Andalusite and kyanite might look different, but they're actually made of the exact same atoms, in the exact same ratio: just aluminum, silicon, and oxygen. The difference is how they're arranged.
If we zoomed in to see the aluminum, silicon, and oxygen atoms,
kyanite would look like this...
and andalusite would look like this.
I'm only showing a few thousand atoms here, but these patterns continue for hundreds of millions of atoms. It's quite pretty.
But, I want to ask: how would you describe the way these atoms repeat? How would you describe what makes kyanite's pattern of repeating atoms different from andalusite's pattern of repeating atoms?
Drag to rotate the 3D model yourself and take a look. What can we use to describe how andalusite and kyanite repeat? It's not easy.
The number of atoms won't help - both crystals have the same number (and ratio) of atoms, and there's billions of them.
Looking at one atom at a time won't help - each atom behaves similarly to all others. For example, in both crystals, every oxygen atom has three bonds, and every silicon atom always connects to four other atoms.
So do andalusite and kyanite repeat in the same way? Or a different way? What would "repeating in a different way" even mean?
So do andalusite and kyanite repeat in the same way? Or a different way? What would "repeating in a different way" even mean?
We don't really have the words to describe this in everyday language, so mathematicians invented their own: the language of group theory.
Group theory is a powerful way to think about repetition, symmetry, and the way things can combine and undo. In this presentation, we'll try to figure out the difference between andalusite and kyanite, and along the way we'll learn about group theory visually using these colorful graphs.
When I first learned group theory, it was taught in a very self-contained way, and I had a hard time seeing why mathematicians want to use it everywhere. Sure, it was interesting to think about a way to define symmetry, but what kind of problems would lead someone to think about such specific actions?
As it turns out, group theory is the perfect language for talking about repeating patterns, like crystals.
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