Mariners had told tall tales of giant waves coming from nowhere and sinking ships. No one believed them. Until 1995, when an oil rig measured a wave 26 metres high, coming from nowhere. Now mathematics is helping ships to stay clear of regions where rogue waves are unusually common.
At school we all learn that rainbows are brightly coloured because a prism splits white light into lots of colours, and a drop of water acts like a prism. But there are lots of water drops in a shower of rain... so why don't the colours all smear out? What actually matters is not the colours, but the shape of the rainbow. To work that out, you need geometry.