The symmetries of matter are deeper and stranger than they first appear, and they have driven many of the biggest breakthroughs in particle physics. But have we exhausted their usefulness?

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.

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“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 that produce new symmetries. “There is a series of well-studied particle models whose symmetries live one inside the other, like a set of Russian dolls,” says Nichol Furey at Humboldt University of Berlin. “Symmetries themselves can have symmetries!”

### The explanatory power of symmetry

This richness reflects the importance of the concept in physics. Symmetry has proven particularly useful in developing theories because it is usually an indication that something can be simplified.

That is how we built the standard model of particle physics, which makes sense of matter and its workings. We defined its symmetries and the identity of all the particles it contains. Everything else, such as the dynamics and interactions of all the particles, can be derived from this, says Jonas Lindert at the University of Sussex, UK. “Symmetry is absolutely fundamental to particle physics.”

Physicists are especially interested in symmetries that disappear or “break”. The existence of the quark was predicted thanks to an observation of broken symmetry. And the search for the Higgs boson was motivated by the need to break a symmetry of the early universe in order to account for the mass of the W and Z bosons.

All of which raises the question: why is symmetry so vital to the universe? Du Sautoy thinks it is an indication of the fundamental role mathematics plays in reality. “My belief is that what we see around us is a physicalised piece of mathematics,” he says.

Whether that is true or not is another debate entirely. But Lindert is confident that symmetry is going to be key to future discoveries in physics. “Symmetry will always be absolutely fundamental in any new theory of nature,” he says.

On that point, Sabine Hossenfelder at the Frankfurt Institute for Advanced Studies in Germany, is more circumspect. Ancient symmetry-centric notions that the planets’ orbits should be circular and modern ideas of supersymmetry, where each known particle should have a hidden partner, haven’t stood up to scrutiny. “Symmetry is sometimes misleading,” she says.