Posted by: Dave Richeson | August 18, 2010

## Mathematical surprises

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!

## Responses

1. 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!

2. 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.

3. 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.

4. 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?

5. 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.

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

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

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

9. I really find interesting the Euler-Mascheroni constant.

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

10. multiple differentiable structures on R4?
Come on, when that came out, it was mindblowing!

e^(i*pi) + 1 = 0

5 of the most important numbers in mathematics cleanly linked together.

• That is definitely one of my favourites.

12. 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.

13. 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.

14. Perelman’s solution to the Poincare conjecture?

15. 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!!!

16. 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…

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.

17. 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!!)

18. 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

19. 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…

20. 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

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

22. >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.

23. pi and e are transcendent

24. 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 ?

25. Conway’s Game of Life

Complexity can come from simple rules.

26. Central Limit Theorem

27. * 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!

28. 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

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

30. [...] Mathematical surprises (divisbyzero.com) [...]

31. 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.

32. [...] I was a lunch table discussion leader. Since my mathematical surprises blog post was so popular, I chose that as my table’s discussion topic. I think it went really [...]

33. A blog about Math,s i love it

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

35. [...] exemple, suivre l’exemple de quelques projets (TROP TROP) sympas : 1, 2, 3, 0 [...]

36. 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.

37. 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.

38. 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).

39. 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.

40. The two square roots of the imaginary number i are cos45+isin45,
and cos 225+isin225.

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

42. 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……

43. 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.

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

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