Lie groups and Noether's theorem are the tools that let 20th-century physicists wield *symmetry* as a powerful tool to study crystals, molecules, atoms, and elementary particles. Lie groups let us describe symmetries that can vary continuously, not just discretely. Noether's theorem shows how these continuously varying symmetries give conserved quantities like energy, momentum, angular momentum and others.

But what is a Lie group? This Quanta article does a pretty good job of explaining the basic idea. One quibble: they say

"The group of all rotations of a ball in space, known to mathematicians as SO(3), is a complicated three-dimensional shape that lives in nine-dimensional space."

That makes it sound more complicated than it really is! I'd rather say

"The group of all rotations of a ball in space, known to mathematicians as SO(3), is itself a ball with each point on its surface glued to the opposite point."

If you're gonna scare people, I feel you might as well scare them with a puzzle they can actually chew on. I guess that's why I'm not a journalist. If you say "a complicated shape that lives in nine-dimensional space", most people will feel this is utterly beyond their powers to understand. And in a way, that's relaxing: it means math is only for geniuses, not them, so they can stop thinking.

But if you say "a ball with each point on its surface glued to the opposite point", people may say WHAT THE HELL???

I feel journalists don't want that. As a blogger, that's exactly what I *do* want. You may be able to figure it out (it's easy with a picture). But if you've got questions, I'm happy to answer them.

https://www.quantamagazine.org/what-are-lie-groups-20251203/