A thought about Buffon's Needle

I am delighted that this blog will be hosting the April 2014 Carnival of Mathematics.  To find out more, see previous Carnivals, or suggest an article for inclusion, go to the Carnival of Mathematics page at the Aperiodical.  The Carnival will appear here soon after the deadline for submissions, which is 5 April.

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In my recent talk at Gresham College, I demonstrated a computer simulation of Buffon's Needle - a Monte Carlo method of finding an approximation to π.  The idea is that if one tosses a needle of length l onto a floor made up of planks of width t (with l < t for simplicity), then the probability that the needle crosses a junction of two planks is 2l/tπ.  So if we toss n coins and m of them cross the junctions, then 2ln / tm will give us an approximate value for π.  By choosing suitable values of l and t, or n, and by being slightly lucky or by not stopping until you have the answer you want, you can get the approximation 355/113 by this method (in my lecture I got this value by tossing only two needles!), which demonstrates the power of the method if you know the answer in advance and use that information to full advantage in conducting the experiment.

If I choose numbers which are less designed to give me the answer I want. I find I can get π to reasonable accuracy - I just simulated 10,000,000 needles and got a result around 3.1488.  Since one is essentially sampling, statistical theory can give estimates for the likely proportion of tosses that will cross a line and hence for the accuracy I can expect.  But, it occurred to me, why bother tossing ten million computer-simulated needles?  Why not just calculate the expected value?

There's a good reason.  Suppose the probability that a random needle crosses a junction is p.  (In my lecture, where I chose l to be 710 and t to be 903 - notice the relationships to 355 and 113! - I have p almost exactly 1/2.)  Then the probability of m "hits" out of n tosses can be calculated by the binomial formula.  In particular, the probability that m is zero is (1-p)^n.  So for ten million tosses, I have a (finite) probability of (1-p)^10,000,000 that no needle crosses a line.  In that case, 2ln/tm is infinite. If at least one needle crosses a line, then the value of 2ln / tm is finite.  So when I calculated the expected value of 2ln / tm from ten million tosses, the result is infinite and the expected value of my approximation to π is infinity. Which is some way out.

A prize which shows the diversity of applications of mathematics

A team from the Department of Mathematical Sciences at the University has just won the Guardian University Awards prize for Research Impact.  The team, led by Professor Ed Galea, is from the Fire Safety Engineering Group, and the official press release can be found here.

The work which has been honoured is concerned with signage: how can we make emergency signage more effective? The project involves dynamic signs, which can change depending on circumstances, so that if a potential escape route is blocked or unsafe, then operators can change the signs to divert people to safe routes.  This work is potentially life-saving for people escaping buildings in emergencies.

I'd like to make two general points about the value of mathematics.  First, its applications are extremely diverse: you might not have thought of mathematicians winning major prizes for working on signage!  But the mathematical algorithms underlying these active dynamic signs are quite literally going to make us all safer.  Secondly, mathematics cannot be done in isolation.  Work on projects like this involves collaboration with many other disciplines.  Computing, to implement the algorithms; engineering and architecture, to understand how buildings work; psychology, to understand how people behave in emergency situations and how they react to signs; and many others.

Mathematics really does make our lives better, especially when mathematicians work with others.