Adam Rutherford meets a new creature created by American scientists, the spider-goat. It is part goat, part spider, and its milk can be used to create artificial spider's web. It is part of a new field of research, synthetic biology, with a radical aim: to break down nature into spare parts so that we can rebuild it however we please. This technology is already being used to make bio-diesel to power cars. Other researchers are looking at how we might, one day, control human emotions by sending 'biological machines' into our brains.

Professor Jim Al-Khalili investigates one of the most important concepts in the world today - information. He discovers how we harnessed the power of symbols, everything from the first alphabet to the electric telegraph through to the modern digital age. But on this journey he learns that information isn't just about human communication, it's woven very profoundly into the fabric of reality.

The second episode is about how plants gain their sustenance. Sunlight is one of the essential requirements if a seed is to germinate, and Attenborough highlights the cheese plant as an example whose young shoots head for the nearest tree trunk and then climb to the top of the forest canopy, developing its leaves en route. Using sunshine, air, water and a few minerals, the leaves are, in effect, the "factories" that produce food. However, some, such as the begonia, can thrive without much light. To gain moisture, plants typically use their roots to probe underground. Trees pump water up pipes that run inside their trunks, and Attenborough observes that a sycamore can do this at the rate of 450 litres an hour — in total silence. Too much rainfall can clog up a leaf's pores, and many have specially designed 'gutters' to cope with it. However, their biggest threat is from animals, and some require extreme methods of defence, such as spines, camouflage, or poison. Some can move quickly to deter predators: the mimosa can fold its leaves instantly when touched, and the Venus flytrap eats insects by closing its leaves around its prey when triggered. Another carnivorous plant is the trumpet pitcher that snares insects when they fall into its tubular leaves. Attenborough visits Borneo to see the largest pitcher of them all, Nepenthes rajah, whose traps contain up to two litres of water and have been known to kill small rodents.

brilliant mathematicians, whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide. Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity. Ludwig Boltzmann's struggle to prove the existence of atoms and probability eventually drove him to suicide.

By our third year, most of us will have learned to count. Once we know how, it seems as if there would be nothing to stop us counting forever. But, while infinity might seem like an perfectly innocent idea, keep counting and you enter a paradoxical world where nothing is as it seems. Mathematicians have discovered there are infinitely many infinities, each one infinitely bigger than the last. And if the universe goes on forever, the consequences are even more bizarre. In an infinite universe, there are infinitely many copies of the Earth and infinitely many copies of you. Older than time, bigger than the universe and stranger than fiction. This is the story of infinity.

The Cave of Swallows in Mexico is a 400m vertical shaft, deep enough to engulf the Empire State Building. The Lechuguilla cave system in the USA is 193km long and 500m deep with astonishing crystal formations hanging from its chambers. Although often overlooked, caves are remarkable habitats with equally bizarre wildlife. Cave angel fish cling to the walls behind cave waterfalls with microscopic hooks on their flattened fins. Cave swiftlets navigate by echo-location and build nests out of saliva. The Texas cave salamander has neither eyes nor pigment. Unique access to a hidden world of stalactites, stalagmites, snotites and troglodytes brings a wealth of surprises.