Most of us have a rough idea that computers are made up of complicated hardware and software. But perhaps few of us know that the concept of a computer was envisioned long before these machines became ubiquitous items in our homes, offices and even pockets.

Geometric hallucinations are very common: people get them after taking drugs, following sensory deprivation, or even after rubbing their eyes. What can they tell us about how our brain works?

In 2004 three physicists decided to dabble in a field they knew little about. Within weeks they had developed a new technique that transforms weeks' worth of computer calculations into something that could be done on a single page in an hour. It's used in particle accelerators such as the LHC at CERN.

The natural logarithm is intimately related to the number e and that's how we learn about it at school. When it was first invented, though, people hadn't even heard of the number e and they weren't thinking about exponentiation either. How is that possible?

Georgian school maths: bushels of corn, kilderkins of beer and feeding soldiers. All without algebra!