You know how the Ducks and Kali fractals can fill all of space with detail? Well, turns out you can fill all of space with Julia/Mandelbrot fractal structures, too! You need to iterate a rational map of the complex plane for which the numerator degree is no more than the denominator degree; then the point at infinity does not superattract or, usually, even attract. For general maps of this character there are generally chunks of parameter space where there are no attractors in the dynamics at all -- the Julia set is the full complex plane.
A system I've explored previously, Matchmaker, is defined by the map:
It has two critical points, at
, and a theorem of iterated rational maps of the complex plane is that any attracting basin must contain at least one of the critical points. So, Matchmaker dynamics can include up to two attractors. As it turns out, with two complex degrees of freedom Matchmaker can more or less place each critical point in an arbitrary part of Mandelbrot structure independently of the other, allowing almost any conceivable pair of (connected) Julia sets to feature in the dynamics, which I've used in the past to produce pretty pictures of entwined, filled-in Julia sets, where instead of one Julia shape floating in "empty space" (really, the circle Julia for a superattracting fixed point), two quadratic Julia set shapes are entangled. I've also married a connected Julia set with an interesting disconnected one by picking parameters to put one critical point in a Mandelbrot bud and the other in a valley at the same time. That results in images with the connected Julia's basin components filling most of the complex plane, and the low iteration areas looking like that Julia, while the high iteration areas look like the disconnected Julia.
But the real doozy might be to make two disconnected Julias intertwine that way. Well, I've managed two, three, and
four. The trick isn't combining them -- find parameters that put the critical points all in valleys and boom. The trick is actually making them visible, because any standard escape-based thing will either give you a blank black screen (no attractors!) or obliterate some of the fractal details (orbit traps).
Fortunately, the methods used to visualize Ducks fractals (exponential smoothing and variants) can work -- but with problems dealing with "high iteration" areas (however one defines that; generally, zones "close" to a low-periodicity repelling point, where "close" depends on the period of the point). This image was generated that way:
Hypervortex
We can do better, using Elliptic Harlequin coloring (UF users have access to this via, IIRC, akl-m-math.ucl). The result resembles smooth iteration coloring, without need for convergence to any attractor, and makes visible the fractal details that completely occupy the entire complex plane. It also brings details to the "voids" in the Mandelbrot (parameter-space slice) images in the areas lacking any stable periodic points: dendrites completely fill space around the minibrots.
Without further ado, I present to you a bevy of space-filling Julia and Mandelbrot fractals:
Julia
Desert Palm
Mutara-Class
Between the Light and the Dark
Jungian Jungle
High-Energy Physics
Electrical Storm
Mythological Creatures
Smoky
Electrical Fire
Mandelbrot
Seahorsehead Nebula
Galaxies
Flames
Desert Flora
Squiggles
T Tauri Sector
Algae
Reaching Roots
Trifid Nebula
Partly Cloudy with a Chance of Fractals
This more complex mapping has three critical points and one more critical value (a horizontal asymptote). It can merge up to four Julia sets at a time, and is capable of producing Herman rings to boot:
For hysterical reasons, it's called Triple (rather than Quadruple) Matchmaker (I didn't discover the extra critical value for a while).
Antares Maelstrom (Mandelbrot)
Tissue Sample (Julia)
That last one combines four, count 'em four, recognizable Julia structures. We have:
- The pink swirls are recognizable as from a minibrot Elephant Valley.
- They are attached like fruits to the branch tips of a dendrite with four-way branchings, associated with a period-4 bud attached to the main set. In the normal M-set this is northeast of the center of the cardioid; the Julia is from off the tip of the top branch.
- The triple spiral vortices are from a valley between a bud and a period-3x-higher smaller bud.
- There are also hollow dendrite meshes that are fairly close to becoming Herman rings, but these are harder to spot. Look at the blue mist in the upper left quadrant for a prominent instance.
At 4000 iterations per pixel (and all of them needed for every pixel), with a complex mapping, a complex coloring, and 49x oversampling, that last image took many hours to generate on a six-core FX6100 processor. (On the other hand, that processor's real strength is integer, not floating point, so deep zooms of the standard Mandelbrot are its forte, rather than shallow zooms of much more complex formulae.)
All images subject to CC-zero license (so, not really subject to any license restrictions).
All images available at 1280x960 with right click "View Image".