# 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

I suppose that $\aleph_0$ and $2^{\aleph_0}$ are, like the natural numbers and the unit ball, "too fundamental" as they are fundamental for mathematics as a whole. But the $\omega^{\mathrm{th}}$ cardinal, $\aleph_{\omega}$ already suits us as a fundamental object in set theory and infinite combinatorics.

Can you show that the continuum is NOT $\aleph_{\omega}$ ?

• Yes you can show that the continuum is not $\aleph_{\omega}$. In what sense is $\aleph_{\omega}$ fundamental in set theory and infinitary combinatorics? Apr 19, 2011 at 6:29
• Maybe because it is the simplest example of a singular cardinal... Sep 15, 2011 at 2:13

to understand curves, first study the abel map. and then the torelli map. [perhaps I should expand this rather succinct answer.]

8320 Spring 2010, day one Introduction to Riemann Surfaces

We will describe how Riemann used topology and complex analysis to study algebraic curves over the complex numbers. [The main tools and results have analogs in arithmetic, which I hope are more easily understood after seeing the original versions.]

The idea is that an algebraic curve C, say in the plane, is the image by a holomorphic map, of an abstract complex manifold, the Riemann surface X of the curve, where X has an intrinsic complex structure independent of its representation in the plane. Then Riemann's approach to classifying all complex curves, is to classify such Riemann surfaces, and then for each such surface to classify all maps from it to projective space. Briefly, the Torelli maps classifies complex surfaces, and the abel maps classify all projective models of a given complex surface.

More precisely, we will construct two fundamental functors of an algebraic curve:

i) the Riemann surface X, and

ii) the Jacobian variety J(X), and

natural transformations X^(d)--->J(X), the Abel maps, from the “symmetric powers” X^(d) of X, to J(X).

The Riemann surface X

The first construction is the Riemann surface of a plane curve: {irreducible plane curves C: f(x,y)=0} ---> {compact Riemann surfaces X}.

The first step is to compactify the affine curve C: f(x,y) =0 in A^2, the affine complex plane, by taking its closure in the complex projective plane P^2. Then one separates intersection points of C to obtain a smooth compact surface X. X inherits a complex structure from the coordinate functions of the plane. If f is an irreducible polynomial, X will be connected. Then X will have a topological genus g, and a complex structure, and will be equipped with a holomorphic map ƒ:X--->C of degree one, i.e. ƒ will be an isomorphism except over points where the curve C is not smooth, e.g. where C crosses itself or has a pinch.

This analytic version X of the curve C retains algebraic information about C, e.g. the field M(X) of meromorphic functions on X is isomorphic to the field Rat(C) of rational functions on C, the quotient field k[x,y]/(f), where k = complex number field. It turns out that two curves have isomorphic Riemann surfaces if and only if their fields of rational functions are isomorphic, if and only if the curves are equivalent under maps defined by mutually inverse pairs of rational functions.

Since the map X--->C is determined by the functions (x,y) on X, which generate the field Rat(C), classifying algebraic curves up to “birational equivalence” becomes the question of classifying these function fields, and classifying pairs of generators for each field, but Riemann’s approach to this algebraic problem will be topological/analytic.

We already can deduce that two curves cannot be birationally equivalent unless their Riemann surfaces have the same genus. This solves the problem that interested the Bernoullis as to why most integrals of form dx/sqrt(cubic in x) cannot be “rationalized” by rational substitutions. I.e. only curves of genus zero can be so rationalized and y^2 = (cubic in x) usually has positive genus.

The symmetric powers X^(d)

To recover C, we seek to encode the map ƒ:X--->C, i.e. ƒ:X--->P^2, by intrinsic geometric data on X. If the polynomial f defining C has degree d, then each line L in the plane P^2 meets C in d points, counted properly. Thus we get an unordered d tuple of points L.C, possibly with repetitions, on C, hence when pulled back via ƒ, we get such a d tuple called a positive “divisor” D = ƒ^(-1)(L) of degree d on X. (D = n1p1+...nk pk, where nj are positive integers, n1+...nk = d.)

Since lines L in the plane move in a linear space dual to the plane, and (if d ≥ 2) each line is spanned by the points where it meets C, we get an injection P^2*--->{unordered d tuples of points of X}, taking L to ƒ^(-1)(L).

If X^d is the d - fold Cartesian product of X, and Sym(d) is the symmetric group of permutations of d objects, and we define X^(d) = X^d/Sym(d) = the “symmetric product” of X, d times, then the symmetric product X^(d) parametrizes unordered d tuples, and inherits a complex structure as well.

Thus the map ƒ:X--->C yields a holomorphic injection P^2*--->Π of the projective plane into X^(d). I.e. the map ƒ determines a complex subvariety of X^(d) isomorphic to a linear space Π ≈ P^2*. Now conversely, this “linear system” Π of divisors of degree d on X determines the map ƒ back again as follows:

Define ƒ:X--->Π* = P^2** =P^2, by setting ƒ(p) = the line in Π consisting of those divisors D that contain p. Then this determines the point ƒ(p) on C in P^2, because a point in the plane is determined by the lines through that point. [draw picture] Thus the problem becomes one of determining when the product X^(d) contains a holomorphic copy of P^2, or copies of P^n for models of X in other projective spaces.

The Jacobian variety J(X) and the Abel map X^(d)--->J(X).

For this problem, Riemann introduced a second functor the “Jacobian” variety J(X) = k^g/lattice, where k^g complex g -dimensional space. J(X) is a compact g dimensional complex group, and there is a natural holomorphic map Abel:X^(d)--->J(X), defined by integrating a basis of the holomorphic differential forms on X over paths in X.

Abel collapses each linear system Π ≈ P^n* to a point by the maximum principle, since the coordinate functions of k^g have a local maximum on the compact simply connected variety Π. Conversely, each fiber of the Abel map is a linear system in X^(d). Existence of linear systems Π on X: the Riemann - Roch theorem.

By dimension theory of holomorphic maps, every fiber of the abel map X^(d)--->J(X) has dimension ≥ d-g. Hence every positive divisor D of degree d on X is contained in a maximal linear system |D| , where dim|D| ≥ d-g. This is called Riemann’s inequality, or the “weak” Riemann Roch theorem.

The Roch part analyzes the relation between D and the divisor of a differential form to compute dim|D| more precisely. Note if D is the divisor cut by one line in the plane of C, and E is cut by another line, then E belongs to |D|, and the difference E-D is the divisor of the meromorphic function defined by the quotient of the linear equations for the two lines.

If D is a not necessarily positive divisor, we define |D| to consist of those positive divisors E such that E-D is the divisor of a meromorphic function on X. If there are no such positive divisors, |D| is empty and has “dimension” equal to -1. Then if K is the divisor of zeroes of a holomorphic differential form on X, the full Riemann Roch theorem says:

dim|D| = d-g +1+dim|K-D|, where the right side = d-g when d > deg(K).

This sketch describes the abel maps and their relation to the RRT. The assignment X-->J(X) is the Torelli map, and classifies X by the numerical data in the lattice defining J(X), i.e. periods of integrals of the first kind on X. This assignment gives birth to the whole subject of "moduli" as numerical invariants of complex or geometric structure.a

The Thompson groups, as wikipedia said, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory. For example it is a finite generated finite presented torsion free group with infinite cohomology dimension. For more information see: http://en.wikipedia.org/wiki/Thompson_groups

In Combinatorics, the Dyck paths (systems of bracketings, theory of languages, group codes) and Motzkin paths (systems of bracketings, theory of languages, continued fractions). Particular cases of
Lattice paths.

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.

• It seems close to the classic pie division procedure. The cutter cuts and the other chose first the part he wants. May 13, 2010 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.

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).

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

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.

• Well, it is rather dull :D Jan 10, 2010 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.

• 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. Feb 20, 2010 at 7:33
• Oops! Sorry about that! Feb 20, 2010 at 18:58
• Gil Kalai: It might help to indicate that in your question. Feb 20, 2010 at 21:19
• Dear Vectornaut, no problems, this was my idea but other people had different ideas. Feb 26, 2010 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. Apr 18, 2011 at 18:47

Dirichlet's theorem is the first use of analysis to prove a number theoretic result which does necessarily seem analytic. His proof leads to a lot of ideas about distributions of primes, many of which used analysis. It even leads to an analytic proof of one of the inequalities in class field theory, a result which can also be proved using a good deal of cohomology and which is therefore not exclusively analytic. (I am not counting the prime number theorem, since that is an asymptotic result and thus reeks of analysis as soon as it is conjectured. It was also proven later.)

The parallel-or functional is a fundamental example in denotational semantics.

The field of Hyperbolic Systems of Conservation Laws has a paradigm : the Euler equations of an inviscid compressible gas. For those interested, these are the conservation of mass, momentum and energy: $$\partial_t\rho+{\rm div}(\rho u)=0,$$ $$\partial_t(\rho u)+{\rm Div}(\rho u\otimes u)+\nabla p(\rho,e)=0,$$ $$\partial_t(\frac12\rho|u|^2+\rho e)+{\rm div}((\frac12\rho|u|^2+\rho e+p)\rho u)=0,$$ where $\rho$ is the mass density, $u$ the flow velocity, $e$ the specific internal energy and $p$ the pressure, given by an equation of state.

Riemann wrote a deep paper on the $1$-dimensional isothermal (drop the last equation, take $p=A^2\rho$) case, after which the Riemann problem and the Riemann invariants have been coined.

The lady tasting tea: https://en.wikipedia.org/wiki/Lady_tasting_tea was a fundamental example introducing the idea of significance tests.

The Hawaiian earring, the union of a null sequence of circles joined at a common point.

1) Shows that a connected, locally path connected metric space can fail to be locally contractible.

2) (Much less obvious). Amplifies the failure of TOP as the `correct' category in which to do algebraic topology. For example the fundamental group of the Hawaiian earring (with the natural quotient topology inherited from the space of based loops) fails to be a topological group in TOP.

(The true source of pathology is not the Hawaiian earring and its properties, but rather the general failure of quotients and products to commute in the category TOP, (i.e. the quotient of the product might not be the product of the quotients with standard definitions of topological quotients and topological products). Such discrepancy makes the case for the continued relevance of category theory.

Euclid's lemma concerning the infinitude of primes seems fundamental and basic to Number Theory. It also seems to be a catalyst for the many "infinitely many primes of the type...." - problems. Also related is the "reductio ad absurdum" model used to prove the result.

• Dear user3195446, Since the question is very broad as it is we limited ourselves to actual examples (not results or methods) and not to very basic examples (like "the prime numbers"). Oct 20, 2016 at 14:27

3264 is considered "emblematic" of a solution to an instance of the fundamental problem of constructing notions of 'random objects'/'generic object'/'objects in general position'.

"[...] the determination, by Chasles,1 of the number of smooth conic plane curves tangent to five given general conics. The problem is emblematic of the dual nature of the subject. On the one hand, the number itself is of little significance: [...] But the fact that the problem is well-posed---that there is a Zariski open subset of the space of 5-tuples $(C_1,\dotsc,C_5)$ of conics for which the number of conics tangent to all five is constant, and that we can in fact determine that number---is at the heart of algebraic geometry. And the insights developed [...] [for] a rigorous derivation of the number [...] [e.g.] a new parameter space for plane conics, and the understanding why intersection products are well-defined for this space---are landmarks [...] the number 3264 [...] is as emblematic of enumerative geometry as the date 1066 [...] is of English history."

[emphasis added] The Zariski-openness means that there is a meaningful notion of 'general position'.

An introduction (in French):

Étienne Ghys:TROIS MILLE DEUX CENT SOIXANTE-QUATRE. Comment Jean-Yves a récemment précisé un théorème de géométrie. Images des Mathématiques. 2008

1 Joncquières is said to have obtained this result earlier, yet did not publish. None of the 'solutions' seems to be considered rigorous, and a proof, i.e. a correct deduction of '3264' from respected axioms, had to wait until the late 20th century.

Fermat hypersurfaces and their Zeta functions.

Udi de Shalit proposed: One example that comes to my mind is the Zeta functions of the Fermat hypersurfaces, which were studied by Weil and are known to have helped him in formulating the Weil conjectures, later on proved by Grothendieck and Deligne.

The most fundamental equation is $x^2+1=0$.