High-Precision Deep Zoom



A deep-zoom into the cubic Mandelbrot set.

Very few deep-zoom animations into the cubic Mandelbrot set have been made, and this is likely the first really deep zoom ever, with a final frame size of 5.9 x 10-33. See the Comments below for more background.

This video also debuts a few technical firsts, being the first video created with the new 64-bit high-precision arithmetic, and the first to showcase a new colorizing technique.

Sample frames
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Vital Statistics
Date Generated:15-24 Mar 2010
Final Image Size:1.9e-33
Resolution:960x540 (16:9 aspect ratio)
Video Length:2:40 of fractal, 3:14 overall
Rendering Time:200 hours
Method:Frame Interpolation
Audio:Custom crafted using Acid Pro 7


Background of this project

Qbix was initially conceived around November 2009, and various drafts and tests were made over the subsequent months. It soon became clear that rendering this video frame-by-frame at even a modest resolution was impractical, with estimates of computation time running into the 3-6 month range. So the decision was made to use frame interpolation to render this project.

When the 64-bit arithmetic project started, I decided to make Qbix be the first project to showcase that new development. The computation speed increase from this new code resulted in a total rendering time of just a few minutes under 200 hours.

Very few deep zooms of the cubic Mandelbrot set exist. One cubic zoom was published on this site several years ago, but it was not a deep zoom (only 3.2E-11), it was very low-resolution (just 320x240), and it was only 17 seconds long.

The Cubic Mandelbrot Set

The cubic Mandelbrot set, esthetically speaking, is very similar to the regular second-order set, with a few differences. The cubic set has no straight-line fibers, since it has no structure comparable to the antenna that extends along the negative real axis in the regular quadratic set. The cubic set, for some reason, also is much more asymmetrical when it is magnified: it is very difficult to find regions that aren't distorted. You can see this effect in the still-image frame captures from the video in the table above. One side of an image will be larger than the other side, while at a similar magnification, the quadratic set will look more isotropic. The count distribution is different as well, which is why many of the methods for colorizing that worked so well on the quadratic set don't work so well for this flavor of the set.

Nevertheless, most of the familiar structures are still there -- the spirals, radiating patterns, fibers (just not the straight ones) are all the same, but they have 3-fold symmetry instead of 2-fold or 4-fold.


The cubic Mandelbrot set is generated the same way as the usual second-order set, but instead of iterating the quadratic mapping

z = z2 + c

we iterate the cubic mapping

z = z3 + c

Otherwise, exactly the same process is used.

More general third-order polynomials can be written in the form

z = z3 - 3 a2 z + c

which opens up a spectrum of possibilities for new images by changing the parameter a. A great deal of work on the dynamics of iterating this form of complex map was done by John Hubbard and Bodil Branner in the late 1980's. Links to two major papers can be found on Dr. Hubbard's web page (The Iteration of Cubic Polynomials, Parts I and II -- scroll down to the "Selected Papers" section).

A series of parameter morph animations of the fractal resulting from iterating the general cubic polynomial has been published at www.rudyrucker.com


It seems that just about every new video requires a new colorizing technique! This one is no exception. None of the previous colorizing methods worked very well. The problem with this video's data is essentially the same one that's come up many times before: the range of the minimum to maximum counts in an image varies widely as the video progresses, from as small as 1.5 to nearly 1000. If a color map is not designed carefully, the high-range frames will have too many cycles of the color palette and look horribly noisy, while the low-range frames will use only a small fraction of the color map and be nearly all one constant color.

This video is colorized with a method that keeps the count-to-color mapping constant in time (as opposed to dynamically adapting it to the changing fractal data), and rather than trying to use some complicated algorithm to find a supposedly optimal color mapping, I decided to just manually enter a few data points specifying what color goes with a count number, and the software interpolates between them.  

Frame Interpolation

This technique is described in detail in a technical page, and it's been an ongoing area of development at hpdz.net for quite some time. It is commonly used in many commercially available fractal drawing programs, and is certainly not a new invention here.

This technique uses a small number of large images that are digitally processed into a large number of smaller video frames. When done correctly, the computational savings are huge, and there is no loss of video quality. Using frame interpolation, the final 5000-frame, 960x540 video was created from 181 master frames that are almost all 1805x1019 pixels. A few are slightly larger or smaller, but 99% of them are that size.


Matching the music with the video is always an important esthetic consideration in these projects. Since the fractal in this animation has three-fold symmetry, it seemed appropriate to use music with a triple-beat rhythm. Most pre-made royalty-free music is based on a four-beat rhythm, and what is available with a triple meter is mostly classical waltz music, which is nice, but doesn't really match well with a fractal animation. The audio component of Qbix was, therefore, crafted from sound loops and drum clips using Sony Acid Pro 7. It took some searching to find rhythm tracks that worked well with the 3-beat rhythm, and none of the melodic clips I have worked at all, so the music ended up having to be just a mix of rhythm and drones with a few effects here and there.