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Andrew Graham-Dixon continues his exploration of German art by looking at the tumultuous 19th and early 20th centuries, and how artists were at the forefront of Germany's drive to become a single nation. Andrew travels to the north and the coastal town of Griefswald, the birthplace of Caspar David Friedrich, the most influential of the German Romantics, to discover how the Baltic coast impacted his mysterious paintings of the German landscape. He also visits Berlin and explores the art of the powerful Prussian state, which would spearhead the unification of Germany in 1871. The episode ends with the outbreak of World War I and the attempts of the artists Franz Marc and Otto Dix to rationalise the catastrophic experiences of the world's first technological war, a war driven by the Prussian innovations.

The German army launched Operation Barbarossa, the invasion of the East, with supreme confidence in its own superiority. High Command was sure that victory would come as quickly and as easily as in France. At the onset, the troops advanced rapidly and the Panzer forces under General Guderian were unbeatable. There were, however, far more casualties and more wear-and-tear on equipment than anticipated. The invading force got bogged down short of Moscow and the withdrawal from there was a severe psychological blow to the men. When an extremely severe winter set in, morale sank further. The soldiers were not equipped for such low temperatures and their guns and machinery could not cope. Hitler ordered his troops to stand firm.

Among the ranks of the Wehrmacht there was limited opposition to Hitler. Most officers initially felt enthusiastic about the prospect of war and were grateful to Hitler whose war-mongering had furthered their personal careers. Furthermore all soldiers had to make an oath of allegiance to the Führer and, up until the very end, there were many who felt unable to break this pledge. There were, however, some officers who opposed Hitler. For some it was on moral grounds and for others because they thought that his military tactics would lose them the war. There were plots against Hitler and attempted coups, the best known being the July bomb plot led by Claus von Stauffenberg. Some soldiers refused to bear arms and were shot, others deserted, others still carried out quiet acts of sabotage and subversion. By and large, however, the Wehrmacht carried out the orders Hitler issued, even if it involved perpetrating acts of atrocity.

The determination of the German forces to keep on fighting in the face of defeat had disastrous consequences. After the Allied landings in the summer of 1944, the Wehrmacht was on the defensive on all fronts. It was clear to the German generals interned at Trent Park that Germany would soon lose the war. In mid-1944, Gerhard Graf von Schwerin chose to use common sense instead of blindly obeying Hitler's orders. He decided to surrender the city of Aachen to the US army to avoid bloodshed. Other commanders such as Field Marshal Ferdinand Schorner kept on pushing their soldiers to give their all. Despite being outnumbered by the Soviet forces, Schorner forced his soldiers to hold out in Sworbe, a 200 square kilometre peninsula on the coast of Estonia. Thousands of soldiers died. Yet Schorner's attitude and the urging by Hitler and Goebbels to hold out were accepted by a large number of young soldiers. By April 1944, the Ruhr pocket was completely surrounded, yet Field Marshal Model refused to surrender, so that 1.2 million German soldiers and a large number of Allied soldiers died between January and May 1945.

There is a tendency to deny German culture the equal reverence of Italy or Spain, and this enlightening new series provides a wonderful opportunity to explore a great, yet often neglected, artistic tradition whose influence has been just as profound. Andrew Graham-Dixon concludes his exploration of German art by investigating the dark and difficult times of the 20th century. Dominating the landscape is the figure of Adolf Hitler, failed artist, would-be architect and obsessed with the aesthetics of his 1,000-year Reich". In a series of extraordinary building projects and exhibitions, Hitler waged a propaganda war against every form of modern art as a prelude to unleashing total war on the whole of Europe. After the war the shadow of the Third Reich persisted, Germany remained divided and traumatised. How would artists deal with a past that everybody wanted to forget? Journeying through the work of Otto Dix and George Grosz and the age of the Bauhaus to the post-war painters Georg Baselitz, Hilla Becher and the conceptual artist Joseph Beuys is a long and strange journey, but the signs that art has a place at the heart of the new reunited Germany are clearly visible.

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.