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.

Metamorphosis

I have always been fascinated with Escher’s work, in particular his "Metamorphosis" woodcuts.  I’ve wanted to do something along those lines for years and finally came up with an approach.

I wrote my Scribble formula for Ultra Fractal in 2009, with something like this in mind, but I never made good use of it until now.  As you can read from the tutorial, Scribble essentially animates a rose curve along a path.  In these cases, the path was a horizontal line.  In the top image, the transition begins with a three-petal rose on the left, which morphs into a four-petal rose and then into a circle in the center.  From there, back to a three-petal rose and ending up with a four-petal rose curve.  The second image uses just four-petal roses of varying size (except for a circle in the center), and the final image uses only three-petal roses (plus a central circle).  In all, the phase angles associated with the sine and cosine functions are varied smoothly to impart a sense of rotation.

Julia Descending a Staircase

In my travels, I’ve become intrigued by the cubist style of painting and have wanted to make cubist fractals, whatever they are.  This is my first attempt. (In it, “Julia” refers to the Julia set fractal, not a particular woman.)  It was inspired by Duchamp's famous painting,"Nude Descending a Staircase." I created it using the embossing technqiue to get the minimal sense of the Julia sets, like Duchamp’s painting resembles sketches of the woman.  The Julia parameter changes slightly with each step, so the shapes vary down the staircase.  Also, I changed the lighting angle each step, to add more variation and to help distinguish the eight different Julia fractals.  None of them are entirely visible in the stack, like Duchamp's figures all kind of running over each other.  But I went from a more gray palette at the back (left) to black/white at the front (right) to help separate them and to give a sense of motion.