A story from geometry shows how developments in mathematics have fundamentally changed the way we think about the world around us.
We can't visualise it, but we can still think about it! And there are clever ways of glimpsing what it might look like.
If a shape has equal sides with 90 degree angles between them then it's a square, right? Well, not quite...
In normal life higher dimensions smack of science fiction, but in mathematics they are nothing out of the ordinary.
How will the Universe end? In a big crunch? Or a big freeze? It all depends on its shape...
The impossible becomes possible when you move into the third dimension.
Keats complained that a mathematical explanation of rainbows robs them of their magic, conquering "all mysteries by rule and line". But rainbow geometry is just as elegant as the rainbows themselves.
Squares do it, triangles do it, even hexagons do it — but pentagons don't. They just won't fit together to tile a flat surface. So are there any tilings based on fiveness? Craig Kaplan takes us through the five-fold tiling problem and uncovers some interesting designs in the process.
