Marcus du Sautoy talks about football, cryptography, and numbers.
In the first part of this article we let maths set the scene for a free kick. Now we continue the drama, tracing the trajectory of the ball throughout the milliseconds it takes it to reach the goal line.
Free kicks will deliver much of the drama in the football world cup this summer. But how should strikers approach them and how does the design on the ball impact on its behaviour in flight? Maths can give us answers...
Remember Frank Lampard's disallowed goal in the 2010 World Cup match against Germany? The ball hit the crossbar, landed well behind the line but then bounced out again. And it all happened too quickly for the ref to spot it was a goal. How these kind of (non)-goals happen and what can we do about them?