# Fundamental Examples

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.)

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)

• I'm not so sure about that... in my opinion, not every soft-question should be community wiki! Why exactly change this one? Nov 11, 2009 at 9:10
• @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. Nov 11, 2009 at 9:50
• 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. Nov 12, 2009 at 8:03
• If it is both community wiki, and it has an open bounty, how does that work? Nov 21, 2009 at 17:10
• Dear Qiaochu, There are various interpretations of the meaning of "examples" for this question and it is nice to see them all. Nov 24, 2009 at 9:40

The Harmonic series 1+1/2+1/3+1/4+1/5+... and for that matter the Riemann zeta function.

Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

Recursion theory has the halting problem. From it, you obtain the first example of an interesting Turing degree, and generalizing it, you get the Turing jump operator.

The halting problem also plays an important role in (the boundary of) computational complexity theory, since it's one of the main tools for demonstrating the insolubility of computational problems.

The p-adic numbers. Discovered only in 1897.

Dirac's Delta function

It was introduced by 1927 by Paul Dirac. Can be regarded as an important example towards the theory of distribution.

For examples of distributions to keep in mind see this question.

• Though of "central" importance in the space of distributions, the delta function is rather simple and by itself doesn't give rise to the theory at large. (It can be regarded as a measure.) I'd think that the fundamental solution to the wave equation (in all dimensions, described in the question linked above) would be a better candidate, being the prime impetus to Hadamard's partie finie and Riesz's method of analytic continuation. Dec 24, 2017 at 16:43
• I thought Heaviside was playing with Dirac's function (and his own function) before Dirac.
– lcv
Mar 4, 2019 at 18:17

In the theory of holomorphic functions of several variables, Hartogs's theorem that any holomorphic function on a punctured open set of $\mathbb C^n$ ($n\geqslant 2$) can holomorphically be continued through the deleted point really started the subject and is still the most spectacular divide between $n \geqslant 2$ and $n=1$ [where it is completely false: on $\mathbb C^\ast$ look at 1/z or, worse, exp(1/z): these functions clearly can't be continued holomorphically through zero]

• It is interesting, but this is a theorem, not an example right? Nov 11, 2009 at 8:56
• Perhaps this answer refers to a class of examples, the punctured open sets of C^n when n>1, of domains which are not domains of holomorphy. Nov 11, 2009 at 9:52
• Dear Gil and Jonas:thanks for the comments and, yes, you are both right. Hartogs's example-theorem shows that there exist in C^n domains which are not regions of holomorphy,a phenomenon impossible in dimension one.This launched the notion of pseudoconvex domains (which exclude Hartogs-type extensions of holomorphic functions) and ultimately led to the concept of Stein manifolds, central in complex geometry (the analogues of affine varieties in algebraic geometry) Nov 11, 2009 at 10:28
• I see, I guess it is a concrete example! (and even a popular one.) Nov 11, 2009 at 14:09

Smale's horseshoe map in dynamical systems.

The full matrix rings of any order over another ring (and their direct union of row-finite, column-finite matrices) are a fundamental example for Noncommutative Ring Theory: they are simple enough to be easily understood "in a glimpse" but complex enough to highlight many interesting concepts of the theory.

sl2 is the fundamental example of a finite dimensional simple Lie algebra. It forms the basis of the Cartan-Killing classification of complex semisimple Lie algebras, since all of the others can be made by gluing (in some sense) copies of sl2 together. Its representation theory is both straightforward and illuminating, since it points the way to the general theory of highest-weight representations.

For the same reason, SU(2) is the fundamental example of a nonabelian compact Lie group. By Borel-Weil, its irreducible representations can be geometrically realized as the spaces of sections of complex line bundles on the 2-sphere (somewhat easier to handle than a general flag variety).

Various families of orthogonal polynomials can probably be considered fundamental in some way, but here I want to single out the Chebyshev polynomials, which seem to be the most miraculous. They have explicit formulas (T_n(x) = cos(n arccos(x)), for example), and they are closed under composition. Also, the polynomial of a given degree that has given bounds on an interval and grows as fast as possible outside the interval is a translated and scaled Chebyshev polynomial. The Chebyshev polynomials are fundamental in numerical analysis. Among their uses:

• Accurate numerical integration
• Accurate polynomial interpolation (if one is allowed to choose the evaluation points)
• Sometimes the mere existence of polynomials with the boundedness/growth properties of Chebyshev polynomials is useful, e. g., in analyzing the convergence of the method of conjugate gradients

(A little bit more on the relation to conjugate gradients: n steps of CG can be interpreted as optimizing something over the space of polynomials of degree n. The optimality of the Chebyshev polynomials in the sense of being as small as possible on [-1, 1] (stated above in a different form) is not quite equivalent to what CG requires, but it's close enough to imply that the optimal polynomial in the sense of CG must be very good (because it is at least as good as the Chebyshev polynomial).)

$\operatorname{SL}_2(\mathbb{Z})$ and its action on the hyperbolic plane. It is the "minimal" example of mapping class groups and arithmetic groups. And one can already see a lot of the general behavior.

• and the whole business of modular forms. Part of the same story, but less familiar now, has to do with differential equations (Fuchs/Klein/Poincare). Dec 22, 2017 at 3:04

In combinatorics, the (Pascal) Binomial Triangle more or less started the entwining of combinatorics, probability and algebra.

The other two most seminal and ubiquituous triangles of numbers are the Stirling (duals between 1st and 2nd sort, permutation world and set world, the most often generalized family of numbers) and the Eulerian numbers (permutations seen as words on ordered alphabet), the reference statistics on permutations.

For links between combinatorics and number theory, I think the first prize would be the Bernoulli numbers.

The associahedron or Stasheff polytope.

(source: wikimedia.org)

In operator theory, the unilateral shift. This operator is not only the fundamental example of an isometry on a Hilbert space, it can also be shown that this operator contains all other operators, in some sense.

Hyperbolic geometry, which led to the understanding that there are non Euclidean geometries and to many other developments.

Gödel's Constructible Universe, L, is the fundamental example of inner model theory. L was the first inner model, and is the minimal one. One could argue that the point of inner model theory is to build L-like models that do things L cannot.

Some people may disagree that this is an example per se, but I'd put up the Feynman path integral (see Wikipedia), because:

• it provided a completely new physical picture of quantum mechanics

• it led to systematic development of quantum field theory and string theories, both of which have had led to enormous synergistic growth in mathematics

• it uncovered a fundamental similarity between of stochastic processes and deterministic quantum dynamics

• it used the connection between Lie algebras and Lie groups in new and unexpected ways

• questions about the path measure have stimulated much development in measure theory and analysis

• tricks like the Wick rotation not only relate statistical mechanics and quantum mechanics (and the corresponding field theories) to each other, but also have stimulated further research in applications of analytic continuation

...and probably more that I am unaware of.

The torus is THE example in many branches of math. Algebraic Topology: it is the example where you can compute explicitly its fundamental group as well as its covering spaces, universal cover and everything else. Algebraic geometry: provided it is smooth, is a cubic which is not rational. It is a compact Lie group (and then a non so trivial example of a trivial tangent bundle). It is compact in $\mathbb{R}^4$ and has zero curvature (Riemannian geometry). It is a Riemann Surface. Actually one can compute very explicitly the moduli space of such Riemann surfaces as well as (very explicitly) the mapping class group (which is a beautiful example). It is an elliptic curve, an abelian variety... and it has all those properties I don't know about...meaning, an endless amount of properties.

• But in what way has the torus shaped any of these subjects? It's certainly a good example demonstrating many of the features, but I don't see (from your answer) that it has played a significant role in shaping them. Nov 11, 2009 at 9:51
• "Algebraic Topology: it is the example where you can compute explicitly its fundamental group as well as its covering spaces, universal cover and everything else." I don't see how this distinguishes the torus from any other closed surface, from the algebro-topological point of view. Indeed, if you're interested in the fundamental group, the fact that it's abelian in this case makes it highly unrepresentative.
– HJRW
Nov 11, 2009 at 17:36

The Heisenberg group: the group of 3 by 3 upper triangular matrices of the form

$$\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}.$$

When making a first step to non-commutativity? Or beyond planar models? Try the Heisenberg group.

• I'm quite fond of this group, but surely the first step to non-commutativity (in groups) is $S_3$ or $D_8$ or $2\times2$ real invertible matrices. Jun 10, 2016 at 12:39

Tsirelson space (see the Wiki entry for quick facts) in Banach space theory was seminal, in that it generated a stream of further refined and specialized counterexamples (most notably in the work of Casazza, Odell, Schlumprecht, and culminating in the famous examples of hereditarily indecomposable spaces by Gowers and Maurey (apologies for the countless others not mentioned here, but the list would be too long). Several long-standing conjectures (even dating back to Banach) have been proved or disproved using these examples, and the flow is not over even now.

The trefoil knot in knot theory.

Fundamental examples of coalgebraic structures:

• Homology-of-a-space Hopf algebra
• Hopf algebra representing a linear algebraic group
• Group rings
• Spheres as cogroups in the homotopy category of spaces

In geometric/combinatorial group theory, Grigorchuk's 2-group is a fundamental example. It originally was constructed as a particularly elegant infinite finitely generated torsion group. Then Grigorchuk showed it had intermediate growth, answering Milnor's problem. For a time it was seen as the universal counterexample in group theory. But now it has spawned a theory of groups acting on rooted trees, self-similar groups and branch groups. Pierre de la Harpe has an entire chapter of his book devoted to this group.

the Fischer-Griess Monster.

see:

http://en.wikipedia.org/wiki/Monster_group

Theory of Algebraically Closed Fields (ACF for short) is an important example in Model Theory. This was a motivation example to develop theory of stable and simple theories (leading, eventually, to a beautiful proof of Mordell-Lang conjecture by Ehud Hrushovski).

The rational parametrization of the locus of the equation $X^2+Y^2=1$ by $(\frac{t^2-1}{t^2+1},\frac{2t}{t^2+1})$. It can be viewed geometrically by taking a line that intersects the unit circle at one rational point and then considering all possible (rational) slopes of the line (including infinity), which are in correspondence with (rational) points of the circle. This is the most basic example of using a geometric idea to find solutions to a diophantine equation, and it leads to very deep mathematics.

• It's also a nice way to prove the trigonometric change of variable formulas with $t=\tan(\alpha/2)$. When I saw those in high-school, I believe we used brute force instead! Apr 18, 2011 at 18:44

The irrational rotation on the torus $\mathbb{T^2}$: $T_t (z,w) = (e^{2 \pi i \alpha }z, e^{2 \pi i \beta} w)$, where $\alpha, \beta \in \mathbb{R}, \frac{\alpha}{\beta} \notin \mathbb{Q}$. It is perhaps the first nontrivial example of an ergodic dynamical system. By considering its orbits $t \mapsto T_t (z,w)$, one gets examples of one-to-one immersions (with dense image) which are not embeddings, immersed submanifolds which are not closed/regular submanifolds, Lie subgroups which are not closed subgroups, non-Hausdorff quotient space and similar phenomena.

I think the seven bridges of Königsberg initiated a lot of modern discrete mathematics.