Imagine an upward surge of fluid like this:
As the dark fluid surges upward, the light-colored fluid around it gets pulled inward, creating a “neck” as shown above at right, and the dark fluid bifurcates at the top. You can compare this shape to the generators for the Hilbert curve and the Peano Sweep. Since these are plane-filling fractal curves, they carry this dance of surging fluids down to smaller and smaller levels, until the fluid regimes are completely mixed. Below are the first few iterations of this mixing.
The Hilbert curve is conveniently self-avoiding: no matter how much the fluids mix, the curve never touches itself. But the Peano Sweep has overlapping segments. This can be alleviated using a technique to separate the overlaps, which makes it topologically similar to the Hilbert curve, in terms of its intricate branching structure.
This comparison is described in more detail here, and it is illustrated below.
One key difference is that the Hilbert curve is a “node replacement curve” while the Peano Sweep is an “edge-replacement curve”. Edge-replacement curves were used by Mandelbrot in the introduction to fractals in his book. (He referred to this as Koch construction).
The Hilbert curve and Peano Sweep both conform to a four-square tiling scheme in which the main square is replaced with four smaller squares, and the two bottom ones are rotated inward.
Notice that the Hilbert curve requires connective links at every iteration. This is a required step for node-replacement curves, as shown in this variation of Mandelbrot’s Quartet that I discovered:
There are many more examples of plane-filling curves that use different tiling schemes, including a series of curves that Peano discovered, using nine inner-squares.
Several people have used plane-filling curves like this to define an image as a squiggly line. Imagine drawing the Mona Lisa with only one line, and the line gets more squiggly when it needs to cover a darker area:
(the above image is based on the Peano Sweep and it is explained here)
Stay tuned for more adventures with plane-filling curves!