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

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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!

By:

Dan Pon August 18, 2010at 1:28 pm

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.

By:

Jeffoon August 18, 2010at 1:28 pm

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.

By:

Geoffon August 18, 2010at 2:19 pm

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?

By:

Richard Elweson August 18, 2010at 3:35 pm

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.

By:

Jeremy Hon August 18, 2010at 4:13 pm

Not sure if statistics results count, but my favorite is Simpson’s paradox.

http://en.wikipedia.org/wiki/Simpson's_paradox

By:

Jim R. Wilsonon August 18, 2010at 4:55 pm

Thanks, everyone, for your surprises. Keep them coming. I’m going to update my list as I think of new items to add.

By:

Dave Richesonon August 18, 2010at 5:10 pm

Banach-Tarski paradox; Russel’s paradox (at the very least it surprised Frege).

By:

mpershon August 18, 2010at 8:25 pm

I really find interesting the Euler-Mascheroni constant.

Surprising for me is also the existence of sequences with multiple limits (in non Hausdorff spaces)

By:

Olackon August 18, 2010at 10:17 pm

multiple differentiable structures on R4?

Come on, when that came out, it was mindblowing!

By:

Gregon August 19, 2010at 12:44 am

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.

By:

Danon August 19, 2010at 9:57 am

That is definitely one of my favourites.

By:

Sebon November 16, 2010at 7:54 am

Shouldn’t we do this over at http://www.mathoverflow.net ?

By:

vonjdon August 20, 2010at 7:06 am

Probably. I’ve not done too much at MO yet. I’ll keep that in mind next time.

By:

Dave Richesonon August 20, 2010at 3:14 pm

Eversion of the sphere.

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

By:

SteveBrooklineMAon August 20, 2010at 3:09 pm

Oooh! Good one. Smale is my (academic) grandfather. I should have thought of this.

By:

Dave Richesonon August 20, 2010at 3:15 pm

Perelman’s solution to the Poincare conjecture?

By:

Rick Meeseon August 21, 2010at 10:54 am

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!!!

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vonjdon August 21, 2010at 3:12 pm

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…

By:

vonjdon August 21, 2010at 4:31 pm

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.

By:

vonjdon August 22, 2010at 5:10 am

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!!)

By:

Barryon August 22, 2010at 9:56 pm

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

By:

John Goldenon August 25, 2010at 4:43 pm

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…

By:

Jon Hintonon August 27, 2010at 6:44 am

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

By:

rob ghriston August 27, 2010at 8:38 pm

[…] love that! It is one of a bunch of surprises that Dave Richeson lists at Division by Zero. (Mathematical surprises is the […]

By:

Carnival of Mathematics 69 « JD2718on September 3, 2010at 11:34 am

>Dropping needles on a hardwood floor to approximate π (Buffon’s needle)

Should probably be to approximate pi.

Great list!

By:

Max Shrhonon September 17, 2010at 8:39 am

D’oh, bad font rendering, that is a pi. Ignore me.

By:

Max Shrhonon September 17, 2010at 8:41 am

pi and e are transcendent

By:

Julian Davieson September 17, 2010at 8:55 am

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!

By:

Andrew Goodsellon September 17, 2010at 10:38 am

Except that it is wrong since you are dividing by zero.

By:

lvlephon September 17, 2010at 10:30 pm

Really? And there’s me thinking that 1 actually is equal to 2!

By:

Andrew Goodsellon September 20, 2010at 3:14 am

UCL ?

By:

P Ton October 24, 2010at 12:26 pm

http://en.wikipedia.org/wiki/Braess's_paradox

By:

RCon September 17, 2010at 12:01 pm

Conway’s Game of Life

Complexity can come from simple rules.

By:

Sean C.on September 17, 2010at 12:30 pm

Central Limit Theorem

By:

anon31416on September 17, 2010at 12:54 pm

* 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!

By:

Dan Asimovon September 17, 2010at 1:35 pm

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

By:

Johnon September 25, 2010at 9:15 pm

Neat. I’d never seen the book-maker feature before.

By:

Dave Richesonon September 28, 2010at 9:53 pm

[…] A nice post on mathematical surprises. […]

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de Moivre’s formula | The Centre for Complex Analysison November 1, 2010at 9:07 am

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.

By:

fred pollackon November 4, 2010at 9:12 pm

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Odds and ends: the genius of Euler, Bulgarian solitaire, and mathematical surprises « Division by Zeroon November 7, 2010at 2:04 pm

A blog about Math,s i love it

By:

fix-m23on November 17, 2010at 6:27 am

[…] A list of mathematical surprises. […]

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C’est facile de sauver le monde | toussaquoion February 13, 2011at 6:19 pm

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.

By:

Peter L. Griffithson June 30, 2012at 12:33 pm

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.

By:

Peter L. Griffithson July 13, 2012at 1:14 pm

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).

By:

Peter L. Griffithson July 19, 2012at 10:28 am

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.

By:

Peter L. Griffithson July 23, 2012at 12:00 pm

The two square roots of the imaginary number i are cos45+isin45,

and cos 225+isin225.

By:

Peter L. Griffithson December 13, 2012at 2:54 pm

For the next lesson, what exactly are the two square roots of the imaginary number -i ?

By:

Peter L. Griffithson January 28, 2013at 12:38 pm

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……

By:

Peter L. Griffithson March 17, 2013at 2:05 pm

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.

By:

Peter L. Griffithson March 23, 2013at 2:31 pm

Can you prove the following, tan6=(tan12) (tan24) (tan48).

By:

Peter L. Griffiths.on May 6, 2013at 10:56 am

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.

By:

Peter L. Griffithson July 28, 2013at 9:30 am

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.

By:

Peter L. Griffithson August 2, 2013at 12:26 pm

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.

By:

Peter L. Griffithson December 2, 2013at 1:30 pm

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?

By:

Peter L. Griffithson January 16, 2014at 2:54 pm

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.

By:

Peter L. Griffithson January 23, 2014at 2:53 pm

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.

By:

Peter L. Griffithson January 24, 2014at 1:20 pm

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.

By:

Peter L. Griffithson February 11, 2014at 12:27 pm

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.

By:

Peter L. Griffithson July 15, 2014at 12:25 pm

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.

By:

Peter L. Griffithson September 22, 2014at 12:33 pm