Take a letter. Say F means “draw a line forward,” + means “turn left,” - means “turn right.” Now write a rule: every F becomes F+F-F-F+F. Apply the rule once and you get a zigzag. Apply it again and the zigzags zigzag. Again and the zigzags-of-zigzags zigzag. By the fourth generation you have a Koch curve — a shape of infinite perimeter enclosing finite area — and you did it with one rule and five characters. ...
L-systems, formal grammars, and trees that don’t know they’re trees.
Here is a theorem that sounds too beautiful to be true: if you take a map and lay it on top of the territory it represents, there is always exactly one point on the map that sits directly over the point it represents. The map-point and the territory-point are the same point. Brouwer proved this in 1910 — any continuous function from a compact convex set to itself has at least one fixed point. A point where f(x) = x. A point that the function leaves exactly where it found it. ...