# What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what about the last one hundred years? Is it still possible for an amateur to make a significant contribution to mathematics? Can anyone cite examples of important works done by amateur mathematicians in the last one hundred years?

For a definition of amateur:

I think that to make the term "amateur" meaningful, it should mean someone who has had no formal instruction in mathematics past undergraduate school and does not maintain any sort of professional connection with mathematicians in the research world. – Harry Gindi

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@Budney. Maybe not. It is a valid research question in history of mathematics. And not a trivial one. It has connotations to epistemology in mathematics. – Mlazhinka Shung Gronzalez LeWy Oct 30 '10 at 14:49
There's a perfectly good question here: "What recent discoveries have amateur mathematicians made?" My advise to the OP: change the title to this, rather than the current title (which is what people seem to be arguing about, rather than the actual question), make it community wiki, and give it the tag "big list" - this applies even if the list is very short. – Peter Shor Oct 30 '10 at 15:32
Part of the problem is: what is the definition of amateur? If it means someone who does mathematics but doesn't get paid for it, there are plenty out there who arguably make a significant contribution. – Todd Trimble Oct 30 '10 at 16:00
Please make the appropriate changes, (e.g., those suggested by Peter Shor seem acceptable), and flag for moderator attention. – S. Carnahan Oct 30 '10 at 16:23
@Piertro: yes, E.T. Bell was an amateur historian of mathematics. :-) – Todd Trimble Oct 30 '10 at 17:58

About ten years ago Ahcène Lamari and Nicholas Buchdahl independently proved that all compact complex surfaces with even first Betti number are Kahler. This was known since 1983, but earlier proofs made use of the classification of surfaces to reduce to hard case-by-case verification.

At the time, Lamari was a teacher at a high school in Paris. Apparently he announced his result by crashing a conference in Paris and going up to Siu (who had proved the last case in the earlier proof in 1983) with a copy of his proof. Lamari's proof was published in the Annales de l'Institut Fourier in 1999 (link: http://tinyurl.com/2e6cbj5).

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Gunnar, it cool to know that Lamari was a teacher at that time (I was always a bit curious that there were two simultaneous proofs of the result). At the same time as I see according to mathgenealogy genealogy.ams.org/id.php?id=56039 Lamari accomplished his Phd at Paris VII in 1999. So, could we really call him an amateur...? – Dmitri Apr 22 '11 at 23:00
@Dmitri I'm told he was given a PhD for his proof. In any case, he got his PhD in 1999, after the result was published, so surely he still counted as an amateur? I've wondered about what happened to him afterwards... nothing turns up when I try to google him, so I wonder if he went back to teaching. – Gunnar Þór Magnússon Apr 23 '11 at 8:47
it is harder than this. in the french system there is a a configuration, called "professeur aggregé", which is kind of wave-particle-dualism. if you are aggregé, you do hold a teacher position in a high school but can also spend some time at the university and may even decide to switch to academia again, later on. that said, and knowing a the french system a bit, i cannot imagine that someone can be awarded a ph.d. out of blue simply for doing something important. this might possibly happen in germany, say. in france there is a rigid procedure to become a ph.d. candidate in the first place. – Delio Mugnolo Nov 21 '13 at 9:35
If you are paid to teach math, you are not an amateur, in any sense of this term, except maybe in a poetical sense. – Patrick I-Z Nov 21 '13 at 14:31
@PatrickI-Z: That certainly depends. At least in Germany math teachers in school have the sad reputation of not knowing anything about mathematics "past undergraduate" level. (And this seems to be a phenomenon of the last few decades, the "old guard" of teachers seems to be more knowledgable). So in the sense of the question a lot of german school teachers would qualify as amateurs. – Johannes Hahn Nov 21 '13 at 17:11

After Martin Gardner published what one mathematician claimed to be a complete list of convex pentagons that could tile the plane, amateurs (Richard James III, a computer scientist, and Marjorie Rice, who had no mathematical training beyond high school) discovered several more classes of pentagons that could tile.

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One place to read about this is Doris Schattschneider's contribution, In Praise Of Amateurs, to The Mathematical Gardner, edited by Klarner, 1981, pp 140-166. – Gerry Myerson Oct 30 '10 at 21:36
When the Math Tower at Ohio State was built, the elevator lobby on each floor was given a tiling with a mathematical meaning. I believe the top floor exhibits one of Marjorie Rice's pentagonal tilings. As I recall, Ms Rice was invited to Columbus for the dedication of the building, but could not come for reasons of health. – Gerald Edgar Oct 30 '10 at 21:49
More generally, Martin Gardner corresponded with countless amateur mathematicians who made significant contributions to recreational mathematics. Rice's achievements are probably the most spectacular, though. – Timothy Chow Oct 30 '10 at 23:54
If you're ranking amateur mathematicians by their influence on mathematics in the last 100 years, rather than by the importance of the discoveries they made themselves, Martin Gardner is clearly on the top of the list. Among other things, I've met a number of mathematicians (myself included) for whom reading Gardner's Mathematical Games column as a kid was one of the things which led them to choose to go into mathematics. – Peter Shor Oct 31 '10 at 14:32
I agree, I read parts of "The Complete Gardner" at least once a week, and reading his work was part of what persuaded me to choose to study math versus computer science. I already read Paulos, Sagan, Feynman, etc., but I don't know of anyone alive today doing what Martin Gardner did. If there is someone, I would love to know. – Eric Tressler Nov 1 '10 at 5:07

Greg Egan. He's a very renowned science fiction writer who holds a bachelor degree in mathematics. He wrote, as a coauthor, 2 articles which were published in peer-reviewed journals, one of them is with John Baez. The first one was written when he was approximately 40 years old.

There's also more eccentric example of Andrew Beal, which is much more known in the world of poker. He made however one minor conjecture in number theory for whose proof or disproof he offers $100,000. And there's also a list on wikipedia which might be worth going through. - Egan works closely with Baez and Dan Christensen, so one might argue that he doesn't meet Harry Gindi's definition of amateur. But actually, I think this is a very good example and I don't think connections with professional mathematicians make someone a professional (rather than an amateur). – Dan Ramras Oct 30 '10 at 21:30 I like some of the "primary vocations" given on that wikipedia page, e.g., Benjamin Franklin (founding father), and Abraham de Moivre (bon vivant). – Gerry Myerson Oct 30 '10 at 21:48 One guy which is missing from the list is Napoleon, who proved my <a href="mathworld.wolfram.com/NapoleonsTheorem.html">my favorite theorem in geometry</a>. – Łukasz Grabowski Oct 30 '10 at 21:55 On the list and worth mentioning explicitly IMO is Harry Lindgren, who has done some truly amazing work on geometric dissections. – Timothy Chow Oct 30 '10 at 23:52 Igor, Dan specifically mentioned Harry Gindi's definition, which is highlighted in the question statement. It includes the phrase, "does not maintain any sort of professional connection with mathematicians in the research world." I don't think Dan was minimizing Egan's contribution, I think he was pointing out a difficulty with Harry's definition. – Gerry Myerson Oct 31 '10 at 10:39 Kenneth A. Perko Jr. is a lawyer and an amateur topologist (with graduate-level training). In 1974 he found that two knots that were listed as separate knots in C. N. Little's "On knots, with a census for order 10" (1885) and similar tables, were actually identical. Mathoverflow-user Daniel Moskovich recounted earlier on this site: Little (with Tait and Kirkman) compiled his tables combinatorially. He drew all possible 4-valent graphs with some number of vertices (in this case 10), and resolved 4-valent vertices into crossings in all possible ways. He ended up with 210 knots. Then he worked BY HAND to eliminate doubles, by making physical models with string. He failed to bring these two knots to the same position, and concluded that they must be different. It took almost 100 years to find the ambient isotopy which shows that there are the same knot. The book "Knots and Links" by Dale Rolfsen, published two years after Perko's publication, still lists the knots as different, they are knots [; 10_{161} ;] and [; 10_{162} ;] in Appendix C. - Unfortunately that's not completely accurate- the history isn't really that interesting. Perko was a student of Ralph Fox at Princeton. He left math without getting his PhD (although his 1964 senior thesis was quite important), and became a lawyer. 10 years later, in his free time, he messed around with math and did some research. He has 6 papers listed on MathSciNet, all very important, and all post-1974. He could have been a complete amateur to discover the Perko pair- but he wasn't. – Daniel Moskovich Dec 31 '10 at 18:34 We have the following suggested edit (part 1): (The "rope" story is pure myth. In 1973, while I was completing the classification of 10 crossing knots, the duplication turned up quite naturally as the only undistinguishable pair. So I sketched some knot diagrams on a yellow legal pad and found out why. Sorry to disappoint, but Moskovich got it right. I was taking graduate math courses my last two years as a Princeton undergraduate, taught by the world's top knot theory topologists. – S. Carnahan Oct 15 '13 at 13:15 (part 2) Also, your second drawing is wrong. It's actually the mirror image of Rolfsen's 10-163, which some overly fastidious knot tabulators have re-named 10-162, adding a dose of confusion to an already difficult subject. --Ken Perko, October 12, 2013) – S. Carnahan Oct 15 '13 at 13:16 (part 3): Contrary to misinformation appearing elsewhere on Mathoverflow, I am not now, nor have I ever been, a Ph.D. --Ken Perko – Carlo Beenakker Oct 27 '13 at 14:05 There are many interesting discoveries made by mathematical distributed computing projects. Their discoveries don't have an impact in the same way that theorems do, but from time to time resolving a theorem boils down to computation, and most of the participants are probably interested amateurs. - @Eric : This is a nice answer in a direction I hadn't thought of, thanks! What is Ramsey@Home about? Trying to improve the known bounds for small Ramsey numbers? – Andrés Caicedo Oct 30 '10 at 19:36 @Andres: Yes. Quote from the site: "Ramsey@Home hopes to raise the lower bounds of R(3,3,3,3,3) or R(4,4,4,4,4)." – Harun Šiljak Oct 30 '10 at 20:43 Is a computer an amateur mathematician? – darij grinberg Oct 31 '10 at 18:57 @darji: phil.stackexchange.com – Eric Tressler Nov 1 '10 at 4:24 Along this line it is also interesting to mention Riecoin, a decentralized virtual currency that broke the record on November 17, 2014 for the largest prime number sextuplet: riecoin.org/Press%20release%202014-11-21.pdf – Shamisen Jan 6 '15 at 15:18$K_n$is not planar for$n \geq 5.$One may ask: what is the minimum Euler characteristic$\gamma(K_n)$among all compact orientable surfaces into which$K_n$may be embedded? It is a nice exercise to embed$K_5,K_6,$and$K_7$into the torus. The final result was that$\gamma(K_n) = 2 \lfloor \frac{n (7 - n)}{12} \rfloor.$In 1968 this theorem had been proven for "all cases except$n = 18,20,$and$23.$The proof was completed, at the end of the sixties, by Jean Mayer, a professor of French literature (!), when he found embeddings for these three values." (Surface topology, Firby and Gardiner, p. 111). - Do you mean "maximum Euler characteristic"? Also, hi! – David Hansen Nov 19 '10 at 5:30 I think Escher qualifies. See Doris Schattschneider, The mathematical side of M. C. Escher, Notices of the American Mathematical Society 57 (2010) 706-718, http://www.ams.org/notices/201006/rtx100600706p.pdf - Bill Gates co-authored the following paper in the 1970s with Christos Papadimitriou: "Bounds for sorting by prefix reversal," Discrete Mathematics 27 (1979), no. 1, 47–57, MR0534952. Not sure if Gates counts as an amateur, but he is at least a college dropout. :) The only reason I know this is because once I ran across a book or article that discusses the results in this paper and then says something like, "Yes, this is THE Bill Gates." I was almost certain the book or article was by Knuth, but now I can't find the reference in any of my Knuth books. If someone else knows the reference I'm talking about, I would be grateful if they would post it as a comment to my answer. (It now bothers me that I can't find that reference. :) ) - Mike, would it be hofprints.hofstra.edu/43 , or jstor.org/pss/10.4169/194762110X489242 ? – J. M. Sep 4 '11 at 5:10 @J.M.: No, although that paper pretty much says the same thing that the one I'm thinking of does. The reference I remember was longer ago than February 2010. Thanks anyway. :) – Mike Spivey Sep 5 '11 at 20:48 Ah yes, Bill Gates' only contribution to computer science. – JeffE May 13 '12 at 18:52 to JeffE : You mean the only positive contribution. – Jérôme JEAN-CHARLES Mar 9 at 0:16 While this is on the front page again, I wanted to make mention of Joan Taylor, who discovered an aperiodic single tile, which she published with Joshua Socolar of Duke University in 2010. This is her bio blurb as it appears on their article in The Mathematical Intelligencer: JOAN M. TAYLOR took up mathematics in 1991 at age 34 after being inspired by a magazine article on quasicrystals featuring Penrose’s rhombus tiling. She began but did not complete a degree, preferring to conduct her own research. Since then she has pursued tiling and related topics in abstract algebra and number theory including original work on constructible polygons. She likes to unwind with knitting and reading. - Robert Ammann had some extremelly important contributions to the study of aperiodic tilings, and to Quasi-crystals. - The important artists Anthony Hill and John Ernest proposed an upper bound for the crossing number of complete graphs (published by Richard Guy in 1960). Hill made other contributions to graph theory and was elected to the London Mathematical Society in 1979. - how about saul kripke? kripke-platek set theory "is used all over the place, in recursion theory and set theory, [b]oth in classical results, and in fairly recent ones." modern philosophers (of science, mathematics, language etc. -- analytical philosophers) are probably a rich source of the list you seek. many do not have above-undergrad training in math, although i would use the definition of 'amateur' that we think of when we think of the ancients. that is, people who are distinctly in another field but make contributions to mathematics as part of their work or hobby. as for other philosophers/logicians (off the top): 1. putnam 2. frank ramsey (i don't have the rep to post more links--had a bunch for this question) i know that by the time we get to someone like ramsey, everyone's like: "surely that's not an amateur mathematician" but by the definition given in the question, i think he fits. at any rate, you can probably find the names you're looking for in analytical philosophy, (mathematical/computational) economics/biology/linguistics, and so on. the problem, i suspect, will be (in addition to the definition of 'amateur' which is not too difficult in my opinion, as long as you are satisfied with it for your list) the definitions of 'important' and 'discovery'. for example, i've known about kripke's contributions for a while but i don't know, even now, whether this community considers them as important. - I would call Kripke's contributions to mathematical logic important (for what it's worth), but of course he is anything but an amateur. Rather, he is a professional from a closely related field. So I think your answer really addresses a different question, namely: "What are the important contributions of physicists, philosophers, computer scientists etc. etc. to mathematics?" – Christian Remling Jun 10 '14 at 18:06 I don't know if this really qualifies, but I would say that Scott Draves can be viewed as an amateur mathematician for inventing/discovering the fractals known as Flame fractals. His work is more towards art, but there is a decent amount of math behind to optimize the aesthetically properties of the fractals. (The "nicest" fractal dimension is 1.52, for example). - – Artem Kaznatcheev Nov 2 '10 at 0:27 There was a nice bit of mathematical modeling carried out by a professor of Law, John F. Banzhaf III. He (re)created a power index in voting, which is better discussed on its wikipage: Banzhaf power index. For Banzhaf's own work, see his paper: Banzhaf III, J. F. (1964). Weighted voting doesn't work: A mathematical analysis. Rutgers L. Rev., 19, 317. Incidentally, his particular approach was first formulated by Penrose (1946), then by Banzhaf (1965), and then yet again by Coleman (1971), thereby making it a potential candidate for the MO post about re-discovered (and re-published) works. The mathematics around voting (the pinnacle being, perhaps, Arrow's impossibility theorem) provides other examples of amateur mathematicians at work. Consider, in particular, approaches to apportionment due to United States politicians John Quincy Adams, Alexander Hamilton, Thomas Jefferson, and Daniel Webster. (I omit further discussion of these latter examples, since they are arguably not "recent.") - Eugène Ehrhart was a high school teacher when he discovered the so called Ehrart polynomial, at the age of 55. He got his PhD at the age of 60. Let$\Delta $be a polytope with integral vertices in$\bf R^d$. Then there exists a polynomial$P$such that for every integer$nP(n)$is the number of integral points in$nP$. This polynomial satisfies a duality property :$(-)^dP(-n)$is the number of integral vertices in the interior of$P$. This duality property has been interpreted has Serre duality on toric varieties by Khovanskii in the 80's. The geometric interpretation of the coefficient of$P\$ is still an open problem despite a huge literature. See ihttp://icps.u-strasbg.fr/~clauss/Ehrhart.html for a short bibliography, and https://en.wikipedia.org/wiki/Ehrhart_polynomial, for an introduction to the subject.

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The composer Kieren MacMillan has written a few papers (see his postings on the arXiv) which prove some identities involving power sums by elementary means.

I suppose that it is in collaboration with a professional mathematician, but I think that he reasonably still counts as an amateur.

(Full disclosure: He is also a friend of mine)

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The American Institute of Mathematics, a nonprofit organization, was founded in 1994 by Silicon Valley businessmen John Fry and Steve Sorenson, longtime supporters of mathematical research.

That's Fry, as in Fry's Electronics, a retail chain in California.

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Have Fry or Sorenson made any noteworthy mathematical contributions, in addition to their monetary contribution? – Nate Eldredge Nov 3 '10 at 18:24
Beal, Escher, and Gardner are all on the list (with 10+ votes each), so I figured "impact on mathematics" qualified. – Kevin O'Bryant May 13 '12 at 4:43

## protected by François G. Dorais♦Jun 10 '14 at 18:44

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