Winding Number, Space Filling Curve

Space Filling Curve

Even though your pencil is infinitely thin, it is possible to scribble over an entire region in the plane in a finite amount of time. Your pencil never leaves the paper (continuous), yet the tip wiggles about in an infinite number of infinitely small vibrations. To say your hand is shaky is an understatement; it is downright fuzzy, almost like quantum mechanics. It's hard to believe. Even after you read the proof below, it's still hard to believe.

Peano (biography) was fascinated by Cantor's work, especially the counterintuitive notion that there are just as many points in the unit interval as the unit square. What better way to demonstrate this than to map the former onto the latter. Do it continuously, and you have a space filling curve.

Peano's curve is based on the cantor set, which I will denote as C. Remember that C maps continuously onto the closed interval [0,1]. Apply this map in two coordinates, and the topological product C*C maps onto the unit square. Then, apply the fact that C*C is homeomorphic to C, and build a continuous map from C onto the unit square.

Of course C is not the same as the closed interval [0,1], but the map can be extended. Let x be a point in [0,1] that is not in C. It has been excised, as part of an open interval. The endpoints of this interval, call them a and b, map to two points in the unit square. Map all points in between, including x, into the unit square linearly. Thus, if x is halfway between a and b, then f(x) is the midpoint of f(a) and f(b). The result is a continuous map from the entire closed interval [0,1] onto the unit square.

Since Cⁿ is homeomorphic to C, a similar result holds in n dimensions. A space filling curve can cover the unit hypercube in n space.

Space Filling Loop

Map the top have of the unit circle onto the unit square, as described above, and map the bottom half onto a line segment from 1,1 to 0,0. The curve starts and ends at the origin - hence it is a "loop", or a "closed curve". Like every other loop in the plane, it has an outside. However, this loop has no inside. Every point is on the loop, or outside the loop.