In AD 856, a massive fleet of Viking ships appeared off the coast of East Anglia, heralding a change in Viking tactics - from raiding to invasion. Richards charts the years of attempted conquest that followed.

Mathematician Dr Hannah Fry explores the mystery of maths. It underpins so much of our modern world that it's hard to imagine life without its technological advances, but where exactly does maths come from? Is it invented like a language or is it something discovered and part of the fabric of the universe? It's a question that some of the most eminent mathematical minds have been wrestling with. To investigate this question, Hannah goes head first down the fastest zip wire in the world to learn more about Newton's law of gravity, she paraglides to understand where the theory of maths and its practice application collide, and she travels to infinity and beyond to discover that some infinities are bigger than others.
In this episode, Hannah goes back to the time of the ancient Greeks to find out why they were so fascinated by the connection between beautiful music and maths. The patterns our ancestors found in music are all around us, from the way a sunflower stores its seeds to the number of petals in a flower. Even the shapes of some of the smallest structures in nature, such as viruses, seem to follow the rules of maths. All strong evidence for maths being discovered. But there are those who claim maths is all in our heads and something we invented. To find out if this is true, Hannah has her brain scanned. It turns out there is a place in all our brains where we do maths, but that doesn't prove its invented.
Experiments with infants, who have never had a maths lesson in their lives, suggests we all come hardwired to do maths. Far from being a creation of the human mind, this is evidence for maths being something we discover. Then along comes the invention of zero to help make counting more convenient and the creation of imaginary numbers, and the balance is tilted in the direction of maths being something we invented. The question of whether maths is invented or discovered just got a whole lot more difficult to answer

Japan's role in World War II gets a whole new perspective in this consisting entirely of full colour footage, including colour films from Japan that were recently discovered. As the visuals of the world war take on a new vivid immediateness, the story of the rise of the militarists in Japan is told through the personal writings of the Japanese themselves. From the first overconfident tastes of victory, to the devastating losses that led to an unthinkable defeat amidst the ruins, the Pacific Theater of World War II is told through the Japanese's eyes.

The documentary is to tell the dramatic story of the greatest natural disaster to shake the ancient world, a disaster that triggered the downfall of a civilisation and spawned a legend. Around 1620 BC a gigantic volcano in the Aegean Sea stirred from its 19,000-year slumber. The eruption tore the island of Thera apart, producing massive tsunamis that flooded the nearby island of Crete, the centre of Europe's first great civilisation – the Minoans. This apocalyptic event, many experts now believe, provided the inspiration for the legend of Atlantis. Based on the work of leading scientists, archaeologists and historians, this drama immerses viewers in the exotic world of the Minoans.

Soldiers and officers continually came into conflict with their consciences. How much freedom did individuals have? Were they executing Hitler's criminal plans, or was it the Wehrmacht's war? There were crimes against humanity, against civilians, prisoners of war, and there was the Holocaust itself. The Trent Park records discovered and analysed by historian Sönke Neitzel, author of Tapping Hitler's Generals, show that the 84 German generals who were interned at Trent Park were aware of the severity of the war crimes they had been involved in and that some discussed them almost compulsively. They included the Commissar Order, to kill any Soviet commissar, and Rathenau's order to kill the Jews, including children.

Between the 1920s and the 1960s the world's great powers sent vast military-style expeditions to conquer the peaks of the Himalayas, with Everest at their head. This was a great game played - camera in hand - by Imperial Britain, Nazi Germany and superpower America. As a result, Himalayan mountaineering's most iconic, epic and tragic moments didn't just go down in history, but were caught on film - from the deaths of Mallory and Irvine on Everest in 1924, to Everest's final conquest in 1953 by Hillary and Tensing. Using footage never before seen on British television, this is the story how of how film-makers turned the great peaks into great propaganda.