Space-Filled Popcorn

Pickover’s Popcorn algorithm is an iterative formula that doesn’t generate fractals in the typical way.  Because it typically uses trigonometric functions of real variables, instead of polynomials, there isn’t the standard inside/outside dichotomy of points.  Typically, none of the orbits diverge to infinity; they are either periodic or wander around a relatively small part of the plane.  Consequently, the Popcorn algorithm can be looked at as a warping of the complex plane.  This characteristic pairs nicely with the geometry of space-filling curves (like the Hilbert curve); they can be rendered to stay within a set region of the plane.

Here is the development of the image at the top of the screen, using this idea.  The first panel shows just a few iterations and the underlying Hilbert curve is clearly visible.  Then, with more and more iterations, the Popcorn swirls take over, obscuring any trace of the Hilbert curve.

Another approach uses a variation on the Hilbert curve.  The curve is based on triangles instead of squares, and starts like this.

Here’s the development of the curve, when combined with the Popcorn fractal:

And here’s the final image.