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It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)

I'd love to learn about further basic or central examples and I think such examples serve as good invitations to various areas. (Which is why a bounty was offered.)

Related MO questions: What-are-your-favorite-instructional-counterexamples, Cannonical examples of algebraic structures, Counterexamples-in-algebra, individual-mathematical-objects-whose-study-amounts-to-a-subdiscipline, most-intricate-and-most-beautiful-structures-in-mathematics, counterexamples-in-algebraic-topology, algebraic-geometry-examples, what-could-be-some-potentially-useful-mathematical-databases, what-is-your-favorite-strange-function; Examples of eventual counterexamples ;

To make this question and the various examples a more useful source there is a designated answer to point out connections between the various examples we collected.

In order to make it a more useful source, I list all the answers in categories, and added (for most) a date and (for 2/5) a link to the answer which often offers more details. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery, year? means that I am not sure if this is the correct year and ? means that I do not know the date. Please edit and correct.) Of course, if you see some important example missing, add it!

Logic and foundations: $\aleph_\omega$ (~1890), Russell's paradox (1901), Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970), The theory of Algebraically closed fields (ACF) (?),

Physics: Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator,(?) Dirac's delta function (1927), Heisenberg model of 1-D chain of spin 1/2 atoms, (~1928), Feynman path integral (1948),

Real and Complex Analysis: Harmonic series (14th Cen.) {and Riemann zeta function (1859)}, the Gamma function (1720), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in 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, The spaces $\ell_p$, $L_p$ and $C(k)$, Tsirelson spaces (1974), Cuntz algebra,

Algebra: Polynomials (ancient?), Z (ancient?) and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[\sqrt -1]$) (1832), $Z[\sqrt(-5)]$,$su_3$ ($su_2)$, full matrix ring over a ring, $\operatorname{SL}_2(\mathbb{Z})$ and SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, cyclotomic fields, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups,

Number Theory: conics and pythagorean triples (ancient), Fermat equation (1637), Riemann zeta function (1859) elliptic curves, transcendental numbers, Fermat hypersurfaces,

Probability: Normal distribution (1733), Brownian motion (1827), The percolation model (1957), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),

Dynamics: Logistic map (1845?), Smale's horseshoe map(1960). 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, The configurations of Desargues and Pappus, 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), 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?), Mobius strip (1858), Cantor set (1883), Projective spaces (complex, real, quanterionic..), Poincare dodecahedral sphere (1904), Homotopy group of spheres, Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), pseudo-arcs (1948), Discrete metric spaces, Sorgenfrey line, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)

Graph theory: The seven bridges of Koenigsberg (1735), 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(?))), Pascal's triangle (China and Europe 17th), Catalan numbers (18th century), (Fibonacci sequence (12th century; probably ancient), 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's dilemma (1950) (and also the chicken game, chain store game, and centipede game), the model of exchange economy, second price auction (1961)

Statistics: the Lady Tasting Tea (?1920), Agricultural Field Experiments (Randomized Block Design, Analysis of Variance) (?1920), Neyman-Pearson lemma (?1930), Decision Theory (?1940), the Likelihood Function (?1920), Bootstrapping (?1975)

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I think that this should be community wiki. – Loop Space Nov 11 '09 at 7:55
@Jose: Hard to say exactly. My instinct is that the kind of answers that this question will garner are those that didn't involve much actual thought, and the votes up or down will be more an assessment of whether the voter liked the example rather than whether the voter liked the answer (which, ideally, should contain an explanation of why that example shaped the discipline); both of these indicate that the answerers should not gain reputation for their answers, hence community wiki. – Loop Space Nov 11 '09 at 9:50
I've hit this with the wiki hammer. – Scott Morrison Nov 11 '09 at 19:34
I can't imagine a counterexample to the following rule: Any question whose purpose is to produce a sorted list of resources (i.e. the question includes, or should include, "one per post please") should be community wiki. – Anton Geraschenko Nov 12 '09 at 8:03
Why does this question have a bounty anyway? – Kevin H. Lin Nov 21 '09 at 17:33

136 Answers 136

Dirichlet Function is a fundamental example in Calculus where Riemann integral does not work. It is also a function which is discontinuous everywhere. The function D(x) is defined as D(x) = 1, if x is a rational number; otherwise D(x) = 0.

For more information, see for example:

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Ravi Vakil gives interesting examples in algebraic geometry: "The existence of some of these pathologies is ``common knowledge'', but I had never known what they were.".

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Although this may fall foul of the criticism that it perhaps it has not shaped a subject yet, I'll give it the benefit of the doubt that it may still shape a future subject. In a way the answer touches two answers already given: the Platonic solids and also the quaternions.

I am talking about the ADE classification, which appears in the theory of Lie algebras, finite subgroups of $SU(2)$ (McKay correspondence), representation theory of quivers (Gabriel's theorem), singularity theory (Du Val), classification of conformal field theories,...

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(...and cluster algeras) This is definitely a good example. – Gil Kalai Nov 21 '09 at 21:06
How does it touches the Quaternions? – Gil Kalai Nov 22 '09 at 22:04
Finite subgroups of the quaternions are labelled by the ADE Dynkin diagrams. It's the classification of finite subgroups of SU(2) thought of as the unit-norm quaternions. (PS: I didn't know about the cluster algebras. Thanks!) – José Figueroa-O'Farrill Nov 23 '09 at 5:43
Here is a link for ADE classification: and here are links for the A-B-C-D-E-F-G root systems and finite Coxeter groups via Dynkin diagrams – Gil Kalai Nov 23 '09 at 17:49

The arithmetics of conics with pythagorean triples is since long been used as toy model for the beautifull combination of arithmetics, analysis and geometry in the study of algebraic curves, but Lemmermeyer's "Conics - a Poor Man's Elliptic Curves" and his subsequent arxiv articles pushes the "toy" into the direction of a "fundamental example" for some fascinating issues.

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Spheres (of various dimensions) are the fundamental examples of (compact) Riemannian manifolds (or even Alexandrov spaces) of curvature > 0. Several major theorems of Riemannian geometry were motivated by the question of how to recognize a sphere. Most recently this culminated in Brendle and Schoen's proof of the differentiable sphere theorem.

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Three related fundamental examples in random matrix theory (in mathematical physics and probability) are the Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble. Much of random matrix theory has been devoted to determining the properties of these families of random matrices and proving that other families exhibit the same behaviors.

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Borromean rings

Borromean rings are important in several places. For example, they appear in computations of homotopy groups of the 2-sphere, where they corresponds to the Hopf fibration.

alt text

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The discovery of Transcendental numbers, or numbers that are not the root of any finite polynomial with rational coefficients.
Also, the proof that e and π were transcendental, the latter via the proof that ea is only algebraic for transcendental values of a (and e*i*π = -1 is algebraic, as is i, so therefore π must be transcendental). Their discovery, as well as the first explicitly created example, the Liouville number, sparked what's called "Transcendence theory".
And as it turns out, any randomly chosen real number is "almost surely" transcendental. In other words, the density of transcendental numbers among the real numbers is 1!!

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$\mathbb{Q}_{p}$. The field of p-adic numbers brings the study of local methods. Hensel's lemma is a great example. It is also interesting that p-adic integers is the projective limit of the rings $\mathbb{Z}/p^{n}\mathbb{Z}$.

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I presented the emerging list of examples over my blog and several people suggested a few more examples. I will mention them together:

Tom LaGatta proposed to add the percolation model (1854), John Sidles made several suggestions and in particular proposed several examples from Control theory such as the Nyquist criteria, Christian Blatter proposed adding the Peano curve, and Mark Meckes proposed adding the fundamental Banach spaces L_p/l_p and C(K).

Joe Malkevich proposed several basic examples of games in addition to the prisoner dilemma (chicken, chain store game, and centipede) and the Gale-Shapley model of two-sided market model (the model in the famous Gale-Shapley stable marriage theorem). I thought that we should probably add a basic economic model of exchange markets (like the Arrow-Debreu model).

I also thought the configurations of Desargues and Pappus should be added.

There was also some critique on the classification of examples, and an interesting suggestion By Michael Nielsen that "Distilled and expanded, it could form the basis for an excellent book. Perhaps: 'Examples from the book'." (This refers to Aigner and Ziegler's book "Proofs from the book". (In fact, a similar idea by Ziegler and me have motivated the question itsef.)

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The pseudo-arc in continuum theory.

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The cyclotomic field $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ is the most basic example of a field extension in which splitting of primes depends on an obvious congruence condition. Specifically, if $\ell$ is another prime, then the Frobenius of $\ell$ is $\ell \mod p \in (\mathbb{Z}/p)^\times = \mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$. In particular, $\ell$ splits in the field iff its Frobenius is trivial, and this is true iff $\ell \equiv 1 \mod p$. We can then relate other congruences to splitting in subfields of $\mathbb{Q}(\zeta_p)$, etc. The theorems of global class field theory show that this basic concept holds in a very general case, although the general case is much harder to prove. This basic example, does, however, motivate the ideas in class field theory, which have greatly influenced modern number theory and related areas. (As an added note, the fact that the Artin reciprocity law is true for cyclotomic fields is actually a key ingredient in the proof for general abelian extensions!)

Answered by Davidac897

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Heisenberg model of 1-D chain of spin 1/2 atoms, solved exactly by Bethe in 1931, is where Bethe Ansatz was born, and with it the field of integrable models in statistical and quantum mechanics.

Answered by Mio

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The semicircular law and the Marchenko-Pastur distribution are fundamental examples of probability distributions in random matrix theory.

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The hyperfinite $\mathrm{II}_1$ factor $\mathcal{R}$ and its ultrapower $\mathcal{R}^{\omega}$ are fundamental examples in von Neumann algebras and Connes' embedding problem.

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The free group factors $L(\mathbb{F}_{n})$, which are the closer in the weak operator topology of the left regular representation of the free group $\mathbb{F}_n$, are fundamental examples in von Neumann algebras. The isomorphism question is the root of the so important Free Probability theory of Voiculescu.

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Schwarzschild metric as a prototype of black hole was a fundamental example in the development of General Relativity (for instance, it is often referred to when "defending" the ADM mass as a natural concept of mass in General Relativity).

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The example that launched category theory: (co)homology, for example simplicial homology and Čech cohomology. The various maps linking (co)homology groups for different 'resolutions' of a topological space (by triangulation or open sets resp.) were I think the first examples of natural transformations. This necessitated defining functors, and hence defining categories.

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Linear Algebra: In linear algebra the symmetric group $S_n=Bij(\{1,,n\})$ is a classical and important example if you learn basics in group theory.

Operator-algebra/ functional-analysis: Fundamental examples of $C^*$-algebras are $C(X)$ ( continuous functions on a compact Haudorff space X), $C_0(X)$ ( continuous functions vanishing at infinity on a localcompact Haudorff space X), $B(H)$ (bounded linear maps on a Hilbert space H). By Gelfand-Naimark you know a lot of abstract $C^*$algebras if you know these concrete examples.

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Margulis's expanders: This class of 8 regular graphs is the first explicit example for a family of expanders. The vertices are pairs of integer modulo m, the neighbors of (x,y) are (x+y,y), (x-y,y), (x,y+x), (x,y-x), (x+y+1,y), (x-y+1,y), (x, y+x+1), (x,y-x+1). All operations are modulo m.

Expanders were first discovered and constucted probabilistically by Pinsker. The Ramanujan graphs of Lubotzky, Philips and Sarnak are expanders with extremely good properties. This paper by Hoory, Linial and Wigderson contains much more information.

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Second price auction (or Vickrey auction): an auction in which the bidder who submitted the highest bid is awarded the object being sold and pays a price equal to the second highest amount bid.

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It seems close to the classic pie division procedure. The cutter cuts and the other chose first the part he wants. – ogerard May 13 '10 at 17:12

Two unrelated examples: The configuration of 27 lines on a cubic surface ; (See also here and here) The regular heptadecagon (17 sides polygon) and its geometric construction.

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The Delaunay triangulation is fundamental in computational (Euclidean) geometry. For a finite point set S in general position, it can be defined in several ways: (1) as the unique triangulation in which every simplicial cell is Delaunay (i. e., its circumsphere does not contain any points of S in its interior), (2) as the uniqe triangulation in which every facet (of any dimension) of every simplicial cell is Delaunay (meaning it has some empty circumsphere), or (3) as the dual of the Voronoi diagram (which is also fundamental). In the plane, the Delaunay triangulation has the additional property of maximizing the smallest angle of all its triangles, among all triangulations.

The Delaunay triangulation is usually the most obvious candidate for "the right" triangulation of a given point set, and most simplicial mesh-generating methods seem to be based on it. It doesn't hurt that there are reasonably fast and elegant algorithms for constructing it (very fast in the plane, but unfortunately (and necessarily) exponential in the dimension in the worst case).

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The solution of Kirkman's Schoolgirl Problem is the archetypal example of a resolvable triple system. This example essential shaped the entirety of Design Theory.

We might also consider Euler's 36 Officers Problem to be one of the fundamental counter-examples within this field. Answer by Disonnant

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I suppose the discrete metric space is a crucial example in the metric spaces theory and in the introductory mathematical analysis. It shows many aspects and pathological behavior of metric spaces in general.

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Well, it is rather dull :D – Mariano Suárez-Alvarez Jan 10 '10 at 20:51

Taking introductory topology, I got the impression that the real line is the fundamental example of a topological space. I wouldn't be surprised if the open and closed intervals of $\mathbb{R}$ were the prototypical examples of open and closed sets, and I think many important topological properties---including compactness, connectedness, and Hausdorffness---first arose because you need them to prove obvious facts about $\mathbb{R}$ and its subsets.

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There are several examples that I would regard as "too fundamental" for the list like: 0, 1,2, $\sqrt 2$, the real numbers, the natural numbers, the prime numbers, the triangle. I also consider Alef_0 and Alef as "too fundamental" and chose Alef_\omega to start the set theory examples. – Gil Kalai Feb 20 '10 at 7:33
Oops! Sorry about that! – Vectornaut Feb 20 '10 at 18:58
Gil Kalai: It might help to indicate that in your question. – Jonas Meyer Feb 20 '10 at 21:19
Dear Vectornaut, no problems, this was my idea but other people had different ideas. – Gil Kalai Feb 26 '10 at 9:36
Actually, the real line might be a little too simple to constitute a fundamental example in Topology, though it is very helpful to get a good grasp on it nevertheless. – Thierry Zell Apr 18 '11 at 18:47

To make this question and the various examples a more useful source this is a designated answer to point out connections between the various examples we collected. please indicate only strong, definite, nontrivial, and clear connections.

1) The Petersen graph is obtained by identifying antipodal vertices and edges in the graph of the dodecahedron - one of the five platonic solids. Such an identification gives a polyhedral complex realizing the real projective plane. Applying this operation to the icosahedron leads to a 6-vertex triangulation of the real projective plane.

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I think no one's pointed Lorenz equations.

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Can you explain or add a link, please? – Gil Kalai Feb 21 '11 at 19:54

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