken.

Both subjects have been recently linked experimentally.

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Thistlethwaite's discovery of links with trivial Jones polynomial (MR1831681) should be mentioned here. He apparently discovered them "during the course of a routine computer ennumeration." Later he and Eliahou and Kauffman (MR1928648) were able to recognize these links as part of an infinite family of links with trivial Jones polynomial. Further, it's my impression that the calculation that these Jones polynomials are trivial doesn't use computers.

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If I am not mistaken, the connection of multiple zeta values to renormalized Feynman integrals was discovered through computations (by David Broadhurst). Here is a survey of ongoing computer research around this field (from 1996).

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John Conway initially thought that his cellular automata game of Life could not lead to an unlimited number of active cells starting from a finite number of cells.

Computer experimentation by Bill Gosper led to the discovery in 1970 of the Gosper Glider Gun, a finite collection of cells which leads to a recurring state which continues to emit "spaceships" or "bullets" which continue out into the periphery infinitely. I can't find a link to the specifics of the discovery beyond the remark that Gosper won a 50$US reward from Conway for making this finding.

The glider gun was also used in constructions involving the manipulations of the "bullets" in such a way as to form a computational apparatus, ultimately proving the Turing-completeness of Conway's Life. I do know that a lot of random configuration discovery was done for the Life cellular automaton and for other CA by creating random patterns, then masking the lattice with a pattern which blanked out the peripheral cell values and left the random pattern alone in the center of the screen, and then allowed the CA rules to run. Escapees such as gliders would continue their motion into the blanked areas and be easily discerned in a simulation.

The first glider-type pattern was discovered by tracing the evolution of the finite starting patterns.

David Eppstein also appears to have written a program that helps in finding these spaceship types of pattern, which should also count as mathematical discoveries and exploration via computer programs.

:gfind A program by David Eppstein which uses de Bruijn graphs to search for new spaceships. It was with gfind that Eppstein found the weekender, and Paul Tooke later used it to find the dragon. It is available at http://www.ics.uci.edu/~eppstein/ca/gfind.c (C source code only).

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I'm hardly the only one to have written programs to search for CA patterns. See ics.uci.edu/~eppstein/ca/search.html for a partial listing of others. –  David Eppstein Sep 2 '10 at 17:21

I was recently at this workshop at Fields on 'discovery and experimentation in number theory': you can get audio and slides from many of the presentations here although the one I wanted to recommend - David Bailey's talk - appears to be audio only. It depends what you mean by 'major mathematical advance' but there certainly seem to be many problems in number theory where even if the proof doesn't depend on heavy computation, gaining insight into what to set about proving did. I personally logged several CPUweeks trying to get to grips with my own thesis topic.

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Recently Burton, Rubinstein and Tillmann proved the conjecture that the Seifert-Weber dodecahedral space is not Haken.

http://front.math.ucdavis.edu/0909.4625

Quite a bit of experimentation led up to their algorithms, as you would see by reading their paper.

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The paper by Mark Haiman: Conjectures on the quotient ring by diagonal invariants J. Algebraic Combin. 3 (1994), no. 1, 17-76 starts with:

It has recently been discovered, mainly on the basis of evidence obtained using the computer algebra system MACAULAY, that there seem to be unexpected and profound connections between a certain natural ring and some fundamental and much-studied aspects of combinatorics and algebraic geometry.

Haiman's conjecture that he discovered by a computer was eventually proved by himself using (human) but deep algebraic geometry.

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According to this obituary of Edward Lorenz in Plus Magazine, he discovered the seemingly chaotic behaviour of the Lorenz attractor when a small change (due to rounding) in the boundary conditions in a numerical simulation gave rise to hugely different solutions.

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There's also a fairly comprehensive website experimentalmath.info where, I guess, some further examples of EM in action can be found.

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I think there may be an additional category, though relatively little-known outside my own field (design automation): genetic programming applications, or more generally the application of metaheuristics. Spector et al's "Genetic Programming for Finite Algebras" (2008) is a good example of the methodology, which seems to me to be much less about the automation of proof as such, and more about enhancing mathematicians' capacity for well-defined combinatorial searches.

That said, you may want to include it, and the growing number of other GP papers in this domain, in your survey categories under (4) or (5).

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Weari-Phelan structure provides a counterexample to Kelvin's conjecture that the minimal area partition of space is realized by an appropriately curved bitruncated cubic honeycomb tesselation. It was discovered by computer simulations of foam in 1993. The problem of finding the minimizing partition is still open, but nowadays you can easily toy with it yourself using Surface Evolver. Who knows, maybe you will discover a better solution!

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The results on Chebychev's bias is a great example. Chebyshev in 1853 noted that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. This is called Chebychev's bias. Littlewood proved in 1914 that this bias exists and also that it gets violated infinitely often, if you go far enough. The distance to be covered for the violation seemed to grow rapidly with each step.

Rubinstein and Sarnak proved in 1994 that the violations have nonzero density. Here the density means the "logarithmic density", as defined in that paper.

This was a quite interesting result guided at each step from considerations of experimental mathematics and computation. For more information have a look at the Rubinstein-Sarnak paper.

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In some simple cases it is even possible to both guess some results and prove them too experimentally. I have done this with some Hankel determinants. Suppose you want to compute the Hankel determinant $\det \left( {a_{i + j} } \right)_{i,j = 0}^{n - 1} $ of a sequence $(a_n )$ . If no Hankel determinant vanishes then you can compute polynomials $p_n (x)$ which are orthogonal with respect to the linear functional $F$ on the polynomials defined by $F(x^n ) = a_n .$ By Favard's theorem there exist numbers $s_n ,t_n $ such that $p_{n + 2} (x) = (x - s_{n + 1} )p_{n + 1} (x) - t_n p_n (x).$ In some cases it is easy to guess a formula for these numbers after computing some of them. Thus also the Hankel determinant can be computed. In order to show that this guess gives the correct answer we can do the following: Define $a(n,j) = a(n - 1,j - 1) + s_j a(n - 1,j) + t_j a(n - 1,j + 1)$ with $a(0,j) = \delta _{j,0} .$ By the underlying theory it suffices to show that $a(n,0) = a_n .$ In order to do this we again compute $a(n,j)$ for small values and try to guess a closed formula for them. In many cases we succeed. Then it suffices to verify that $a(n,j) = a(n - 1,j - 1) + s_j a(n - 1,j) + t_j a(n - 1,j + 1)$ holds for the conjectured formula.

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Didn't the work of Candès and Tao on compressed sensing begin by a computer experiment by Candès that gave results "too good to be true"? See: http://www.ams.org/samplings/math-history/hap7-pixel.pdf

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The Shimura-Taniyama-Weil conjecture (now a theorem) and the Birch and Swinnerton-Dyer conjectures were based on examples.

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