## Articles

John Barrow gives us an overview, from Aristotle's ideas to Cantor's never-ending tower of mathematical infinities, and from shock waves to black holes.

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?

Horses, like all animals, have a number of different gaits. But how can they perform these complicated leg movements without having to stop and think? And why do they switch to a new gait when they want to go faster? Mathematics can shed some light on these questions.

Does it pay to be nice? Yes, it does. And we're not just talking about that warm fuzzy feeling inside, it pays in evolutionary terms of genetic success too. We talk to Martin Nowak about how the mathematics of evolution prove that being nice is unavoidable.

The holy grail for 21st century physics is to produce a unified theory of everything that can describe the world at every level, from the tiniest particles to the largest galaxies. Currently the strongest contender for such a theory is something called M-theory. So what is this supposed mother of all theories all about?

Some things are so familiar to us that they are simply expected, and we may forget to wonder why they should be that way in the first place. Sex ratios are a good example of this: the number of men and women in the world is roughly equal, but why should this be the case? A simple mathematical argument provides an answer.

The number 1 can be written as a sum of unit fractions, that is fractions with 1 in the numerator. But how long can we make such a sum?

This is the first part of the lecture given by Astronomer Royal Martin Rees at Stephen Hawking's birthday symposium.