Dr Hannah Fry travels down the fastest zip wire in the world to learn more about Newton's ideas on gravity. His discoveries revealed the movement of the planets was regular and predictable. James Clerk Maxwell unified the ideas of electricity and magnetism, and explained what light was. As if that wasn't enough, he also predicted the existence of radio waves. His tools of the trade were nothing more than pure mathematics. All strong evidence for maths being discovered.
But in the 19th century, maths is turned on its head when new types of geometry are invented. No longer is the kind of geometry we learned in school the final say on the subject. If maths is more like a game, albeit a complicated one, where we can change the rules, surely this points to maths being something we invent - a product of the human mind. To try and answer this question, Hannah travels to Halle in Germany on the trail of perhaps one of the greatest mathematicians of the 20th century, Georg Cantor. He showed that infinity, far from being infinitely big, actually comes in different sizes, some bigger than others. This increasingly weird world is feeling more and more like something we've invented. But if that's the case, why is maths so uncannily good at predicting the world around us? Invented or discovered, this question just got a lot harder to answer.

Every summer in the seas off Alaska humpback whales, sea lions and killer whales depend on an explosion of plant life, the plankton bloom. It transforms these seas into the richest on Earth. But will these animals survive to enjoy the great feast? The summer sun sparks the growth of phytoplankton, microscopic floating plants which can bloom in such vast numbers that they eclipse even the Amazon rainforest in sheer abundance of plant life. Remarkably, it is these minute plants that are the basis of all life here. But both whales and sea lions have obstacles to overcome before they can enjoy the feast. Humpback whales migrate 3,000 miles from Hawaii, and during their 3 month voyage lose a third of their body weight. In a heart-rending scene a mother sea lion loses her pup in a violent summer storm, while another dramatic sequence shows a group of killer whales working together to kill a huge male sea lion.
In late summer the plankton bloom is at its height. Vast shoals of herring gather to feed on it, diving birds round the fish up into a bait ball and then a humpback whale roars in to scoop up the entire ball of herring in one huge mouthful. When a dozen whales work together they employ the ultimate method of co-operative fishing - bubble net feeding. One whale blows a ring of bubbles to engulf the fish and then they charge in as one. Filmed from the surface, underwater and, for the first time, from the air, we reveal how these giant hunters can catch a tonne of fish every day.

American troops launch widespread 'search and destroy' operations; body count, not territory, becomes the measure of success in Vietnam. Charles Brown fights for survival on the bloody slopes of Hill 875 and at home, the American public begins to question U.S. military strategy.

The enemy gains ground when the massive Tet Offensive catches the Americans by surprise. At Khe Sanh and Pleiku, U.S. troops are under siege. Americans mount a counteroffensive but the shock of the assault ignites anti-war fervour at home.

The turmoil of 1968 helps Nixon win the presidency. Troop strength in Vietnam peaks and the draft accelerates. Karl Marlantes endures bitter jungle fighting on Hill 484, and draftee Arthur Wiknik gets a grim introduction to combat on 'Hamburger Hill'. But shortly after winning both hills, US troops abandon them.

Troop withdrawals begin as the Americans train their ARVN allies to take over the war. Gery Benedetti patrols the hostile waters of the Mekong Delta. James Anderson leads a battalion into Cambodia after Nixon's controversial order. Don DeVore struggles to survive a fierce night attack, and Anne Purcell gets promising news about her husband.