It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples?
I’d love to learn about further basic or central examples and I think such examples serve as good invitations to various areas.
I asked this question over mathoverflow and it yielded around 100 examples. They are not equally fundamental and they are not equally suitable to be regarded as “examples,” but overall it is a very good list. If you see some important example missing please, please add it. Here are the examples classified to areas. (Of course, sometimes, the same example may fit several areas.)
Logic and foundations:
Physics:
Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator (?), Dirac’s delta function (1927), Feynman path integral (1948),
Real and Complex Analysis:
Harmonic series (14th Cen.) and Riemann zeta function (1859), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in $latex C^n$ (Hartog’s theorem *1906 ?)
Partial differential equations:
Laplace equation (1773), the heat equation, wave equation, Navier-Stokes equation (1822),KdV equations (1877),
Functional analysis:
Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra:
Z and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers (![Z[\sqrt -1]](https://s0.wp.com/latex.php?latex=Z%5B%5Csqrt+-1%5D&bg=ffffff&fg=333333&s=0&c=20201002)
Number Theory:
Conics and pythagorean triples (ancient), Fermat equation (1637), eliptic curves, Fermat hypersurfaces,
Probability:
Normal distribution (1733), Brownian motion (1827), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),
Dynamics:
Logistic map (1845?), Mandelbrot set (1978/80) (Julia set), cat map, (Anosov diffeomorphism)
Geometry:
Platonic solids (ancient), the Euclidean ball (ancient), The configuration of 27 lines on a cubic surface, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle (19th century), Fano plane (early 20th century ??), cyclic polytopes (1902), Delaunay triangulation (1934) Leech lattice (1965), Penrose tiling (1974), the noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)
Topology:
Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Cantor set (1883), Poincare dodecahedral sphere (1904), Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in 
Graph theory:
Petersen Graph (1886), two edge-colorings of K_6 (Ramsey’s theorem 1930), K_33 and K_5 (Kuratowski’s theorem 1930), Tutte graph (1946), Margulis’s expanders (1973) and Ramanujan graphs (1986),
Combinatorics:
Tic-tac-toe (ancient Egypt(?)) , The game of nim (ancient China(?)), Fibonacci sequence (12th century; probably ancient), Catalan numbers (mid 19th century), Kirkman’s schoolgirl problem (1850), surreal numbers (1969), alternating sign matrices (1982)
Algorithms and Computer Science:
Newton Raphson method (17th century), Turing machine (1937), RSA (1977), universal quantum computer (1985)
Social Science:
Prisoner dillema (1950), second price auction (1961)
The (partial) links are to the answers over MO which often have more information and external links (additions are moset welcome). I tried to find dates for the various examples and this was not easy. Corrections and additions are welcome! A date with a * like *1970 refers to a date when this example had become important in view of a certain development. (E.g 3SAT (*1970) refers to the discovery of P,NP, and NP-completeness.)
Combinatorics and more 
Gil,
Probability should include Percolation and the Ising Model too. One can argue that these are more in the realm of Mathematical Physics, but that isn’t an option in your list!
One could also argue that Brownian Motion is part of Physics, whereas the Wiener Process/Measure is the concept properly belonging to Probability.
Dear Tom,
The Ising model is part of the list (In the “physics” category). Certainly the classification into areas is not perfect, not by itself and not in the way I associate examples to areas.
Gil – This is a wonderful list. From the quality of the examples I already know that are on this list, I’m sure the other examples are also excellent. Distilled and expanded, it could form the basis for an excellent book. Perhaps: “Examples from the book”.
Gil, please let me agree that this is a terrific list!
From an engineering point of view, control theory was revolutionized by stability criteria arising from complex analysis (Routh-Hurwitz, Root-Locus, and Nyquist criteria, for example) … Norbert Weiner wrote a novel (The Tempter) in which this revolution is one theme.
More recently, shotgun sequencing algorithms (with roots in combinatoric theory) have revolutionized the way genomic data is acquired.
Now synthetic biology and synthetic chemistry are creating an ongoing revolution in wet-bench biology and chemistry (and even physics too) … these tools have deep roots in symplectic and metric geometry.
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Dear John, You are right that control theory is not represented in the list and more generally I would love to see a few more examples coming from applied mathematics. (There is a very easy fix to this problem: add an example). Meanwhile, the wikipedia article for control theory and for Nyquist stability criterion seem quite good.
Dear Michael, Indeed after the amazing success of Aigner and Ziegler’s “Proofs from the book” Guenter Ziegler and I played a little with the idea of “A book of examples” and the MO question is related to this old idea with Guenter. Going ahead with such a book will require much work: selecting a few dozens of basic examples, studying the historical background, finding a way to describe the interconnections between them, finding the right level of explanation, polishing everything, adding drawings and cartoons,…
But the rough unpolished MO list also has some charm.
Probably, a little more can be done there. As one user, Andrew Stacy, said, ideally, an answer should contain a short explanation of why that example shaped the discipline. (Perhaps also connections with other examples, and of course links.) At present, everybody can edit every answer. But, as I said, the unpolished, incomplete, present form is sort of nice and can be useful.
What about counter-examples, like a Peano curve or a ring without unique prime factorization?
In functional analysis, you can’t possibly list Tsirelson’s space but leave out L_p/l_p and C(K) spaces!
Dear Christian, sure, counter examples are very good examples;![Z[\sqrt {-5}]](https://s0.wp.com/latex.php?latex=Z%5B%5Csqrt+%7B-5%7D%5D&bg=ffffff&fg=333333&s=0&c=20201002)
is on the list, Peano curve is not there yet. Dear Mark, I agree and there is an easy fix for this.
It also occured to me that the Pappus and Desargues configurations should be mentioned. Of course, when you zoom in on a single area you often see more important examples.
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I too think this compendium is very valuable. In terms of thinking of the role of fundamental examples and the way that they have shaped various parts of mathematics I agree that “grain size” and the name used for the area is important. For example, from the beginning of the list on MO I did not like the “social science” designation. One could easily have a separate list of fundamental examples which have arisen when mathematics has been applied in other disciplines. So in the area of “business” the linear programming and integer programming examples/models have been transformational. I would prefer to see Social Science changed to game theory, and a separate list of applications areas for mathematics.
One beautiful and for me fundamental example (does this show up in the MO list somewhere?) is the two-sided market model for which David Gale and Lloyd Shapley developed their justly celebrated deferred acceptance algorithm. This example has resulted in lots of wonderful “pure” mathematics and applications ranging from the resident matching program in the US to school choice models. So this example in my opinion belongs as fundamental in mathematical economics and game theory too.
Now if one lists the social sciences/behavioral sciences separately, economics, anthropology, psychology, political science, different fundamental examples come to mind, and sometimes they are the same as things that showed up in the other list. Thus, in game theory, I would list prisoner’s dilemma, chicken, chain store game, and centipede, while under political science my list would not include these last two.
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Dear Joe, I agree that some model of economics (for example the basic model of exchange economics) should be included. Of course, we can have a special math overflow question devoted to example on a specific topic like examples from game theory. (But I am not sure how much such a question will be welcome; there is a certain sentiment in MO against such questions in spite of them being very successful in terms of MO internal quantitative measures.)
At some point I will expand the answers to the original questions to include some suggestions from this post. Actually the Gale Shapley example is a good answer to this very nice MO question: http://mathoverflow.net/questions/14782/what-are-some-applications-of-other-fields-to-mathematics
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I looked up examples of fundamental and not all of this.