Chaos: Making a New Science

by James Gleick

On reflection, it becomes apparent that the Koch curve has some interesting features. For one thing, it is a continuous loop, never intersecting itself, because the new triangles on each side are always small enough to avoid bumping into each other. Each transformation adds a little area to the inside of the curve, but the total area remains finite, not much bigger than the original triangle, in fact. If you drew a circle around the original triangle, the Koch curve would never extend beyond it. Yet the curve itself is infinitely long, as long as a Euclidean straight line extending to the edges of an unbounded universe. Just as the first transformation replaces a one-foot segment with four four-inch segments, every transformation multiplies the total length by four-thirds. This paradoxical result, infinite length in a finite space, disturbed many of the turn-of–the-century mathematicians who thought about it. The Koch curve was monstrous, disrespectful to all reasonable intuition about shapes and—it almost went without saying—pathologically unlike anything to be found in nature.