What is Weyl's criterion?

Weyl’s criterion is like having a special rule that tells you when a group of dancers moves in perfect harmony.

Imagine you're at a dance party where all the kids are wearing numbered badges from 1 to 10. Now, suppose they start dancing in a circle, but not just any old way: each time the music changes, they shift positions based on their badge numbers. If we watch them closely for a long time and see that every possible pair of dancers has had a chance to be side by side exactly as often as they should, then we know the dance is perfectly balanced, just like Weyl's criterion tells us about numbers.

In real life, this could look like a fair spinner. If every number on the spinner has an equal chance of coming up, say you have 10 sections and each one is equally likely, then the dance is balanced.

But if some numbers come up more often than others, it’s like having dancers who get to move around more, making the rhythm feel off-kilter. Weyl's criterion helps us know for sure when everything is just right, evenly spread out and in harmony.