If you had a sloppy maths teacher at school you might have grown up with the idea that the number is equal to Now that is completely wrong. Writing those numbers out in decimal gives while . There’s a difference in the third decimal place after the decimal point!
How accurately do we need to know the value of π?
Surely this small inaccuracy doesn't matter? Well, as the following extract from a longer article by Chris Budd shows, it really does.
The point is that is not any number. It lies at the heart of any technology that involves rotation or waves, and that is much of mechanical and electrical engineering. If rotating parts in, say, a typical jet engine are not manufactured to high tolerance, then the parts simply won’t rotate. This typically involves measurements correct to one part in 10,000 and, as these measurements involve , we require a value of to at least this order of accuracy to prevent errors. In medical imaging using CAT or MRI scanners, the scanning devices move on a ring which has to be manufactured to a tolerance of one part in 1,000,000, requiring an even more precise value of .
However, even this level of accuracy pales into insignificance when we look at modern electrical devices. In high frequency electronics, with frequencies in the order of 1GHz (typical for mobile phones or GPS applications), electrical engineers require a precision in the value used for in the order of one part in 1,000,000,000,000,000.
So, the modern world needs and it needs it accurately!
The Klein bottle is a very strange shape, discovered in 1882 by the mathematician Felix Klein. It intersects itself, only has one side (as opposed to an inside and an outside) and you can turn a pattern drawn on the bottle into its mirror image simply by sliding it along. You can find out more here. This picture of the Klein bottle was created by Charles Trevelyan and rendered partially transparent so that we can see the inside.
We have just learnt a really nice fact about the game of tic-tac-toe. As you may know, tic-tac-toe always results in a draw if both players make the best moves possible at every step of the game. To force a draw you need to follow a particular strategy, which isn't too hard, but tedious to write down (see here).
But now imagine you play tic-tac-toe the other way around, so that the first person to get three in a row loses the game. Again the game will end in a draw if both players play optimally. But this time the strategy for the first player is really easy: at your first move, choose the central square, and after that simply mirror your opponent's moves. By "mirror" we mean that you choose the square that is diametrically opposite the last square the opponent chose: if the opponent chose the top right corner, you choose the bottom left one, if they chose the middle top one, you choose the middle bottom one, and so on.
The same works in a version of tic-tac-toe that has, not 3x3, but nxn squares, where n is an odd number (if n is even there is no central square). Again we play the version in which the player who first gets n in a row loses. If the first player adopts the strategy above, they will never lose.
It's not too difficult to see why this strategy works. Let's call the player who goes first A and the other one B. After A takes the central square at the first move, B goes next and A mirrors B from then on. This means that whatever configuration of squares A has taken, B has taken the mirror image of the same configuration first. In particular, if A was forced to take n squares in a row, then B must have taken n squares in a row at the previous move, so B would have lost before A even got to make the supposed losing move. Nice!
Can we always find order in systems that are disordered? If so, just how large does a system have to be to contain a certain amount of order? In this video Imre Leader of the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge gives an equation free introduction to a fascinating area of maths called Ramsey theory.
This talk was originally given to an audience of Year 12 A-level maths students (aged 16-17) in June this year and formed part of a mathematics enrichment day organised by the Millennium Mathematics Project with a special focus on encouraging the development of mathematical thinking.
"This talk will mostly be stories ... I want to tell you about Albert Einstein, and about his theory of relativity — what it is, why he was thinking about it and also about some of the very latest developments that have happened just this year."
This video contains a talk by one of our favourite physicists, David Tong at the Department of Applied Mathematics and Theoretical Physics at Cambridge. It was originally given to an audience of Year 12 A-level maths students (aged 16-17) in June this year and formed part of a mathematics enrichment day organised by the Millennium Mathematics Project with a special focus on encouraging the development of mathematical thinking.