A montage of fractal animation projects
Over the years, quite a few projects have been completed that just didn't manage to get published. Sometimes it's because they are just too short, but other times it's just that other things took priority. Finally, these are getting published in this 12-minute montage. Or is it a collage?
The individual animations with carefully selected individual musical themes for each were lovingly joined together with Sony Vegas Pro 10 and rendered into the following downloadable files.
In order to conserve limited server hosting space, the larger versions of this video are only available as MP4 files.Trouble Downloading?
|MP4 Files (QuickTime player)|
|Extreme Quality||1.3 GB 960x540 15 Mbps|
|HD Quality||691 MB 960x540 8 Mbps|
|Good Quality||180 MB 960x540 2 Mbps FastStart|
|Low Bandwidth||59 MB 480x270 600 Kbps 15 fps FastStart|
|Minimal bandwidth||25 MB 480x270 200 Kbps 15 fps FastStart|
|WMV Files (Windows Media Player)|
|Minimal-bandwidth||29 MB 480x270 200Kbps 15fps|
This is the first montage published on HPDZ. It won't be the last, but the next one will be quite a long time from now. It's a lot of work to assemble so many projects into one video file.
The montage begins with a visually quiet title page lasting a couple of seconds, followed by the title of the first animation. Meanwhile, a fast, energetic music track is getting started. The first video is a high-def deep-zoom into the Magnet fractal, with a rich complex color palette.
Next is a Tricorn animation, one of only a few ever published. Somber, foreboding music matches the simple, dark color palette, which has only a few variations of off-white set against deep black.
Following that is a HyperNova morph animation. The coloring is simple, with white/gold details on a blue/black background and some green highlights. Rhythmic background music enhances the graceful flowing motion of this morph.
A deep zoom Mandelbrot set animation follows, rendered with the distance estimator method, with those sensuous curves it makes. A quiet, meditative musical accompaniment features a simple drum beat with and a mysterious melodic wind instrument.
Things then get odd with a secant/cosine morph animation and an appropriately odd, primirively rhythmic little segment of music.
A zoom animation of a Nova-based fractal follows. Stark RGB and white colors provide the visual texture accompanied by a quiet mysterious musical background.
A morph animation based on the secant method is next, wtih a curious bubbly soundtrack.
The second Mandelbrot set deep-zoom animation has a faster-paced musical background. The stereo-panning percussion is always fun.
Next to last is a slow-moving morph animation based on the Secant method applied to finding the cubic roots of 1. This fractal performs its graceful whirling dance to an ethereal, nonmelodic musical background.
Finally, a short morph animation of the Mandelbrot set with a cool percussive musical accompaniment that ends with the closing text and a nice bell ding.
There are more options than usual here. The reasons are explained below, and get kind of deep into minutiae, but I will say this: The gigantic 8 Mbps encoded Hi-Def files are FANTASTIC. And the even more ginormous 15-17 Mbps extreme quality files are STUNNING! If you have the patience to downlod either the WMV or the MP4 Extreme files, you will be glad you did. The video playback is flawless, almost exactly as the uncompressed files look, but with no hesitation from limits on disk drive throughput speed (the 15 Mbps files are about 1/10 the size of the uncompressed raw AVI files, which have bit rates in the range of 200-750 Mbps, enough to challenge even a very fast system).
These are encoded at 30 fps, so you might want to set your monitor to a refresh rate of 60 Hz to avoid judder if your monitor is set at 72, 75, or 80 Hz refresh rate. Those of you who have made the move to LCD (hopefully a 16:10 model) won't need to mess with this. But anyone still living on a CRT (like me, as of this writing) should really consider setting the refresh rate to 60Hz (sorry to the Europeans and other 50Hz folks) for silky smooth playback. You can always set it back after you are done watching this amazing video!
You'll need a reasonably good video controller for smooth playback of the HD and Extreme files. It's doubtful that most native motherboard video hardware will deliver the required decoding speed, but maybe some can.
The things to look for to appreciate the difference between the HD/Extreme encodings and the other encodings are the smoothness of the gradients, the lack of "boxing" artifacts, and, well, something long regarded as a flaw in fractal animations--the "sparkle" artifacts. Due to the nature of the MP4 and WMV encoding process, smooth gradients are transformed into boxy checkerboard-like textures. As the bit rate increases, this effect diminishes. The 8 Mbps encodings have a very subtle hint of boxing artifact. The 15 Mbps videos have almsot no discernible boxing artifact at all. And the "sparkle" will be washed out at lower bit rate encodings. The Extreme versions have sharp, crisp detail, and every little sparkle noise pixel is clear.
The reason for the plethora (yes, 5 is a plethora, while 4 is normal) of download options is something a few techies might want to know about. I had a set plan for four encodings of this video in each format (WMV and MP4). But the usual tools (GSpot and "Details" in Windows explorer) showed odd results for bitrates, which misled me at first. One encoding that was supposed to be at 8 Mbps was showing as 24 Mbps, which made no sense based on simple arithmetic of file size and time. The only tool that showed consistently correct bit rates for these files was MediaInfo. In all the confusion, I ended up creating a lot more files at many more bit rates than usual. Most of them got tossed. But I kept five in each format. Another reason is that at nearly 12 minutes, this video is significantly longer than anything I've published before, and so the file size even at a low bit rate is going to be very large. So I felt it was appropriate to offer lower-quality download options for those who want to just have a quick look.
The magnet fractal is a computational challenge, since the mathematical formula that generates it is significantly more complicated than the one that generates the Mandelbrot set. When HPDZ acquired the new Core i7 980X system, one of the first things that got attention was the Magnet fractal.
Unfortunately, the first published deep-zoom animation of this fractal on this site was a bit of a disaster because of serious limitations in the high-precision arithmetic code that created artifacts that looked like real structures.
The animation in this montage predates the MagnetZoom fractal project by a couple of weeks, and was one of the first projects rendered on the new Core i7 980X system back in October 2010. Since it is not a deep-zoom, it has no artifacts induced by the faulty arithmetic code in use at that time. The problems with the high-precision arithmetic code have since been fixed, and deep-zoom animations of fractals with complicated formulas involving division are now possible.
This animation zooms to a size of 2.4e-12 and consists of 3600 frames of 960x540 video rendered frame-by-frame (no interpolation) in 7 hours 26 minutes on 13 Oct 2010 using a Corei7 980X overclocked to 4.0 GHz.
The Tricorn fractal is the ugly sibling of the Mandelbrot set. It is generated by interating not zn+1 = znzn, but rather zn+1 = znzn*, where * indicates the complex conjugate. This seemingly trivial change has a huge effect on the appearance of the resulting fractal. It's hard to find attractive images in this fractal, but there are some.
Because mathematicians often represent the complex conjugate of a variable by writing a bar over that variable, this fractal is sometimes called the "Mandelbar" fractal.
A bogus image of the Tricorn fractal appeared in the unfortunate MagnetZoom video published in October 2010.
The modestly deep zoom animation in this montage is the first animation of the Tricorn fractal published on HPDZ.NET. It zooms to a final size of 4e-15. These 1800 frames of 640x480 video were rendered in 53 hours on a quad-core Core2 at 2.4 GHZ from 28-31 Oct 2010.
The HyperNova fractal is an invention of HPDZ.NET named after the well-known Nova fractal. Some still images of the Nova fractal have been published on HPDZ, but no animation. Essentially, the Nova family of fractals is based on Newton's method with a "relaxation" parameter and the addition of the coordinates of the image point. Most images of this huge family of fractals that have been published here or elsewhere use
as the underlyiing polynomial. The HyperNova fractal is based applying the Nova technique to finding the roots of the polynomial equation
(z2+p)(z3+q) = 0
with the relaxation parameter set to 1. The iterated function is
zn+1 = c + zn - (zn2+p)(zn3+q)/(5zn4+3pzn2+2qzn).
The first realization of a HyperNova fractal was published on HPDZ.NET as a test video for helping to troubleshoot video playback problems.
The 1800-frame 640x480 morph animation in the present video was generated in 3.5 hours on 11 Apr 2009. The values of p and q are fixed at (2,0) and (1,0), and the center point of the frame is fixed at (-0.24356312625250504, -0.00001202404809641). The size is 6.0e-3. The morph is created by varying z0, the initial value for the iteration, from 0.67 to 0.72.
This is a deep-zoom animation into one of the branches of the western fiber of the Mandelbrot set. It was rendered in slightly over 17 hours on 22 Apr 2008 using the Distance Estimator method. These 3500 frames at 640x480 zoom to a final size of 1.7e-42.
The secant method for solving equations is similar to Newton's method, but its much slower convergence creates fractals with more interesting visual texture.
The short morph in this video is based on using the secant method to find the complex roots of the cosine function, varying the real part of the initial guess at the solution from 1e-3 to 1e+3. These 600 frames of 800x450 video were generated in slightly under 30 minutes on a quad-core Core2 at 2.4 GHz on 27 Feb 2009.
Two previous full-length animations based on the secant method have been published on HPDZ. SecantAnim1 is a big tour-type animation of a fractal very similar to the one in this short morph sequence. SecantAnim2 is a morph animation based on a polynomial.
This is a short zoom into a modified version (see next paragraph) Nova fractal finding the cubic roots of 1. This animation was generated with code that had a slight bug in it, but the bug doesn't create any visual difference from the correct code for the particular parameters chosen in this animation.
This is not exactly the standard Nova fractal, but rather a slight modification I made. The Nova fractal has an initial seed value, z0, which is usually set to something around (1,0). When z0=(0,0), the equation has a division by zero, so (0,0) is an illegal starting value. After finding a slight bug in my initial code for the Nova fractal, I realized it made sense to trap this condition and replace it with z0=c, where c is the point on the image. That modification creates the three-fold symmetric shape seen here, which is quite different from the usual Nova fractal.
These 1200 frames of 640x480 video zoom to a final size of 2.2e-10 and were generated in slightly over 6 hours on 23 Jul 2008. This was created before the software made detailed log files of animations, so it's not clear which system this was rendered on, but it was probably the quad-core Core2 that was acquired in April 2008.
Very similar to the Secant Cosine morph, this short 600-frame 500x300 video is based on the secant method applied to finding the roots of the function f(z)=cosh(z)/z2. The fractal looks very similar because the Cos and Cosh functions are very similar when their arguments are complex numbers.
Multiple parameters are varied to generate the morphing. Rendering was completed in 84 minutes on 4 Jul 2010.
This is a deep zoom into the northern trifurcating fiber area of the Mandelbrot set. The final size is 2.0e-43. These 4800 frames of 640x480 video were rendered in 32.5 hours from 6-8 May 2008. This was before the software kept a log of which system ran an animation rendering, but it was probably rendered on the quad-core Core2.
Another morph animation based on the secant method, this time using the polynomial z3--1=0 that is used to make the classic images of the Newton method fractal. The initial guess in this morph is varied from (0,-1) to (0,1). Rendering these 2400 frames at 640x480 on the Core2 system took 88 minutes from 28-29 Jan 2009.
The final animation of this production is a morph animation of the Mandelbrot set, varying the starting seed value, z0, from (3,0) to (0,0) to (0,3). This never gets old, does it? This 200-frame segment was rendered in 9 minutes on 5 July 2010.