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.

Born in Naples, Bernini was an exceptional talent from an early age and went on to dominate the art world of 17th century Rome. His work epitomised the Baroque style and his sculpture, church interiors and exteriors and town planning could be seen everywhere. Bernini worked under successive Popes; Pope Gregory XV made him a knight and Pope Urban VIII made him as his best friend. He was revered in his time until a jealous rage caused him to have the face of his mistress slashed after discovering her romance with his brother. His reputation fell further after his bell towers for the Cathedral of St Peter's started cracking in 1641. He redeemed himself and kick started his career again with what is arguably his most famous work, The Ecstasy of St Theresa, in 1652.

There could be an undiscovered species on Earth unlike anything we’ve ever known. Not in the ocean, but the ocean itself! Its body spans thousands of miles; its heart beats with a one-thousand-year pulse. It could even have an immune system capable of annihilating all other life on earth. Just as our bodies function through the interaction of water with individual cells, including bacteria and other microorganisms, the ocean’s residents might collectively form a super-organism. A recent discovery suggests the ocean is a living being capable of thought. If so, what is the ocean thinking about us?

Beneath Jupiter's swirling clouds lie our solar system's deepest secrets: from its violent youth, through the birth of life to the death of the sun. Now, scientists are unlocking these secrets and discovering that every living thing exists thanks to Jupiter.

Humans are so clever they’ve built their own habitat, designed to keep wild nature out. Yet as we discover the urban environment is totally dependant on the natural world. Over the three years of filming for Human Planet, the teams have spent time with over seventy different communities in countries across the planet. Despite seeing many different ways of living, there are some aspects of family life that remain the same whether you live in a tree house in West Papua, or a brick house in Bolton. These similarities became very clear when the urban team went to film on a rubbish dump in Kenya. Here on the Kibarani dump on the outskirts of Mombasa, Ali, Ashe and their family live and survive amongst the rubbish, and even here, just like anywhere on the planet, everything stops for a nice cup of tea!

Since the dawn of civilization, humans have wondered who or what created the universe. Religion offers a spiritual answer, but do the latest discoveries in physics show evidence of a transcendent intelligence, or simply that the laws of physics by themselves could have led to the universe in which we live? This episode embarks on a mind-bending scientific search for God, asking physicists and theologians if the seemingly miraculous way the universe has been calibrated to support life is evidence of a creator...whether string theory will eventually be able to rule out the existence of God...why Stephen Hawking says the universe could have been created spontaneously...and how an advanced civilization in another universe could have conceivably created our own.