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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
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_02.jpg~original)

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
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_01.jpg~original)

Mutara-Class
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_03.jpg~original)

Between the Light and the Dark
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_04.jpg~original)

Jungian Jungle
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_05.jpg~original)

High-Energy Physics
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_06.jpg~original)

Electrical Storm
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_07.jpg~original)

Mythological Creatures
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_08.jpg~original)

Smoky
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_09.jpg~original)

Electrical Fire
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_10.jpg~original)


Mandelbrot

Seahorsehead Nebula
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m01.jpg~original)

Galaxies
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m02.jpg~original)

Flames
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m03.jpg~original)

Desert Flora
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m04.jpg~original)

Squiggles
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m05.jpg~original)

T Tauri Sector
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m06.jpg~original)

Algae
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m07.jpg~original)

Reaching Roots
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m08.jpg~original)

Trifid Nebula
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m09.jpg~original)

Partly Cloudy with a Chance of Fractals
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sfmm_m10.jpg~original)


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)
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sftmm_m01.jpg~original)

Tissue Sample (Julia)
(http://i1248.photobucket.com/albums/hh496/Paul_Derbyshire/Spacefilling%20Mandelbrot%20and%20Julia%20Fractals/sftmm_m11b.jpg~original)

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".

Those are amazing!
Thanks for the really nice explanation of how to get them. Could you maybe shed some more light on that?

So this is essentially combining the ideas of normal Möbius transforms with generic Mandelbrot/Julia sets? Essentially resulting in higher order Möbius-transforms?

Hi!

I posted about these a while ago:
http://www.fractalforums.com/new-theories-and-research/space-filling-2d-patterns-without-mirrors/
http://www.fractalforums.com/mandelbrot-and-julia-set/dense-julia-sets-on-the-sphere/
See also these blog posts for more information about these fractals:
http://www.algorithmic-worlds.net/blog/blog.php?Post=20110710
http://www.algorithmic-worlds.net/blog/blog.php?Post=20120316
http://www.algorithmic-worlds.net/blog/blog.php?Post=20130428
A really cool feature they have is that they naturally live on the sphere (complex plane plus one point at infinity). If you have Java enabled, you can see one of them panoramically on the whole sphere in the second blog post above.

In a nutshell, the orbits of rational conformal maps of the plane fall into two categories. The "regular" orbits are such that a neighborhood of the original point stays close to it along the orbit. The irregular orbits are the ones which display a chaotic behaviour. The Julia set is the set of points which have irregular orbits. It can happen that essentially all the orbits are chaotic, so that the Julia set fills the whole sphere (complex plane plus point at infinity), although this does not occur for standard Julia sets. In this case, you get dense fractal pattern of the type you mention. The "Julia/Mandelbrot" feel they have comes from the fact that we are using honest conformal transformations, and not mirror transformations as in the Ducks-like formulas.

You can find many fractal patterns of this type in my galleries:
http://www.algorithmic-worlds.net/expo/expo.php
Each picture can be seen panoramically on the whole sphere using the same java applet, look for the link at the bottom of the page of each image.

I should also mention that Dan Wills has been exploring this type of fractal patterns for even longer than I have, check his blog:
http://ultraiterator.blogspot.ch/

Best,

Sam

Awesome images!

Thanks!

There is an interesting tradeoff among exponential smoothing, average magnitude, and elliptic harlequin coloring. Exponential smoothing has issues with the centers of spirals, average magnitude seems to produce noisier images, and elliptic harlequin produces flame-shaped artifacts at, one presumes, preimages of infinity or some such distinguished locales.

lol s31415: "I posted about these before: check these links: [link collection]"
Upon checking links: s31415: "check these links: [link collection]"

Are you trying to do a hyper-link fractal? 'cause that already exists and is called the Internet :P
Anyway, really interesting stuff.

So essentially, all you need to do to get these to work is to add in an inversion, e.g. to make it all a rational map. (And then to find nice, interesting patterns)
Now this process can easily be done in higher dimensions too, but I wonder: How would you visualize such things in 3D? It would seem like that's a rather non-trivial challenge.
I suppose one could try a density-based approach but if this really uniformly fills the plane, it might just result in a perfectly uniform-looking image.

So I propose a challenge: Come up with an interesting way to produce the equivalent of these images in 3D using not-connected quaternion- or triplex-Juliasets (or things similar to that).
I suggest to first try this with only two Julia-sets. That's probably easier.
The tricky part here will not at all be the calculation but much rather the visualization so it doesn't look all uniformly filled or something.

There is now a video, which is one of my contest entries this year. I took your idea of "algorithmic worlds" to its logical conclusion: spheremapped some and rendered them as alien planets. (The background starfields are also such images.)

http://www.fractalforums.com/film-b240/strange-new-worlds/

The 720p HD setting is to be greatly preferred, of course.