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

the final episode deals with plants that live in hostile environments. Attenborough visits Ellesmere Island, north of the Arctic Circle, to demonstrate that even in a place that is unconducive to life, it can be found. Algae and lichens grow in or on rock, and during summer, when the ice melts, flowers are much more apparent. However, they must remain close to the ground to stay out of the chilling wind. In the Tasmanian mountains, plants conserve heat by growing into 'cushions' that act as solar panels, with as many as a million individual shoots grouped together as one. Others, such as the lobelia in Mount Kenya, have a 'fur coat' of dense hairs on their leaves. The saguaro cactus in the Sonoran Desert flourishes because of its ability to retain vast amounts of water, which can't be lost through leaves because it has none. Many desert dwellers benefit from an accelerated life cycle, blooming rapidly within weeks after rainfall. Conversely, Mount Roraima is one of the wettest places on Earth. It is a huge sandstone plateau with high waterfalls and nutrients are continuously washed away, so plants have to adapt their diet if they are to survive. A bladderwort is shown invading a bromeliad. Inhabitants of lakes have other problems to contend with: those that dominate the surface will proliferate, and the Amazon water lily provides an apt illustration. Attenborough ends the series with an entreaty for the conservation of plant species.

The film reveals the secret story behind one of the greatest intellectual feats of World War II, a feat that gave birth to the digital age.
In 1943 Bill Tutte, a 24-year-old maths student and a GPO engineer called Tommy Flowers combined to hack into Hitler's personal super code machine - not Enigma but an even tougher system, which he called his 'secrets writer'. Their break turned the Battle of Kursk, powered the D-day landings and orchestrated the end of the conflict in Europe. But it was also to be used during the Cold War - which meant both men's achievements were hushed up and never officially recognised.

Dive with David Attenborough into a world where a single life can last a thousand years. See things no eye has ever seen, and discover the dramatic, beautiful plant life of Earth. In the first episode, Sir David Attenborough shows how more kinds of plants are crammed together in the tropical rainforests than anywhere else on Earth. The result is astonishing beauty and intense competition - a plant battleground.
New filming techniques allow us to enter the plants’ world and see it from their perspective and on their timescale. From fast-growing trees to flowers that mimic dead animals, this is a journey into a magical world that operates on a different timescale to our own.

After the murder of his brother Frank, Al Capone is out for revenge against William Dever and the Chicago police. He plots to kill every undercover cop that was in Cicero on the day of Frank’s death. Before he can set his plan in motion, however, Johnny Torrio returns from Italy and reclaims his control of their criminal empire. Torrio forbids Capone from killing cops, fearing it would spark an all-out war. Despite Capone’s personal loss, Torrio returns to a booming business that’s raking in $10 million a month. However, as Torrio and Capone’s empire grows, it becomes harder to keep smaller factions under control. After a small group of Italian mobsters attacks one of Dean O’Banion’s distilleries, O’Banion blames Torrio and decides to end their business partnership. He announces his retirement to Torrio and offers Torrio an opportunity to buy him out. Torrio meets O’Banion at the Seine Brewery on May 19, 1924, with the buyout money. Chicago police raid the brewery and arrest Torrio while letting O’Banion walk free: O’Banion had set the whole thing up. With prior convictions, Torrio serves a mandatory prison sentence but is first released on bail. He vows to kill O’Banion as payback and gives Capone his blessing to kill O’Banion. Capone spends months planning the hit. Capone knows that catching O’Banion off guard is nearly impossible, so he places an order with O’Banion’s flower shop for nearly $20,000 worth of flowers, knowing that O’Banion personally oversees his biggest arrangements. On November 10, 1924, O’Banion works in his shop preparing the flower delivery. Two of Capone’s men enter the shop and fatally gun O’Banion down. Four days later, Chicago’s most powerful Irish gangster is laid to rest in one of the most lavish funerals Chicago has ever seen. More than two dozen cars are required to transport floral arrangements as a crowd of 10,000 mourns the death of the infamous gangster. Torrio and Capone attend the funeral, infuriating O’Banion’s top lieutenants, Hymie Weiss and Bugs Moran. Chaos breaks out as smaller gangs through the city begin a turf war. It’s the beginning of an era of bloodshed in Chicago known as the Beer Wars. Over 300 people are murdered at a rate of 75 to 100 murders a year.

The story of the British Library's Royal Manuscripts collection reaches its end with the last great flowering of illumination, in the magnificent courts of the Tudors. Dr Janina Ramirez investigates astrological texts created for Henry VII, and unwraps his will - still in its original, extravagantly-decorated velvet and gold cover. She hears music written for Henry VIII, which went unperformed for centuries; and reads love notes between the king and Anne Boleyn, written in the margins of a prayer book. Nina also visits Bruges, the source of many of the greatest manuscripts, where this medieval art form collided with the artistic innovations of the Renaissance.