Julia Set Maps

[HPDZ]

OK, this is not exactly breaking new artistic ground here, but I thought it was time to publish some Julia set maps. These are overlays on the complex number plane of tiny thumbnails of Julia sets corresponding to each point in the plane. In the limit of infinitely many infinitely small Julia sets for a given function, you'd kind of get what the Mandelbrot set for that function looks like.

Far from being infinitely dense, these are 21x21 grids with 64x48 pixel mini-Julia thumbnails. So far, I've made maps for the quadratic and cubic functions, the Burning Ship, and the Magnet type 1.

On the corresponding page on www.hpdz.net/StillImages/JuliaSetMaps.htm, each thumbnail is a hyperlink to a 1280x960 big image. I've got the quadratic and cubic Julia sets, the Burning Ship, and the Magnet type 1 fractal Julia sets. The Burning Ship one was really inspired this project; the others just seemed compulsory. I'm working on a Magnet type II as I write this.

Click on the images below -- each is a link to the HPDZ website for each fractal type.

Quadratic Julia Sets


Cubic Julia Sets


Magnet Type 1 Fractal


Burning Ship Fractal (second-order)

[Fractal Ken]

Really cool idea!

Here's a quick thought (which may or may not make sense ) about the discreteness issue. Instead of preparing a fixed number of Julia set pictures, just present the user with an image of the Mandelbrot fractal. When he clicks at a particular location, dynamically generate and display the corresponding Julia set as a popup. This approach would, of course, involve a good deal of programming.

[bib]

I love this concept!
<a href="http://www.youtube.com/v/H603qZfhTNM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/H603qZfhTNM&rel=1&fs=1&hd=1</a>

[lycium]

This is basically the first thing everyone learns about the Mandelbrot and Julia sets, and on this very form, back in its halcyon days, a very modest guy named Julius Ruis decided he'd invented it and named it after himself: http://www.fractal.org/Julius-Ruis-Set.pdf

[Kali]

Cool video, bib... I think that abrupt zoom "flashes" at the end are a result of aliasing, it happened to me once.

[Aexion]

If you unwind this sphere in the right way, you will get a Mandelbrot Set..the Julia C values aren't chossed randomly..
this was an old (2007) render.


The center of the spiral is the center of the mandelbrot set, thats why most of the julias near it turns into circular towers..

[Pauldelbrot]

I've generated such images myself. One thing they can be useful for is visualizing the M-sets of multi-variable functions, such as discrete Volterra-Lotka and the Henon map, that aren't susceptible to the "critical points determine the full dynamics" trick that applies to single-variable functions.

[bib]

Aexion, that was a very beautiful image

[Aexion]

Aexion, that was a very beautiful image

Many Thanks Bib.
It was an early experiment for visualizing the julia sets in the mandelbrot set.

[Pauldelbrot]

Here's a Henon parameter-space map I made nearly ten years ago by plotting an 800x600 array of 10x10 Julias and downsampling. The Julia interiors (finite attractor basin(s)) become thin and weedy near the boundaries of the set, and this causes the set to "fade" into its surroundings in a nifty way:



Since then, I've made more sophisticated visualization algorithms for these types of maps, involving sampling a lot fewer than 100 points in a less regular, but more useful for characterizing the dynamics, arrangement and determining pixel color in more complex ways -- e.g. a layer for parameters for which any of the dynamic-plane sample points is trapped in a 2-cycle, colored by the average Lyapunov exponent of all such sample points, a layer for ... 3-cycle ... sample points, and such, with catch-all layers for long cycles and for ergodic points, and a layer for exterior points (all samples escape). These give the set sharp boundaries, at the cost of little "nibblings" around the edges where the thin, weedy nearly-unstable Julia interiors slip between the sample points without managing to hit any of them. The internal higher-period stripes, where multiple periods coexist, also get "nibblings" when the higher-period basins miss all the sample points. I posted a Henon image generated this way here in 2009, named Nimbus.