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## 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. – Franklin Oct 30 2010 at 14:49
@Franklin, I believe this not to be a history of math research question. It appears to be a request for advice. Besides, if one simply wants a list of examples, we do have one already: mathoverflow.net/questions/3591/… – Ryan Budney Oct 30 2010 at 15:03
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 2010 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 2010 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 2010 at 16:23

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 2011 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 Magnusson Apr 23 2011 at 8:47

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.

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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.

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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. :) )

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Mike, would it be hofprints.hofstra.edu/43 , or jstor.org/pss/10.4169/194762110X489242 ? – J. M. Sep 4 2011 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 2011 at 20:48
Ah yes, Bill Gates' only contribution to computer science. – JeffE May 13 2012 at 18:52

$K_n$ 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).

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 Do you mean "maximum Euler characteristic"? Also, hi! – David Hansen Nov 19 2010 at 5:30

Kenneth A. Perko Jr. is a lawyer and an amateur topologist. In 1974 he found -while manipulating loops of rope on his living floor- 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 actual 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.

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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 2010 at 18:34

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 2010 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 2012 at 4:43

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).

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 a link: en.wikipedia.org/wiki/Scott_Draves – Artem Kaznatcheev Nov 2 2010 at 0:27

Robert Ammann had some extremelly important contributions to the study of aperiodic tilings, and to Quasi-crystals.

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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

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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.

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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 2010 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 2010 at 21:48
One guy which is missing from the list is Napoleon, who proved my <a href="mathworld.wolfram.com/… favorite theorem in geometry</a>. – Łukasz Grabowski Oct 30 2010 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 2010 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 2010 at 10:39

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.

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@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? – Andres Caicedo Oct 30 2010 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 2010 at 20:43
Is a computer an amateur mathematician? – darij grinberg Oct 31 2010 at 18:57
@darji: phil.stackexchange.com – Eric Tressler Nov 1 2010 at 4:24

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 2010 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 2010 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 2010 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 2010 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 2010 at 5:07