We've all heard of some mindblowing phenomena involving the sciences, such as the doubleslit experiment. I was wondering if there are similair experiments or phenomena which seem very counterintuitive but can be explained using mathematics? I mean things such as the Monty Hall problem. I know it is not exactly an experiment or phenomenon (you could say a thoughtexperiment), but things along the line of this (so with connections to real life). I have stumbled across this interesting question, and this are the type of phenomena I have in mind. This question however only discusses differential geometry.

8community wiki? – JohnPhteven Dec 24 '12 at 20:31
11 Answers
Off the top of my head, I can think of two astonishing phenomena which were observed experimentally before they were explained mathematically.
The first is Benford's law. This was discovered by an astronomer in the 19th century who noticed that the early part of his book of logarithmic tables was more worn than the later part, suggesting that numbers with first digit 1 or 2 appear more frequently in nature than numbers with first digit 8 or 9. In fact, the distribution of first digits from most data sets (including numbers lifted from the New York Times!) tend to obey a fairly consistent logarithmic distribution. This pattern has a variety of different explanations depending on the context  it is known to happen for power law data or data coming from a variety of different distributions, for instance. Some explanations involve serious mathematics, such as the ergodic theorem.
The second is Feigenbaum's constant. Begin with a simple oneparameter dynamical system, such as $f(x) = ax(1x)$. To say that this is a dynamical system means that we are going to pick a value of $x$ (between $0$ and $1$), plug it into $f$, plug the output back into $f$ again, and so on. The question is: what will happen as we keep iterating this procedure? The answer depends on the constant $a$. If you start off with $a$ between $3$ and $1 + \sqrt{6}$ then you get an attracting $2$cycle, meaning the system bounces back and forth between two points. For a range of values slightly larger than $1 + \sqrt{6}$, you get an attracticing $4$cycle, meaning he system oscillates between four points. As you keep sliding $a$ upward, you get $8$cycles, $16$cycles, $32$cycles... until finally, for values of $a$ around 3.6 or higher, you just get chaos. The question is: what is the period doubling rate, i.e. the rate at which you go from an $2^k$cycle to a $2^{k+1}$cycle? The period keeps doubling faster and faster, but the ratio is asymptotically a constant: about 4.6692. The crazy thing is that you can repeat this analysis for a variety of similar dynamical systems, such as $f(x) = a  x^2$, and you get the same phenomenon: accelerating period doubling followed by chaos. The ranges of $a$ for which you see a given cycle vary from system to system, but the period doubling rate of $4.6692$ appears over and over again. Feigenbaum discovered this experimentally and then proved that it holds for any oneparameter dynamical system $f(x)$ with a single quadratic maximum.
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If you let $a_1=a_2=a$, and $a_{n+1}=20a_n19a_{n1}$ for $n=2,3,\dots$, then it's obvious that you just get the sequence $a,a,a,\dots$. But if you try this on a calculator with, say, $a=\pi$, you find that after a few iterations you start getting very far away from $\pi$. It's a good experiment/demonstration on accumulation of roundoff error.
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There's the famous Buffon Needle problem: if you drop a needle of length $a$ at random onto a floor made of parallel wooden strips of identical width $l \gt a$, what is the probability that the needle will lie across a line separating two strips? The answer involves $\pi$, and you can actually use this process to experimentally approximate $\pi$.
One that definitely blew my mind when I first heard about it was Khinchin's constant: if you take just about any real number (excluding the rationals and a few others) and write down its (unique) continued fraction expansion,
$$x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + ...}}} = [a_0;a_1,a_2,a_3,...]$$
and then form the geometric mean of the first n coefficients,
$$\left(\,\prod_{k=1}^n a_k\right)^{1/n}$$
you can ask what happens if you do it again with larger and larger n, i.e. in the limit as $n \rightarrow \infty$. The intuitive guess is that this depends on the $a_k$ (meaning, on the $x$ you picked.) Surprisingly, it turns out that this isn't true: basically, no matter what $x$ you pick (i.e. almost always), this infinite product will converge to the same value, which is called Khinchin's constant.
Weirdest of all, while it's really easy to check this numerically (just pick a real number at random and start computing), and while there are several different proofs that this happens almost always, it hasn't been proven for any one particular case: for example, $\pi, \gamma$ and even Khinchin's constant itself all seem to have this property, but no one has a proof.
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Here are few such examples
1.The Smale's paradox which informally says it is possible to turn a sphere inside out in a threedimensional space with possible selfintersections but without creating any crease. Smale's Paradox
2.The BanachTarski paradox, which says it is possible to break a ball into pieces and create two balls which are identical to the first one. Banach Tarski Paradox
3.The existence of undecidable problems is also very counter intuitive to me,even though the proof is very simple. Undecidable problems
4.Borsuk–Ulam theorem which very informally says that at any given time on the earth's surface, there exist 2 antipodal points with the same temperature and pressure.
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3http://www.youtube.com/watch?v=BVVfs4zKrgk here's a nice video about turning sphere inside out – Adam Dec 24 '12 at 12:45

4For BanachTarski paradox, it may be important to mention that you need only actually break it into just a few (less than $10$) pieces, not some large number (and absolutely not infinitely many). – tomasz Dec 24 '12 at 13:34

4It's also worth mentioning that there is no way to craft an "experiment" illustrating the BanachTarski paradox because it relies on nonconstructive techniques. – Paul Siegel Dec 24 '12 at 14:04

2How would one demonstrate Smale's paradox or the existence of undecidable problems via experiment either? These are all surprising mathematical facts, but not surprising "experiments or phenomena" like the question asks for. – Dec 24 '12 at 17:08


5I'd be reluctant to describe the BanachTarski paradox with a phrase like "break a ball into pieces" that would suggest to an uninformed audience that the "pieces" are connected or otherwise nice enough to visualize. – Andreas Blass Dec 24 '12 at 19:07

2@Adam here's the original (480p!) video http://www.youtube.com/watch?feature=player_embedded&v=wO61D9x6lNY – raindrop Dec 24 '12 at 21:53
One experiment I remember being amazed by when I first saw it goes as follows:
Draw a triangle and label the corners 1, 2 and 3 also mark a point halfway along one edge.
Pick a random number out of 1, 2 and 3 (via dice or otherwise).
Mark a new point exactly halfway between the point you last drew and the corner whose number you just rolled.
Go back to step 2.
If you do this for long enough you obtain something looking like the Sierpinski gasket! It turns out the randomness is a bit of a red herring, as performing this half the distance procedure will always give you a point on the gasket. So it will give you a similar picture with non random sequences provided the distribution is "okay". There are loads of other cool Sierpinski ones too!
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4This is known as the [chaos game](https://en.wikipedia.org/wiki/Chaos_game). You can make any [IFS](https://en.wikipedia.org/wiki/Iterated_function_system) fractal with this method. – Dec 25 '12 at 06:34
For general audience simply show that $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ have same cardinality. To someone without university level studies on mathematics it is really counterintuitive.
You could also compare unit ball and unit cube in different dimensions. Start from unit circle inside unit square, and put 4 smaller circles to corners of square. Then go to 3D and put 8 smaller balls to corners of cube. Continue to 4D  how big are now "smaller" hyperballs compared to "big" hyperball? You could also try to explain something in probability with this example.
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I like hexaflexagons since I watched Vihart lovely video (also fell in love with her in the process) http://www.youtube.com/watch?v=paQ10POrZh8
Building and playing with the Möbius strip is also super mindblowing http://youtu.be/BVsIAa2XNKc?t=5s
Topology puzzles are also very fun http://www.youtube.com/watch?v=S5fPwE7GQOA
Calculate the day of the week a person was born can also be surprising Calculate which day of the week a date falls in using modular arithmetic
I would recommend checking out the list of Wiki experimental mathematics applications and examples as it shows a nice list.
There is even a Journal of Experimental Mathematics.
I would also look at the controversial Wiki computeraided proofs, including a university that will sell you a proof with your name on it!
Lastly, I would say that field of Computational Science is a direct blending of mathematics with the typical science related fields to solve problems.
Regards
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simpson's paradox explains why it is possible to manipulate the way a data is presented to get contradicting trends ( for example gender bias in University Admissions )
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The 100 Prisoner Problem is one candidate for such an experiment.
In it you have 100 prisoners on death row, who are given a chance to survive:
There is a cupboard, with 100 drawers. Each drawer contains the name of a prisoner. Each prisoner now has 50 tries to try and draw his name. Only if all 100 prisoners succeed, they'll be pardoned.
No information may be exchanged, etc.
The astonishing thing is now that there actually exists a strategy with viable survival chance ($>30 \%$) for the prisoners.
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