In the fourth episode, Professor Marcus du Sautoy concludes his investigation into the history of mathematics with a look at some of the great unsolved problems that confronted mathematicians in the 20th century. After exploring Georg Cantor's work on infinity and Henri Poincare's work on chaos theory, he sees how mathematics was itself thrown into chaos by the discoveries of Kurt Godel and Paul Cohen, before completing his journey by considering some unsolved problems of maths today, including the Riemann Hypothesis.

Chaos theory has a bad name, conjuring up images of unpredictable weather, economic crashes and science gone wrong. But there is a fascinating and hidden side to Chaos, one that scientists are only now beginning to understand. It turns out that chaos theory answers a question that mankind has asked for millennia - how did we get here? In this documentary, Professor Jim Al-Khalili sets out to uncover one of the great mysteries of science - how does a universe that starts off as dust end up with intelligent life? How does order emerge from disorder? It's a mindbending, counterintuitive and for many people a deeply troubling idea. But Professor Al-Khalili reveals the science behind much of beauty and structure in the natural world and discovers that far from it being magic or an act of God, it is in fact an intrinsic part of the laws of physics. Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern. And the best thing is that one doesn't need to be a scientist to understand it. The natural world is full of awe-inspiring examples of the way nature transforms simplicity into complexity. From trees to clouds to humans - after watching this film you'll never be able to look at the world in the same way again.

Four and a half billion years ago, the young Earth was a hellish place, a seething chaos of meteorite impacts, volcanoes belching noxious gases, and lightning flashing through a thin, torrid atmosphere. Then, in a process that has puzzled scientists for decades, life emerged. But how? Join mineralogist Robert Hazen as he journeys around the globe. From an ancient Moroccan market to the Australian Outback, he advances a startling and counterintuitive idea—that the rocks beneath our feet were not only essential to jump-starting life, but that microbial life helped give birth to hundreds of minerals we know and depend on today. It's a theory of the co-evolution of Earth and life that is reshaping the grand-narrative of our planet’s story.

After the murder of his brother Frank, Al Capone is out for revenge against William Dever and the Chicago police. He plots to kill every undercover cop that was in Cicero on the day of Frank’s death. Before he can set his plan in motion, however, Johnny Torrio returns from Italy and reclaims his control of their criminal empire. Torrio forbids Capone from killing cops, fearing it would spark an all-out war. Despite Capone’s personal loss, Torrio returns to a booming business that’s raking in $10 million a month. However, as Torrio and Capone’s empire grows, it becomes harder to keep smaller factions under control. After a small group of Italian mobsters attacks one of Dean O’Banion’s distilleries, O’Banion blames Torrio and decides to end their business partnership. He announces his retirement to Torrio and offers Torrio an opportunity to buy him out. Torrio meets O’Banion at the Seine Brewery on May 19, 1924, with the buyout money. Chicago police raid the brewery and arrest Torrio while letting O’Banion walk free: O’Banion had set the whole thing up. With prior convictions, Torrio serves a mandatory prison sentence but is first released on bail. He vows to kill O’Banion as payback and gives Capone his blessing to kill O’Banion. Capone spends months planning the hit. Capone knows that catching O’Banion off guard is nearly impossible, so he places an order with O’Banion’s flower shop for nearly $20,000 worth of flowers, knowing that O’Banion personally oversees his biggest arrangements. On November 10, 1924, O’Banion works in his shop preparing the flower delivery. Two of Capone’s men enter the shop and fatally gun O’Banion down. Four days later, Chicago’s most powerful Irish gangster is laid to rest in one of the most lavish funerals Chicago has ever seen. More than two dozen cars are required to transport floral arrangements as a crowd of 10,000 mourns the death of the infamous gangster. Torrio and Capone attend the funeral, infuriating O’Banion’s top lieutenants, Hymie Weiss and Bugs Moran. Chaos breaks out as smaller gangs through the city begin a turf war. It’s the beginning of an era of bloodshed in Chicago known as the Beer Wars. Over 300 people are murdered at a rate of 75 to 100 murders a year.

Andrew Marr tells the story of the first empires which laid the foundations for the modern world. From the Assyrians to Alexander the Great, conquerors rampaged across the Middle East and vicious wars were fought all the way from China to the Mediterranean. But this time of chaos and destruction also brought enormous progress and inspired human development. In the Middle East, the Phoenicians invented the alphabet, and one of the most powerful ideas in world history emerged: the belief in just one God. In India, the Buddha offered a radical alternative to empire building - a way of living that had no place for violence or hierarchy and was open to everyone. Great thinkers from Socrates to Confucius proposed new ideas about how to rule more wisely and live in a better society. And in Greece, democracy was born - the greatest political experiment of all. But within just a few years, its future would be under threat from invasion by an empire in the east...

The final part examines political revolution and how art was at the forefront of throwing out 1,000 years of royal rule, from its earliest revolutionary days of enthusiasm and optimism when painting died, the poster was king and the machine-made triumphed over the handmade to the dead hand of Socialist Realism. Andrew roots out great portraits of Stalin now hidden in museum storerooms and never on public view, looks at the transformation of the Moscow metro into a great public art gallery and visits the most stunning creation of post-war Communist rule, the Space Monument. Finally, he comes to the confusion and chaos of Russia today and how it is producing some of the world's strangest art - from heroic sculptures of Russian leader Vladimir Putin to the insides of a giant erotic apple; from the recreation of the Imperial royal family facing the firing squad to sculpture in liquid oil; from Russia's embrace of the commercial art market to a return to Socialist Realism. Russia seems to stand on another brink of revolution.