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I've been searching for such a thing for a long time, and it's nice to see the responses here. But I'd ask for something similar for Group Theory. What is your "Brief, nice thing that makes people go Wow!" for Group Theory?
I'm not a group theorist, nor a mathematician, so I'm the wrong person to ask. I do find the Solovay-Kitaev thm wonderful: informally, if you take products of elements in (many) compact Lie groups, they fill in the group exponentially quickly, & surprisingly near to uniformly
A picture is better to explain what this means. And probably best explained concretely using, e.g., rotations and a few similar examples.
I don't quite understand it well enough to make the attempt, but I'll bet there's a mindblowing 5-minute explanation of what Gromov's theorem on groups of polynomial growth says.
That is a lovely result, but I was hoping for something more accessible and elementary, something I can explain - at least in outline - to high school students. Why is Group Theory "A Thing(tm)"? Why is it interesting, useful, and how does it tell us new things?
An approach I like is to start with a problem that doesn't involve group theory, and show how it almost forces you to discover group theory. I don't know of such a problem (but then, I've never thought much about group theory!)
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