hpdz.net

Magnet Zoom

The first HPDZ into a magnet fractal

NOTE: The structures in this video are largely due to artifacts in the basic arithmetic functions in my software. They are interesting to look at, but they are not an accurate representation of the true nature of the magnet fractal. I have rewritten the main page on this video. This page here contains the original published version and links to the video.

Background

A brief description of the origin of the magnet fractals, along with the formulas for them, is on the short still images page for them on this site.

After rendering Prima Luce II, I wanted something truly different to showcase the new Core i7 980X system. I have been wanting to do a deep zoom into a magnet fractal for a long time, and this seemed like the perfect thing to set this system to work on.

The magnet fractals are largely unexplored because they take so long to calculate, and this video is one of only a very small number of zoom animations into one of the magnet fractals.

This animation is based on the type I magnet formula. Below is an overview of this fractal. Click for a high-resolution image.

Surprises

After having a lot of experience with the Mandelbrot set, exploring the magnet fractal led to some surprises. Maybe these features have been discovered before, but I've not found them described anywhere else on the internet. Except for the multiple-connectedness, the surprises here are artifacts.

Connectedness

The magnet type I fractal is not a simply connected set, unlike the Mandelbrot set. Simply connected basically means that there is only one way to connect two points in the set using paths that are entirely in the set. More formally, it means every path between any two points can be continuously deformed into every other path, with every point on the path remaining in the set. A circle, for example, is simply connected. But if you remove part of the center of the circle, so you have a ring (an "annulus") then it's not simply connected anymore, since there are now two ways to connect any two points in the ring, and you can't smoothly transform them into each other without crossing the center part that's not in the ring.

You first start to get a sense of this around 22 seconds into this video, and it becomes really clear by about 25 seconds that something is going on that is quite different from what you see in the Mandelbrot set. One of the arms of the spiral is reaching out and touching the big flower next to it. You see this again at around 42 seconds.

The following is an artifact and not part of the real magnet fractal structure

Furthermore, the type I magnet fractal is not a connected set, which means there are parts of it that cannot be reached from other parts without passing through points that are not in the set. The Mandelbrot set is connected, so every point in it can be reached from any other point by a path that is entirely within the set. The magnet fractal has isolated sets that are disconnected from the rest of the set. You start to see this around 1:03-1:05, and the video continues into the center of one of these disconnected islands. At 1:20, we see a much more complex structure with many small disconnected islands around it. By 1:30 we see that the isolated islands do not always come in pairs, as the animation continues into one at the center of the whole structure.

Tricorn Cameo This is also an artifact

By 1:58 in this video, you start to see something different from the two-fold and four-fold symmetry that's dominated the animation up to this point. The maroon ring in the center looks slightly deformed into a three-fold symmetric shape, and two seconds later, a tricorn fractal starts to emerge!

The tricorn fractal is the shape that results from replacing z_n² with z_nz_n* in the Mandelbrot set function, where the * designates complex conjugation. The resulting fractal is also called the "mandelbar" fractal, because sometimes a bar is used to designate complex conjugation. It is something of an ugly duckling, and I have never previously published an animation featuring it, or, oddly, even a still image.

It is very common for small Mandelbrot sets to appear in other fractals, because there are many points that locally look like the Mandelbrot set's iterated map. But I have never seen a tricorn fractal appear in a fractal that also has regular mini-Mandelbrot sets! The endpoint of this animation terminates on an embedded tricorn structure. Maybe it should not be too surprising, since the tricorn fractal itself has embedded mini-Mandelbrot sets.

Video

In order to conserve limited server hosting space, the larger versions of this video are only available as MP4 files.

Note on Videos

Sorry, no true-HD quality WMV file is available. The HD MP4 file looks fairly decent but it is really seriously huge, and it is probably not worth the trouble to download. Try the 4 Mbps half-size image, and view it with 2X magnification in your player, and you'll probably have about the same experience as downloding the 553 MB file. See comments, below.

Comments

Video Options

This fractal is very noisy, and this video was not rendered with any noise-reducing oversampling. As a result, the MP4 and WMV encoders I use have a very hard time encoding this video full-size at anything less than 10 Mbps, and even then they do not do a great job. Below this rate, frames get dropped as the encoder struggles to compress all the noise in the fractal. So the full-size 1600x900 files are very large, and actually look a little worse than the half-size 4Mbps files.

There is no good way to deal with this problem once the raw data has been created. Any sort of smoothing filter applied to the video data is essentially equivalent to reducing the resolution, so that's what I've chosen to do here, providing smaller 800x450 files at the lower bit rates. You can simulate 1600x900 by viewing it at 200% size.

Frame Interpolation

Even with the serious computing power of a 4GHz, six-core CPU like the 980X, the magnet fractals are very difficult to calcualte, so frame interpolation was necessary to make this project approachable.  This video was constructed by interpolating data from 355 master images.

Deeper

The formula that is iterated to generate this fractal involves division. The high-precision arithmetic functions in the software currently works in a fixed-point representation, with a 32-bit integer value and an arbitrary number of digits of fraction. This works well for functions like the Mandelbrot set's function, but obviously division can be a problem because this system can express numbers whose reciprocal cannot be represented. For example, this number system can easily represent 5e-100, but cannot represent its reciprocal, 5e+100. For some parts of the magnet fractal, like where this video zooms, it turns out that no very large quotients appear. Other parts do generate large quotients, especially at deeper zoom levels, so the current software cannot draw them. Work is underway to extend the high-precision arithmetic to include full floating-point capability.