I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way—Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise—Andrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.
I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below!
- Gödel’s incompleteness theorems
- The discovery of irrational numbers by the Pythagoreans
- Cantor’s theorems—nondenumerability of the continuum and the cardinality of the power set of A is greater than the cardinality of A
- The rational numbers are countable
- The continuum hypothesis can neither be proved nor disproved in ZFC
- The existence of a continuous nowhere differentiable function
- Euler’s solution of the Basel problem
- The existence of non-Euclidean geometries
- The insolvability of quintic equations
- The Monty Hall problem
- Fermat’s non-prime (Euler proved that
is composite)
- The shape of a hanging chain is a catenary
- The existence of space filling curves
- The Banach-Tarski theorem
- The relationship between the complex numbers and the primes (E.g., Riemann zeta function)
- The prime number theorem
- Aperiodic tilings
- Arrow’s impossibility theorem
- Ulam’s spiral of primes
- Andrew Wiles’ proof of Fermat’s Last Theorem
- The use of a computer to prove the four color theorem
- Russell’s paradox
- The Cantor set
- Euler’s polyhedron formula
- The five Platonic solids
- The Brachistochrone problem
- Noncircular figures of constant width
- 0.999…=1
- Lorenz’s “butterfly effect”
- Period 3 implies chaos (and Sharkovsky’s theorem)
- The fundamental theorem of calculus
- Descartes’ discovery of analytic geometry
- Discovery of complex numbers (and their real-world applications)
- Hamilton’s discovery of the quaternions
- There exists a flow in 3-space with closed orbits of every knot and link type
- 19-year-old Gauss’ ruler-and-compass construction of a 17-gon (and its relation to Fermat primes)
- Proving the impossibility of squaring a circle, trisecting an angle, and doubling a cube
- The Euler line
- A complex function that is once differentiable on a disk is infinitely differentiable
- Liouville’s theorem—a function that is bounded and differentiable at every point in the complex plane is constant
- Thomae’s function—a function that is continuous at every irrational number, discontinuous at every rational number
- The elementary linear algebra behind Google’s pagerank
- Kuratowski’s closure-complement theorem
- Surprisingly open problem: does every triangular billiard table have a periodic orbit?
- Surprisingly open problem: the Collatz conjecture/3n+1 problem
- Surprisingly open problem: Goldbach conjecture
- Dirac’s belt trick
- Benford’s law on the distribution of leading digits
- The short proof of the solution to the art gallery problem
- Dropping needles on a hardwood floor to approximate π (Buffon’s needle)
- Robert Conelly’s flexible polyhedron
- The many equivalent interpretations of the Catalan numbers
David Richeson: Division by Zero 
A surprise so monstrously implausible that it has ‘monstrous’ in its name: http://en.wikipedia.org/wiki/Monstrous_moonshine
The strange numerical coincidence was enough to justify the name ‘monstrous’. But when Borcherds completed his proof of the result he used mathematics borrowed from String Theory. Doubly monstrous!
Thue’s discovery of the existence of arbitrarily long strings over three symbols with no identical sequences in immediate succession (a.k.a. square-free strings).
ABCACBABCBACABCAC. . .
Maybe the surprise comes from the setup, in which one usually notes that any string over two symbols of length four must have a square.
When pi shows up when summing an infinite series of seemingly “ordinary” fractions, like:
1/1-1/3+1/5-… = pi/4
or
1/1^2 + 1/2^2 + 1/3^2 + … = pi^2 /6.
The divergence of the harmonic series.
Khinchin’s constant.
Heech’s tiling. (It surprised Hilbert.)
The non-verification of Hales’ proof of Kepler’s sphere-packing conjecture.
The Leech lattice.
And… Goodstein’s theorem – why not?
The Halting Problem and Rice’s Theorem. Granted, it’s equivalent in some sense to Godel’s Theorem, but I think it still deserves its own mention.
Two other’s from Theoretical CS:
-The PCP theorem
-That “P vs NP” (and “P vs PSPACE”, for that matter) is still open.
Not sure if statistics results count, but my favorite is Simpson’s paradox.
http://en.wikipedia.org/wiki/Simpson's_paradox
Thanks, everyone, for your surprises. Keep them coming. I’m going to update my list as I think of new items to add.
Banach-Tarski paradox; Russel’s paradox (at the very least it surprised Frege).
I really find interesting the Euler-Mascheroni constant.
Surprising for me is also the existence of sequences with multiple limits (in non Hausdorff spaces)
multiple differentiable structures on R4?
Come on, when that came out, it was mindblowing!
What about Euler’s identity (to add to the euler list)?
e^(i*pi) + 1 = 0
5 of the most important numbers in mathematics cleanly linked together.
That is definitely one of my favourites.
Shouldn’t we do this over at http://www.mathoverflow.net ?
Probably. I’ve not done too much at MO yet. I’ll keep that in mind next time.
Eversion of the sphere.
http://en.wikipedia.org/wiki/Eversion_of_the_sphere
Oooh! Good one. Smale is my (academic) grandfather. I should have thought of this.
Perelman’s solution to the Poincare conjecture?
Ito’s lemma for sure! The fact that random/stochastic behaviour turns into deterministic behaviour under certain well defined circumstances and that you have to use the second derivatives term to integrate an stochastic process – WOW!!!
This might be of interest to you: I am just reading “Darf ich Zahlen?” from the well known mathematician Günther Ziegler (TU Berlin). He has a seperate chapter on surprises (“Über Überraschungen”, p. 187 f.) There he writes about the Göttinger mathematics-philosopher Felix Mühlhölzer who has worked out a scheme of mathematical surprises on the basis of “Bemerkungen über die Grundlagen der Mathematik” from Ludwig Wittgenstein. Basically he differentiates between “R-Überraschungen” and “F-Überraschungen”: R stands for Repräsentation, so that the surprise is only based on the representation. F stands for Fakt, so that the fact itself is a surprise. Wittgenstein says that real F-Überraschungen shouldn’t exist in mathematics. Perhaps this is a good starting point for further investigations and some ordering scheme…
Addendum:
It is actually “Günter” (without the extra “h”) Ziegler.
And the article from Mühlhölzer seems to be in English: “Wittgenstein and Surprises in Mathematics”, in: Wittgenstein and the Future of Philosophy: A Reassessment after 50 Years (Proceedings of the 24th International Wittgenstein-Symposium, Kirchberg am Wechsel, 2001), hg. v. Rudolf Haller and Klaus Puhl, öbv&hpt Verlagsgesellschaft, 2002, S. 306-315.
Hope this helps.
The nonexistence of a pair of 6×6 Latin Squares. (Euler was proven correct and it only took around 100 years.) However, he was wrong when it was proved that if n>6, n=2k, and 2 doesn’t divide k, then there is a pair of orthogonal Latin squares of order n. (It only took 178 years to prove him wrong!!)
Euler relation: e^(i*pi)+1=0
Does aperiodic tilings include quasiperiodic tilings? Otherwise I’d add the Penrose rhombs or kites and darts.
Minkowski geometry
Atiyah-Singer Index Theorem
A fantastic list. My personal favourite is Goodstein. Getting back to basics… my eight-year son finds it really surprising that the product of two negative numbers is positive. And despite my best efforts to explain why I’m not sure he entirely believes me…
a few more topological surprises:
* milnor’s construction of exotic 7-spheres
* donaldson’s theorem on 4-manifolds, leading to…
* exotic R^4’s (homeomorphic but not diffeomorphic to standard R^4)
* the proof of infinitely many primes using only point-set topology (proof by furstenberg)
* all the borsuk-ulam type theorems
back down to earth:
* rearrangements of divergent series into anything you want
>Dropping needles on a hardwood floor to approximate π (Buffon’s needle)
Should probably be to approximate pi.
Great list!
D’oh, bad font rendering, that is a pi. Ignore me.
pi and e are transcendent
Proof that 1=2.
x=y therefore x-y=0
also 2x=2y therefore 2x-2y=0
If follows that (2x-2y)=(x-y)
Divide each side by (x-y)
2=1
That is a surprise!
Except that it is wrong since you are dividing by zero.
Really? And there’s me thinking that 1 actually is equal to 2!
UCL ?
http://en.wikipedia.org/wiki/Braess's_paradox
Conway’s Game of Life
Complexity can come from simple rules.
Central Limit Theorem
* There is a collection of subsets of the integers, totally ordered by inclusion, that is uncountable.
* The tautochrone: No matter where a marble is placed on a bowl that’s a cycloid of revolution, it will reach the bottom at the same time.
* Every differentiable function C -> C must be defined by a power series.
* The existence of multiple differentiable structures: The topological 7-sphere has 28 distinct ones.
4-dimensional Euclidean space has uncountably many distinct ones!
Carleson’s theorem (1966): L^2 convergence of Fourier series (integrals) implies pointwise convergence almost everywhere. Generalized a couple years later to L^p for 1 < p < infinity.
http://en.wikipedia.org/wiki/Carleson%27s_theorem
Neat. I’d never seen the book-maker feature before.
Euclid’s axiomatization of plane geometry and the resulting deductive system, which still serves as a paradigm for almost all of mathematics (and a great deal of physics), even though Godel’s astounding results dumped Russell and Whitehead’s “Principia Mathematica” into the garbage bin.
A blog about Math,s i love it
Fermat’s Last Theorem can be proved by recognising that for n greater than 2, the binomial expansion of (p+q)^n-(p-q)^n can only have an nth root if p=+q or -q.
Further to my comment of 30 June 2012, for n=2 the Pythagorean Triples can be easily identified from finding that pq has an integer square root.
A little known but very important trigonometric equation is the half angle equation cotu+cosecu=cot u/2. For those still stuck on sines and cosines, sinu=cos(90-u), tanu=cot(90-u) and secu=cosec(90-u).
Some very advanced mathematicians do not seem to know how to compute the two square roots of the imaginary number i, or even that there are two square roots.
The two square roots of the imaginary number i are cos45+isin45,
and cos 225+isin225.
For the next lesson, what exactly are the two square roots of the imaginary number -i ?
Euler appears to have solved the Basel Problem by applying the Newtonian formulae to the infinite series for sines. What is not generally known is that the infinite series for cosines can be similarily used to arrive at the appropriate formula for cosines being[(PI)^2]/8=1+1/3^2 +1/5^2……
Further to my comment of 17 March 2013, a crucial question is where exactly did Isaac Newton first state these Newtonian formulae, the answer is to be found in vol 5 pages 358-359 of D.T. Whiteside’s Mathematical Papers of Isaac Newton. Converting infinite product series into infinite summation series and vice versa seems to be a very rare skill.
Can you prove the following, tan6=(tan12) (tan24) (tan48).
Two crucial formulae are the sinking fund formula s = (1 +r)[(1+r)^n -1]/r and the present value formula p =[1 -{1/(1 +r)^n}]/r. Both are obtained from the difference between two infinite series involving wonderful mathematics apparently too difficult for schools and universities.
Further to my comment of 30 June 2012, I have proved Fermat’s Last Theorem in just over 400 words distinguishing between rational and irrational numbers. The nth root of 2 with n an integer is always irrational, but this irrationality can be corrected in the binomial expansion unless p and q are unequal, thus proving FLT.
On to another subject, The Riemann Hypothesis, Some Doubts.
Near the beginning of his 1859 paper Riemann incorrectly assumes that the complex variable s =(1/2) + ti is a zeta power. Riemann fails to recognise that an expression containing an imaginary number such as (1/2) +ti cannot be a power unless the base is a log base such as e, and also unless t being the coeffiicient of i is a specific angle. The best known example of this is Cotes’s formula cosu + isinu = e^(iu) where u is a specific angle, and it is not possible for e to be replaced by other values,
also e^(1/2) X e^(iu) equals e^[(1/2) +iu]. This means that Riemann is badly wrong in applying as a power s =(1/2) + ti. It also means that practically all the arguments in his 1859 paper are fallacious.
On to yet another subject, 2014 is the 400th anniversary of the discovery of logarithms by Napier probably the most important of all the mathematical discoveries, but present day mathematical societies are curiously reticent about commemorating. Is this because modern mathematicians have no idea how Napier achieved this discovery?
Further to my comment of 16 January 2014, it seems that Napier knew how to prove that sine 75 degrees being 0.9659258 to the power of 10 equals sine 45 degrees, can anyone else prove this, at the moment I can’t.
Further to my comment of 23 January 2014, the way to prove that sine 75 degrees to the power of 10 equals sine 45 degrees is to express sine 75 degrees as sine (45 + 30) which can be expanded to equal (3^[0.5] +1)/(2 X 2^[0.5]) which can be raised to the power of 10 to equal sine 45 degrees which is 0.70703.
Further to my comments of 23 and 24 January 2014, Napier and Regiomontanus before him knew the basic formula for constructing sine and cosine tables, this is sin2u =2sinu.cosu, which can be expressed as sin2u =2sinu.(1- [sinu]^2). This formula can be applied so that Sin30 degrees which is 1/2 which can be bisected to achieve by quadratic equations sin15 degrees which is the cosine of 75 degrees, from which sin 75 can be calculated.
One little known work by Johannes Kepler is the paper Concerning Conic Sections included in his book on Optics published in 1604. In this paper Kepler brings together the 5 conic sections known to the ancient Greeks, the straight line, the circle, the ellipse, the hyperbola, and the parabola and imagines wrongly that they possess in common the focus which he invented and used with pins and thread. It was not until 1618 that Kepler recognised that the common focus was the location of the Sun in relation to the orbiting planets.
Further to my comment of 2 August 2013, let p/q equal r which is rational, so that irrational 2^(1 -[1/n]) which contains 1/r will be reduced by a rational amount leaving another irrational amount, thus confirming Fermat’s Last Theorem.