# Euclidean geometry

If a shape has equal sides with 90 degree angles between them then it's a square, right? Well, not quite...
As we are well over half way through advent, we thought today we'd look towards the future of the Universe
How will the Universe end? In a big crunch? Or a big freeze? It all depends on its shape...
So easy to describe, yet so hard to prove.
The 19th century experienced a geometrical revolution. Find out how the new geometries that were discovered shaped philosophy, science, culture and art.
This article explores how Euclid's ancient geometry interacts with all aspects of human thought and life.
The impossible becomes possible when you move into the third dimension.
Did you learn at school that the angles in a triangle always add up to 180 degrees? If yes then your teacher was wrong. Find out why here.
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.

### The Plus anniversary year — A word from the editors

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.
Runner up in the general public category. Great minds spark controversy. This is something you'd expect to hear about a great philosopher or artist, but not about a mathematician. Get ready to bin your stereotypes as Rebecca Morris describes some controversial ideas of the great mathematician David Hilbert.