Space-Filling Curves Are Not Squares

Spacefilling Curves

Screen Shot 2014-11-15 at 8.20.35 PMWikipedia says:

“In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n-dimensional hypercube“.

The image at right shows the Hilbert curve as an example. Indeed, the Hilbert curve fills a square. And its 3D counterpart fills a cube.

Screen Shot 2014-11-15 at 8.53.04 PMHowever, a space-filling curve (or…to just stick with two dimensions: a plane-filling curve) can be more generally described as a curve that fills a region of the plane that is topologically equivalent to a square (or…a disk). Note that a filled-in square, a disk, and a cone are topologically the same. “A cone?” you may ask. Yes, it has one surface (interior), and one boundary.

Now consider the following space-filling curves:

2000px-Gosper_curve_3.svg

Here we see the famous Dragon Curve, the Gosper Curve, and two curves that I discovered. If you have spent any time studying the Dragon curve, you know that it can…

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