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The big blue is the world's greatest wilderness, far from shore and many kilometres deep. It's a vast marine desert where there is little to eat and nowhere to hide. Yet it's home to some of the biggest and most spectacular creatures on earth. This episode reveals what it takes to survive in this savage and forbidding world. We witness feats of incredible endurance, moments of high drama and extraordinary acts of heart-wrenching self-sacrifice. Every animal in the big blue must find their own unique way to survive. Sperm whales have the largest brains in the world. They live for 80 years, and we are only now beginning to learn the extraordinary complexity of their language of clicks - thought to coordinate the whole family in everything from childcare to hunting. With special pressure-proof cameras, we witness record-breaking feats of endurance as they hunt for squid a kilometre down into the abyss. Only recently have we begun to solve the mystery of where baby turtles disappear to in their early years. They leave the crowded waters of the coast and head to the open ocean, where they use floating debris like logs as life rafts. Here they remain until adulthood, adrift on the high seas in relative safety away from coastal predators. Over half of all animals in the open ocean drift in currents. Jellyfish cross entire oceans feeding on whatever happens to tangle with their tentacles. The jelly-like Portuguese man-of-war can harness sail power to fish with its deadly tentacles. Sometimes there is a brief explosion of food in this marine desert, but ocean hunters must be fast to make the best of this bonanza. We witness super pods of up to 5,000 spinner dolphins racing to herd vast shoals of lanternfish, briefly caught at the surface where it is thought they spawn. New aerial footage reveals, for the first time, the truth to a centuries-old sailors' legend of the 'boiling seas' - the spectacular feeding frenzy of 90kg tuna and dolphins smashing through the lantern fish shoals turning the sea white with foam.

On the coast, two worlds collide. Coasts are the most dynamic and challenging habitats in the ocean - that brings great rewards but also great danger. The extraordinary animals that live here must find ingenious ways to cope with two very different worlds. This episode is a rollercoaster ride of heart-stopping action and epic drama, peopled with characters from the beautiful to the bizarre. We meet fish that live on dry land and puffins that must travel 60 miles or more for a single meal, and witness a life-and-death struggle in a technicolour rock pool. In a secluded cove in the Galapagos, sea lions feast on 60kg tuna. It should be impossible - tuna are usually far too fast for sea lions to catch. But here the sea lions club together to herd their prey inshore. Once trapped in the shallows, these huge fish are easy pickings. As the tide recedes in Brazil, lightfoot crabs leap from rock to rock, desperately avoiding the water - their lives depend on it. Moray eels launch themselves from rock pools, jaw gaping. Then octopuses, too. Both crawl across dry rock to set their ambush. Elsewhere, the ever-changing tides create rock pools. But these temporary worlds are a battleground. Predatory starfish turn a magical garden into the stuff of nightmares.

The years 1825-1918 were bloody and traumatic, a period when four tsars tried - and failed - to deal with the growing pressure for constitutional reform and revolution. In 1861, millions of enslaved serfs were freed by the Tsar-Liberator, Alexander II. But Alexander paid the ultimate penalty for opening the Pandora's box of reform when he was later blown up by terrorists on the streets of St Petersburg. Elsewhere, there was repression, denial, war and - in the case of the last tsar, Nicholas II - a fatalistic belief in the power of God, with Nicholas's faith in the notorious holy man Rasputin being a major part in his undoing. Lucy Worsley also details the chilling murder of Nicholas and his family in 1918, and asks whether all of this horror have been avoided.

Every year grizzly bear families in North America depend for their survival on a spectacular natural event: the return of hundreds of millions of salmon from the Pacific Ocean to the mountain streams where they were born. The salmon travel thousands of miles to spawn and then die. The great run not only provides food for bears, but for killer whales, wolves, bald eagles, and even the forest itself. The question is: will the salmon return in time to keep hungry bears alive?
A mother grizzly and her cubs emerge from their den high in snowy Alaskan mountains. Filming from the air the team capture a TV first, following the bears as they negotiate a near vertical slope on their journey to the coast where they await the return of the salmon. Meanwhile, the salmon are making their way to the to river mouths where they must swim upstream and against the current. The programme reveals how they tackle the torrents and leap over waterfalls, a feat equivalent to a human jumping over a house. Dozens of hungry bears eagerly await the salmon that make it up river. In another TV first, underwater cameras record the ingenuity and fancy footwork they use to collect dead salmon from the bottom of deep pools.

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.

'It's really about terror and intimidation and people basically fighting for survival and often committing extraordinary violence in order to protect themselves or to stay safe. It's kind of like the weak dog in the pack. If others spot weakness, they're gonna pounce on you for a couple of reasons. You pounce on that guy, that gives you a little more status. So, I had to ask myself, are you gonna be a victim? No, I ain't gonna be a victim. Well, that really only left me one choice in my mind. That means I gotta be the victimizer. At some point, for some reason, might be legitimate, might not be, someone's gonna test you.. Even if you lose, you're gonna have to stand up for yourself. '
'A guy comes over, and it's your day to get your package, and he tries to take your package from you that your people sent you, if you let him do it, there's gonna be 10 other dudes, oh, yeah, he let that guy take his package, I'm gonna go get his TV, right down the line until somebody's after your ass. But if you stand up that first time and they see you'll stand up for yourself, even if you lose, people will respect that. Oh, don't mess with him. There's easier prey. Why do you have to go and get that guy and get a couple lumps for it when he can go get that guy's stuff over there, don't cost you nothing. It usually only has to happen once or twice but just as importantly that you didn't go to the man for help... you're gonna be all right.'