Beautiful Abelian Sandpiles

I can't remember where I first saw them, but ever since, I have been unable to forget them: abelian sandpiles. I'm far from the only one. They're remarkably simple, yet produce lovely symmetric patterns. I loved them so much that I adorned the title banner of this blog with an animation of an abelian sandpile. But what exactly are abelian sandpiles? How do they work? And how many pretty,…