The natural numbers, 1, 2, 3, 4, ..., are nice. So what could be nicer than discovering interesting patterns within them?

Like spirals and flowers? Then you'll love polar coordinates and the pretty pictures they allow you to draw!

The ability to see order in chaos has won the mathematician Yakov G. Sinai the 2014 Abel Prize.

There's no doubt that information is power, but could it be converted into physical energy you could heat a room with or run a machine on? In the 19th century James Clerk Maxwell invented a hypothetical being — a "demon" — that seemed to be able to do just that. The problem was that the little devil blatantly contravened the laws of physics. What is Maxwell's demon and how was it resolved?

A Gömböc is a strange thing. It wriggles and rolls around with an apparent will of its own. Until quite recently, no-one knew whether Gömböcs even existed. Even now, Gábor Domokos, one of their discoverers, reckons that in some sense they barely exists at all.

How chickens' eyes solve a subtle maths problem.

A clever statistical technique helped the Kepler mission to find the huge haul of new planets it announced last week.