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
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 P on August 18, 2010
at 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: Jeffo on August 18, 2010
at 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: Geoff on August 18, 2010
at 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 Elwes on August 18, 2010
at 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 H on August 18, 2010
at 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. Wilson on August 18, 2010
at 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 Richeson on August 18, 2010
at 5:10 pm
Banach-Tarski paradox; Russel’s paradox (at the very least it surprised Frege).
By: mpersh on August 18, 2010
at 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: Olack on August 18, 2010
at 10:17 pm
multiple differentiable structures on R4?
Come on, when that came out, it was mindblowing!
By: Greg on August 19, 2010
at 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: Dan on August 19, 2010
at 9:57 am
That is definitely one of my favourites.
By: Seb on November 16, 2010
at 7:54 am
Shouldn’t we do this over at http://www.mathoverflow.net ?
By: vonjd on August 20, 2010
at 7:06 am
Probably. I’ve not done too much at MO yet. I’ll keep that in mind next time.
By: Dave Richeson on August 20, 2010
at 3:14 pm
Eversion of the sphere.
http://en.wikipedia.org/wiki/Eversion_of_the_sphere
By: SteveBrooklineMA on August 20, 2010
at 3:09 pm
Oooh! Good one. Smale is my (academic) grandfather. I should have thought of this.
By: Dave Richeson on August 20, 2010
at 3:15 pm
Perelman’s solution to the Poincare conjecture?
By: Rick Meese on August 21, 2010
at 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!!!
By: vonjd on August 21, 2010
at 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: vonjd on August 21, 2010
at 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: vonjd on August 22, 2010
at 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: Barry on August 22, 2010
at 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 Golden on August 25, 2010
at 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 Hinton on August 27, 2010
at 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 ghrist on August 27, 2010
at 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 « JD2718 on September 3, 2010
at 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 Shrhon on September 17, 2010
at 8:39 am
D’oh, bad font rendering, that is a pi. Ignore me.
By: Max Shrhon on September 17, 2010
at 8:41 am
pi and e are transcendent
By: Julian Davies on September 17, 2010
at 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 Goodsell on September 17, 2010
at 10:38 am
Except that it is wrong since you are dividing by zero.
By: lvleph on September 17, 2010
at 10:30 pm
Really? And there’s me thinking that 1 actually is equal to 2!
By: Andrew Goodsell on September 20, 2010
at 3:14 am
UCL ?
By: P T on October 24, 2010
at 12:26 pm
http://en.wikipedia.org/wiki/Braess's_paradox
By: RC on September 17, 2010
at 12:01 pm
Conway’s Game of Life
Complexity can come from simple rules.
By: Sean C. on September 17, 2010
at 12:30 pm
Central Limit Theorem
By: anon31416 on September 17, 2010
at 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 Asimov on September 17, 2010
at 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: John on September 25, 2010
at 9:15 pm
[...] A nice post on mathematical surprises. [...]
By: Assorted Links « Daily Expositions on October 10, 2010
at 10:19 am
[...] Mathematical surprises (divisbyzero.com) [...]
By: de Moivre’s formula | The Centre for Complex Analysis on November 1, 2010
at 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 pollack on November 4, 2010
at 9:12 pm
[...] 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 [...]
By: Odds and ends: the genius of Euler, Bulgarian solitaire, and mathematical surprises « Division by Zero on November 7, 2010
at 2:04 pm
A blog about Math,s i love it
By: fix-m23 on November 17, 2010
at 6:27 am
[...] A list of mathematical surprises. [...]
By: Most Popular Posts of The First Year « Daily Expositions on December 31, 2010
at 10:23 am
[...] exemple, suivre l’exemple de quelques projets (TROP TROP) sympas : 1, 2, 3, 0 [...]
By: C’est facile de sauver le monde | toussaquoi on February 13, 2011
at 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. Griffiths on June 30, 2012
at 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. Griffiths on July 13, 2012
at 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. Griffiths on July 19, 2012
at 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. Griffiths on July 23, 2012
at 12:00 pm
The two square roots of the imaginary number i are cos45+isin45,
and cos 225+isin225.
By: Peter L. Griffiths on December 13, 2012
at 2:54 pm
For the next lesson, what exactly are the two square roots of the imaginary number -i ?
By: Peter L. Griffiths on January 28, 2013
at 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. Griffiths on March 17, 2013
at 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. Griffiths on March 23, 2013
at 2:31 pm
Can you prove the following, tan6=(tan12) (tan24) (tan48).
By: Peter L. Griffiths. on May 6, 2013
at 10:56 am
Neat. I’d never seen the book-maker feature before.
By: Dave Richeson on September 28, 2010
at 9:53 pm