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Author Topic: Orbit density map (aka anti-buddhabrot) rotation  (Read 3576 times)
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aluminumstudios
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« on: March 06, 2010, 02:32:55 PM »

Notice:  The music has very low bass.  You might not hear it if you have small speakers, or if you have small speakers turned up too loud they might distort.  Don't worry however, it is pleasant music, just heavy on bass.

<a href="http://www.youtube.com/v/58w-OOIjiwA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/58w-OOIjiwA&rel=1&fs=1&hd=1</a>

Please click above to view it larger on youtube, or you can download a higher quality .mp4 file here because youtube's compression has a noticeable, negative effect on the quality:  http://www.aluminumstudios.com/media/wmilberry-mandelbrot4d.mp4

I've been interested in fractals for a long time, but haven't done anything in recent years.  I wanted to refresh my programming skills and write a little mandelbrot program in C++.  While doing that I learned about the buddhabrot, then I wanted to see it from different angles and one thing lead to the next.

I don't care for the word "buddhabrot" because it only describes a specific fractal viewed with a specific range of parameters.  I try to call it an orbit density map because it's a more descriptive term and can apply to more situations.

Even though many here probably know about this, I thought I'd write a description.  Please correct me if anything I say is wrong.

The mandelbrot fractal, created by the equation Z=Z^2+C can be thought of as a 4 dimensional object.  The normal mandelbrot fractal is viewed from the plane formed by the (C-real, C-imaginary) axis (aka the constant plane.)  What you see here how many times you can iterate Z=Z^2+C given a point C.  But while iterating, where does the point go?  If you map out Z (Z-real, Z-imaginary) for each step during the iteration, you are now viewing the mandelbrot fractal from the Z-plane.  This is the "buddhabrot" rendering of the fractal.  The points travel all over and create a cloudy, nebulous, version of the mandelbrot.  Here is an example: (http://www.aluminumstudios.com/photography/photoblog/wmilberry-mandy46.jpg

Usually escaping points are rendered because they take wild and interesting paths creating a stunning image.  Points that don't escape are confined to smooth bulbs and thus don't seem quite so interesting.  However, while looking more closely at these "inside" points and rotating from the Zr, Zi plan to other planes such as those formed by the (Cr, Zi) or (Zi,Ci) axis, I began to to think thing it had it's own aesthetic and realized that I could add a few more frames to my exploration renders and have a nice little animation.

Since there are 4 axis (C-real, C-imaginary, Z-real, Z-imaginary), the mandelbrot set is somewhat like a tesseract (hypercube) when it is rotated.  Some planes rotate to be viewed edge on causing parts of the fractal to seem to collapse and disappear, while other edge-on planes rotate into view.  This can cause the fractal to look like it is morphing or even inverting when in fact it is a fixed, non-changing structure. 

I rendered this with my own threaded software I wrote in C++ on my Mac. Each frame was fully calculated.  The orbits that make up this type of fractal can easily 20+ gigs for a modest sized image.  My software could actually re-calculate each frame faster than trying to read a data set of many gigs over and over from my HD.  Also, a certain amount of variation in each frame caused by my algorithms gives the fractal a "live" feeling that I find aesthetic.

I don't believe that the streaks that you see are due to a defect in my software.  Rather, I've been noticing that the fractal has a kind of pattern texture to it.  I don't believe this is due to my sampling method (although sampling methods certainly can impart a pattern into the grain of this type of rendering.)  I think that there is a natural pattern there, and as it is rotated it's like looking at a folded piece of mesh window screen.  Sometimes the holes and wires of the screen align creating a moire pattern.  I think that this pattern is made very noticeable by the coloring method and thresholds.  Supporting this theory is the fact that as bulbs are viewed straight on there aren't many streaks, but as the are rotated and viewed more from the side (with more visual overlap of the bulbs, like layers of window screens), the streaks become quite noticable.  I rather like the effect, it makes my fractal shiny smiley

I hope you enjoy it.
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kram1032
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« Reply #1 on: March 06, 2010, 02:43:08 PM »

it's a very nice render smiley
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Timeroot
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« Reply #2 on: March 06, 2010, 04:46:55 PM »

That's a very nice video... next step would be a projective view of the entire tesseract!  evil
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
aluminumstudios
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« Reply #3 on: March 07, 2010, 05:52:51 AM »

Thanks for the comments.

That's a very nice video... next step would be a projective view of the entire tesseract!  evil

Thinking about it, it might not be too hard to render the edges of a tesseract around the fractal ...   I think I'll add this to my "to do" list. wink
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Bent-Winged Angel
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« Reply #4 on: April 19, 2010, 04:52:06 PM »

I also enjoyed it! smiley
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Alef
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« Reply #5 on: September 17, 2013, 06:06:28 PM »

4D budhhagram  grin
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