Animation of a mass on a spring. Its position, a blue wave, and its momentum, a red wave, keep changing over time, while a circle showing its total energy stays exactly the same size.
My favourite thing in science is Emmy Noether’s theorem. Einstein described her as the most significant creative mathematical genius since women gained access to higher education, yet most people have never heard of her, because her big idea is hard to picture. So let me try.
Watch the spring. Its position (blue) and momentum (red) are always changing. When it’s stretched furthest it’s momentarily still, and when it whips through the middle it’s at its fastest. But the black circle, the total energy, never changes. Leave it for an hour and come back: same circle, same size.
Why? Because the laws of physics don’t care what time it is. A spring behaves the same today as it will tomorrow. Noether proved that every symmetry like this comes with something that can never change. Rules that don’t change over time give you conservation of energy. Rules that are the same everywhere give you conservation of momentum. Rules that don’t care which way you face give you conservation of angular momentum.
That’s why physicists hunt for symmetries: each one hands you a quantity you can stop tracking, and the maths gets much simpler. Machine learning uses the same trick. A model that knows a cat is a cat wherever it appears in an image doesn’t have to relearn it at every position, which makes it far more efficient. That is the idea behind convolutional neural networks.