In the third episode we will see Europe by the 17th century taking over from the Middle East as the powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was on to discover the mathematics to describe objects in motion. This programme explores the work of Rene Descartes, Pierre Fermat, Isaac Newton, Leonard Euler and Carl Friedrich Gauss. Du Sautoy proceeds to describes René Descartes realisation that it was possible to describe curved lines as equations and thus link algebra and geometry. He talks with Henk J. M. Bos about Descartes. He shows how one of Pierre de Fermat’s theorems is now the basis for the codes that protect credit card transactions on the internet. He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the Leibniz and Newton calculus controversy and the Bernoulli family. He further covers Leonhard Euler, the father of topology, and Gauss' invention of a new way of handling equations, modular arithmetic. The further contribution of Gauss to our understanding of how prime numbers are distributed is covered thus providing the platform for Bernhard Riemann's theories on prime numbers. In addition Riemann worked on the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.

Visit the ghostly villages surrounding Chernobyl (abandoned by humans after the 1986 nuclear disaster), travel to remote islands off the coast of Maine to search for abandoned towns that have vanished from view in only a few decades, then head beneath the streets of New York to see how subway tunnels may become watery canals. A visual journey, LIFE AFTER PEOPLE is a thought provoking adventure that combines movie-quality visual effects with insights from experts in the fields of engineering, botany, ecology, biology, geology, climatology, and archaeology to demonstrate how the very landscape of our planet will change in our absence.

Leading authority on the Middle Ages, Professor Robert Bartlett, presents a series which examines the way we thought during medieval times. In the second episode, Bartlett unearths remarkable evidence of the complex passions of Medieval men and women. The Church preached hatred of the flesh, promoted the cult of virginity and condemned woman as the sinful heir to Eve. Yet this was the era that gave birth to the idea of romantic love.

The Cave of Swallows in Mexico is a 400m vertical shaft, deep enough to engulf the Empire State Building. The Lechuguilla cave system in the USA is 193km long and 500m deep with astonishing crystal formations hanging from its chambers. Although often overlooked, caves are remarkable habitats with equally bizarre wildlife. Cave angel fish cling to the walls behind cave waterfalls with microscopic hooks on their flattened fins. Cave swiftlets navigate by echo-location and build nests out of saliva. The Texas cave salamander has neither eyes nor pigment. Unique access to a hidden world of stalactites, stalagmites, snotites and troglodytes brings a wealth of surprises.

The Arctic and Antarctic experience the most extreme seasons on Earth. Time-lapse cameras watch a colony of emperor penguins, transforming them into a single organism. The film reveals new science about the dynamics of emperor penguin behaviour. In the north, unique aerial images show a polar bear swimming more than 100km. Diving for up to two minutes at a time. The exhausted polar bear later attacks a herd of walrus in a true clash of the Titans.

The rise of alternative rock in the USA. From its early underground days where bands like Black Flag drew inspiration from the DIY ethos of punk. We explore why the bands that emerged from the underground offered an alternative both to the established music industry and the prevailing politics. Bands like R.E.M., The Pixies and Husker Du, REM breaking into the mainstream charts with 'Losing My Religion', Seattle 'grunge' scene, culminating in the success of Nirvana's 'Nevermind'