In the first part of this article we explored Landau's theory of phase transitions in materials such as magnets. We now go on to see how this theory formed the basis of the Higgs mechanism, which postulates the existence of the mysterious Higgs boson and explains how the particles that make up our Universe came to have mass.

Quantum mechanics and general relativity are incompatible — and this has led to a decades-long search for a theory of quantum gravity that could combine the two. But the particle physicist Richard Woodard thinks that the mismatch between the two could be nothing more than an illusion, created by the complicated maths techniques used in attempts to unite them.

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.

A simple question to ask about kayak races is whether having lots of paddlers helps or slows the boat down? The kayak with two paddlers has twice as many "engines" to power it but it also has twice as much weight to drag through the water. Which is the dominant factor?

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.

It does pay to be nice if you repeatedly deal with the same person. Martin Nowak explains why cooperation also wins in matters of reputation, neighbourliness and family. But can evolutionary game theory save the world?

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?

The men's and women's 100 metre freestyle swimming races are set to be two of the most glamorous events of the London 2012 Olympic Games. Much has been made of the swimming events for London 2012 because the previous 2008 Beijing Olympics saw an unprecedented number of new world records, due to the use of controversial swimsuits. So how do these suits improve performance?