YOU might remember learning about symmetry at school. Maybe a teacher showed you a snowflake’s six-fold symmetry and you marvelled at how it looked the same no matter how you rotated it. Well, it turns out that the wonders of symmetry go a whole lot deeper – as any mathematician who has studied it will tell you.

“Instead of being something visual, which is what I responded to as a child, it became something much more abstract and linguistic in nature,” says Marcus du Sautoy, a mathematician at the University of Oxford. “The understanding of symmetry I have now is so much deeper and stranger, and it gives me access to symmetries that are so much more exotic than anything you can see with your eyes.”

For mathematicians, a symmetry is a type of invariance – when something remains unchanged under some kind of transformation, such as flipping it or rotating it. That sounds simple enough, but, as du Sautoy suggests, most symmetries go beyond what is obvious to a casual observer.

Consider antimatter, which is what you get when positively charged particles become negative and vice versa. If no significant effects occur, then the system involved has charge symmetry. The laws of physics as we understand them suggest that the very early universe should have had equal amounts of matter and antimatter and then immediately annihilated itself. The fact that this didn’t happen means there was no charge symmetry in the newborn universe – understanding why is one of the biggest tasks in physics.

Matter’s symmetries aren’t just a laundry list of things that are invariant under some change, however. They can relate to each other in ways …