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

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The harmonic oscillator is a fundamental example in both classical and quantum mechanics.

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    $\begingroup$ Of course, for this question and many others there is no meaning to "correct answer". In fact I liked all the answers to the question and I hope more answers will come along. Jose was the most valuable partner to this endeavor and he contributed several great answers both before and after the boundy was announced. $\endgroup$ Commented Nov 26, 2009 at 12:39
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    $\begingroup$ @Gil, now that the bounty has been delivered you might "unaccept" this answer. $\endgroup$ Commented Nov 30, 2009 at 16:20
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    $\begingroup$ Is it possible? Is it moral? $\endgroup$ Commented Dec 1, 2009 at 14:53
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    $\begingroup$ I think it is somewhat unnatural that a question like this has an accepted answer. $\endgroup$ Commented Oct 24, 2016 at 16:53
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The integral $\int \frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$ for $\lambda\neq 0,1$ essentially launched both complex analysis and algebraic geometry, via Riemann's discovery of the Riemann surface that is the natural domain of a function, leading to both analytic theory of Riemann surfaces and to the study of algebraic curves, leading to...complex analysis including in several variables and complex algebraic geometry, as we now know it.

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    $\begingroup$ This great answer has earned Charles the first gold badge in mathoverflow history. Congratulations, Charles! $\endgroup$ Commented Nov 27, 2009 at 12:38
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Relevant to many areas (but mostly topology) is the Cantor set. It is an example of a set with properties too numerous to list here. To name a few: it is uncountable, compact, nowhere dense and it has Lebesgue measure 0.

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    $\begingroup$ Another good reason: Every compact Hausdorff space is a quotient of the Cantor set!! See Tom Leinster's answer here mathoverflow.net/questions/5357/… It's just incredible - I still can't get over it. $\endgroup$ Commented Nov 20, 2009 at 15:30
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    $\begingroup$ @Peter Arndt: Every compact Hausdorff metrizable (equivalently, second-countable) space is a quotient of the Cantor set. There are compact Hausdorff spaces of greater than continuum cardinality, and these evidently are not quotients of the Cantor set. $\endgroup$ Commented Jan 15, 2010 at 10:30
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    $\begingroup$ @PeteL.Clark: however, every compact Hausdorff space is a quotient of a subspace of a power of the two-element discrete space. (If the power is countable then you don't need to pass to a subspace because every closed subspace of the Cantor set is a retract.) Proof: every compact Hausdorff space embeds in a power of $[0,1]$. $\endgroup$ Commented Jun 10, 2016 at 13:48
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The Petersen Graph in graph theory.

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For a picture that launched a thousand papers, I'd nominate the bifurcation diagram of the logistic map.

alt text

(image via Wikipedia).

Answered by Martin M. W.

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    $\begingroup$ If pictures are allowed then the Mandelbrot set and its Julia sets are also quite fundamental as prime examples of what lies hidden in dynamical systems. Of course, as the Wikipedia page on the Mandelbrot set says, there is a close relation between the Mandelbrot set and the bifurcation diagram of the logistic map. $\endgroup$ Commented Nov 11, 2009 at 18:37
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    $\begingroup$ Roughly speaking, the bifurcation diagram is what you get by going through the Mandelbrot set along the real axis. $\endgroup$ Commented Mar 3, 2010 at 10:54
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    $\begingroup$ What do you get by going through the Mandelbrot set along a different line? $\endgroup$ Commented May 10, 2010 at 20:13
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The Fermat Equation xn + yn - zn = 0.

This has truly been much more than an example in both algebra and number theory: it was one of the main motivations to develop the theory of unique factorization domains, Dedekind domains, class numbers, regular primes, etc. in the 19th century. In the late 20th century it provided a motivation for Wiles to work on modularity of elliptic curves.

In the 21st century, the equation c1 xa + c2 yb - c3 zc = 0 is similarly motivational for things like Q-curves, Galois representations, hypergeometric abelian varieties...

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    $\begingroup$ Caution: I think this example is often overrated in its motivating role, e.g., see mathoverflow.net/q/34806/6518 $\endgroup$ Commented Apr 14, 2015 at 7:02
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Brownian motion has a central role in the theory of stochastic processes

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    $\begingroup$ See the book and several expository papers by Jean-Pierre Kahane for the role of Brownian motion in and outside Stochastic Processes. $\endgroup$ Commented May 9, 2010 at 9:57
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The Prisoner's Dilemma in Game Theory.

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    $\begingroup$ Prisoner's Dilemma is certainly a fundamental example but there are other games which I think have been as rich in encouraging interesting research: Chicken en.wikipedia.org/wiki/Chicken_game Chain store en.wikipedia.org/wiki/Chainstore_paradox centipede en.wikipedia.org/wiki/Centipede_game One reason these are interesting games is that help one try to understand what is "rational" behavior and help distinguish between what people do in practice as compared with some "abstract model" of rationality. $\endgroup$ Commented Dec 25, 2009 at 15:10
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    $\begingroup$ Rock-Paper-Scissors is the fundamental example of a zero-sum game, with a Nash equilibrium which is unique and which involves mixed strategies. $\endgroup$ Commented Jan 13, 2010 at 15:26
  • $\begingroup$ Actually, matching pennies is even easier than R-P-S and more popular as an example. $\endgroup$ Commented Nov 11, 2011 at 14:56
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The Brachistochrone problem. Solved by both Newton/Bernoulli. It is considered to be the fundamental/first problem which led to the formulation of the Calculus of Variations.

Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time.

alt text

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In complex dynamics: The Mandelbrot set

Mandelbrot set

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There's always the venerable normal distribution (for probability theory).

community wiki

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  • $\begingroup$ But there should be added some probability examples NOT related to the normal distribution! $\endgroup$ Commented Sep 16, 2015 at 8:12
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Laplace's equation is the fundamental example of a PDE.

If I could broaden the question to allow a triumvirate of examples, I'd say Laplace's equation, the heat equation, and the wave equation are the canonical examples of PDEs, representing elliptic, parabolic, and hyperbolic equations respectively.

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    $\begingroup$ By all means, go for the triumvirate, even if it violates the stated requirement of one example. $\endgroup$ Commented Nov 11, 2009 at 13:53
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Russell's paradox

Showing that Frege's set theory leads to contradiction.

The example can be described as the set of all sets $A$ such that $A \notin A$. It is related in spirit to Cantor's proof that a power set has larger cardinality than a set and to the ancient "liar paradox".

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    $\begingroup$ In fact, it has shaped Mathematics in its entirety! $\endgroup$ Commented Dec 2, 2009 at 12:19
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Someone has already mentioned tori, but I think elliptic curves in algebraic geometry merit their own separate mention.

Answered by Kevin Lin

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The cotangent bundle is the fundamental example of symplectic manifold/phase space.

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Answered by James : The Platonic solids. They are fundamental, collectively and individually, to many areas of mathematics.

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    $\begingroup$ Can anyone elaborate a little on why they are fundamental? Reading the Wikipedia page didn't enlighten me about this. $\endgroup$ Commented Dec 12, 2010 at 19:48
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    $\begingroup$ Not that Jack Lemon's been around since '12, but for other people, look up ADE classifications on Wikipedia to see the many other things that correspond to the Platonic solids. $\endgroup$ Commented Oct 24, 2015 at 3:01
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SAT (Boolean satisfiability problem) in complexity theory/theoretical computer science -- it's the canonical example of an NP-complete problem not just because it came first, but because it launched a thousand other research papers all on its own. (3SAT is maybe more canonical, but the more general form is better to generalize and study for its own sake.)

Answered by: Harrison Brown

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    $\begingroup$ I would say that it's the canonical example of an NP-complete problem just because it's the one which is "the most obviously" NP-complete. $\endgroup$ Commented Nov 12, 2009 at 19:04
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The ring $\mathbb{Z}[\sqrt{-5}]$ is a fundamental example of non-unique factorization in rings in algebraic integers. Perhaps of more historical relevance is the example of $p=37$ that shows that Lamé's "proof" of Fermat's Last Theorem fails. I'm not sure Kummer used $p=37$ though. In any case, examples of non-unique factorization in rings in algebraic integers lead to the whole theory of ideal numbers and later Dedekind domains and set the tone for algebraic number theory.

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    $\begingroup$ Unique prime factorization fails for the ring of integers underlying the p = 23 case of FLT. Kummer's method of proof works for p = 23, but not p = 37 (as Kummer was well aware of). $\endgroup$ Commented Nov 16, 2009 at 19:35
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    $\begingroup$ @lhf: around the case p=37 Kummer created the concept of irregular prime and found the relation with the prime factors of the Bernoulli numbers. For more details see H. Edwards, "Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory" or P. Ribenboim, "13 lectures on Fermat's Last Theorem", both worth reading. $\endgroup$ Commented May 13, 2010 at 17:26
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1The Fano plane in finite geometry

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

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  • $\begingroup$ The former link was broken, so I replaced it. $\endgroup$ Commented Mar 15, 2015 at 18:31
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Penrose tiling

The Penrose tiling (image from Wikipedia). It's a fundamental example not just for aperiodic tilings more generally, but for Connes' work on noncommutative geometry.

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The Catalan numbers are definitely a fundamental example in combinatorics.

Answered by Qiaochu Yuan

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    $\begingroup$ I agree. I would also add the Fibonacci numbers en.wikipedia.org/wiki/Fibonacci_number . Fibonacci numbers lead to rational generating functions and Catalan numbers to rational generating functions. The Catalan numbers appear in an amazing number of different problems. $\endgroup$ Commented Nov 12, 2009 at 18:40
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    $\begingroup$ *Algebraic (I assume). $\endgroup$ Commented Nov 12, 2009 at 19:41
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The quaternions

Probably the natural numbers, real numbers, and complex numbers are "too fundamental" to count here. But the field of quaternions discovered in the mid 19 century qualifies.

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    $\begingroup$ 14 years late, but the quaternions do not form a field. Or perhaps you mean field as in "field of study"? $\endgroup$ Commented Apr 9, 2023 at 12:28
  • $\begingroup$ Right! I meant a "division ring" or a "skew field". $\endgroup$ Commented Apr 10, 2023 at 16:31
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$2ドル^{\aleph_0} = \aleph_1$$

The Continuum Hypothesis is an example of an undecidable statement par excellence. It is an example of a problem that is:

  • natural;
  • historied -- it was first asked by Cantor himself;
  • celebrated -- it was Hilbert's first of his famous 23 problems; and
  • undecidable in a very deep way -- it's independent of all current large cardinal axioms, and in fact something deeper than this can be said, but that is currently beyond my understanding.

Moreover, the desire to "solve" CH was the main motivation, or at least one of the main motivations, for:

  • Godel's description of the Constructible Universe, which has already been listed as a fundamental example in logic and foundations, and the beginnings of inner model theory; and
  • Cohen's invention of the method of forcing which is now indispensable and ubiquitous in modern set theory, and also won Cohen a Fields medal.
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A lot of algebraic topology was developed with computing the higher homotopy groups of spheres in mind.

Answered by: David Lehavi

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In geometry, group theory and other areas: The Leech Lattice:

community wiki

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The KdV equation in integrable systems. It was through a numerical study of KdV that the word soliton was coined. This numerical study lead to much analytical work, including the development of Lax Pairs. (Answer by Aaron Hoffman)

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  • $\begingroup$ I could not make the link to wikipedia works. $\endgroup$ Commented Dec 13, 2009 at 8:48
  • $\begingroup$ I fixed the link. $\endgroup$ Commented Jan 14, 2010 at 8:56
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I do think Milnor's exotic sphere distinguish differential topology from general topology, but i don't know if this is an example of the kind you want.

Answered by: Yuhao Huang

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It has often been said that if you understand $su_3$ you understand all simple Lie algebras, so that should make it the fundamental example. (Personally I think that it suffices to understand $su_2$!)

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    $\begingroup$ On the same line, SU(2) (compared to U(1), which is abelian) already shows a lot of features of the compact non abelian groups (Peter-Weil theorem, irreducible representations of (every) dimension greater than 1, ...). $\endgroup$ Commented Nov 11, 2009 at 9:11
  • $\begingroup$ (This is going to look like spam since it's the same as my comment on another answer! However, I think it's equally applicable to this one.) In what way has SU_3 shaped any of these subjects? It may well be 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 the subject. $\endgroup$ Commented Nov 11, 2009 at 9:52
  • $\begingroup$ Perhaps I do not understand what it means to "shape" the subject. It is hard to say, for very classical subjects, what object, if any, has shaped it. I took the question to mean which example is emblematic, in the sense you write. $\endgroup$ Commented Nov 11, 2009 at 10:03
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    $\begingroup$ I guess you could say that the heat equation "shaped" functional analysis since so many of the early tools were developed to study just that equation (more so if you say fourier analysis). Similarly (as has been noted) the spheres are currently shaping algebraic topology since so many of the tools are developed to get at the stable homotopy of spheres. I didn't put my comment on your "harmonic oscillator" example because that has played a role in shaping quantum mechanics. Part of it is purely timing: good examples have a chance to shape a subject if they are encountered in its infancy. $\endgroup$ Commented Nov 11, 2009 at 10:33
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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.

(see also this Wikipedia article.)

alt text (source)

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  • $\begingroup$ Here is a nice link for the figure eight knot mathworld.wolfram.com/FigureEightKnot.html . The trefoil knot en.wikipedia.org/wiki/Trefoil_knot is a basic example in knot theory. $\endgroup$ Commented Nov 13, 2009 at 6:56
  • $\begingroup$ The figure eight knot is the only arithmetic knot. $\endgroup$ Commented Jun 15, 2024 at 6:11
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