For me it is Cellular Automata, and more precisely the Game of Life.
Imagine a giant Excel spreadsheet where the cells are randomly chosen to be either “alive” or “dead”. Each cell then follows a handful of simple rules.
For example, if a cell is “alive” but has less than 2 “alive” neighbors it “dies” by under-population. If the cell is “alive” and has more than three “alive” neighbors it “dies” from over-population, etc.
Then you sit back and just watch things play out. It turns out that these basic rules at the individual level lead to incredibly complex behaviors at the community level when you zoom out.
It kinda, sorta, maybe resembles… life.
There is colonization, reproduction, evolution, and sometimes even space flight!
Noether’s Theorem:
https://en.wikipedia.org/wiki/Noether%27s_theorem
https://math.ucr.edu/home/baez/noether.html
Fundamentally, it allows us to logically infer the conservation laws from the laws of motion of a given physical system using relatively simple math. It always applies, no matter if we’re talking about massive systems or quantum ones.
I think the concept is even more beautiful than you described:
A symmetry in a physical system implies a conservation law.
As a physicist, since the beginning of your studies you learn to appreciate and seek symmetries in various systems. At first, it’s mostly on an intuitive way to help you understand or simplify a problem. But at some point you learn about Noether’s theorem and see the even deeper meaning and power of symmetries.
For example, symmetry in movement in space (meaning I can move my entire system and it stays the same) implies conservation of momentum.
And symmetry in time translation (meaning if move the entire system a through a same interval in time and it still behaves the same way) implies conservation of energy.