Exploration of the 4D mandelbrot set on the Z-plane (Buddhabrot)

(creator's note: I am disappointed with youtube's compression quality of my video despite uploading a lossless 720p source video)

The title is a reference to Siddhartha Gautama who meditated intensely under a bodhi tree until he achieved enlightenment and became the Buddha. I seek enlightenment through art and science and my work with fractals.

About one year ago I decided to refresh my rusty C++ programming skills by writing a little program to draw Mandelbrot fractals - something that has always interested me. In doing a little reading into it I discovered the so called "buddhabrot" method of rendering Mandelbrot fractals which consists of mapping out orbits (all of the in-between values created during calculation) rather than simply coloring pixels based on the final count of calculation loops. This essentially shows you the "Z" plane of the Mandelbrot set. The typical rendering method shows you the C (constant plane.) Z and C of course come from the well known equation Z = Z^2 + C. If you consider Z and C as planes, each having a real and imaginary axis, then the Mandelbrot set can be viewed as a 4-dimensional object. This can be rotated and viewed from different perspective and planes which this animation does.

I was hooked on this style of fractal from the moment I rendered my first image. I wanted to see more, so I tried to render a higher resolution version and that's when I hit the wall that I feel has prevented this fractal from being fully explored and appreciated. Because any tiny area that you want to see is potentially drawn from points originating anywhere else in the plane, enlarging and zooming becomes a computational nightmare fast! The number of calculations needed to find points whose orbits pass through the smaller and smaller area of interest as you zoom in rises exponentially.

I did some searching and kept finding the same methods described for rendering this style of fractal over and over. Some of them used statistical analysis and some of them talked about random sampling and some of them dove into Mandelbrot period analysis and some math that is quite honestly a little over my head at this point. I saw shortcomings in all of these methods and sought to develop my own algorithms and methods to allow me to explore and render beautiful images of this fascinating fractal at reasonable computational speeds.

This animation is an early result of my work. While I don't suspect that my methods and software is the fastest out there, it seems quite fast compared to various render times I've seen randomly mentioned on forums. It also avoids statistical biases and potential detail loss found in other methods that I feel has held back visualizing and appreciating the full nature and beauty of this fractal.

This animation project was done over a period of 3 months in a number of separate sections while I continued to improve my software. Because I made several major speed improvements while working on this and didn't render continuously it's difficult to say how much time it actually took to render. The fastest frames rendered in about 30 seconds while the the slowest frames of this project took upwards of 90 minutes (with maybe a few frames hitting 2 hours.) The animation is around 5250 frames rendered at 1080x720 30fps.

This project was done entirely on a single 2.16 GHz Core2Duo MacBook using C++ and Xcode as my editor. I live in Japan and spend my money on traveling these days, so I make due with my little Macbook. My limited hardware was great motivation in trying to find new intelligent ways to approach this computational problem.

In an interview that I saw before (sorry I can no longer sight the source), Beniot Mandelbrot said that he was a visual person and sought to learn or demonstrate things visually. I wonder if this rendering or future ones like it might not reveal hints about how and if things are connected and such. I definitely need to do more research. Noting how it appears to be many copies of the traditional fractal, layered together, with some of the layers almost "curling" up and peeling away and how the sequence starting around 1:38 makes it look as if the border is a continuous thread the goes traces the connects through all of the curling layers makes me hopeful that some little secret might be learned visually.

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The images are (C)2010 All Rights Reserved. Please do not redistribute or use images from my fractal art. If you are interested in using images of mine please contact me and I will happily respond.

Category:

Science & Technology

Tags:

• Science
• Math
• AluminumStudios
• Mandelbrot
• fractal
• buddhabrot

[TiagoTiagoT]

Somtimes it looks like it is not a continuous volume but a bunch of cross-sections at different angles and depths put together; are those artifacts of the rendering or the actual numbers at full resolution really get distributed like that in some places?

[AluminumStudios]

@TiagoTiagoT Those aren't rendering artifacts. I cant explain it well because I'm not a mathematician, however what my software has revealed to me is that the fractal is actually thousands and thousands of Mandelbrot fractals, stacked on top each other, with each one twisting and tilting just a little bit into the various 4 dimensions. When viewed from just the right angle in all 4 dimensions they align and reveal the familiar Mandelbrot shape. Interesting isn't it!

[YesIamJames]

Any way to render this as just a 3D shape? Possibly by rendering only at point 0 of the imaginary C axis. I just made my first buddhabrot renderer today but I don't have the skill to do it in 3D but I'd imagine you can get something just as beautiful as the Mandelbulb with some practice.

[albertlobo1981]

Awesome animation, great job! :D

[boinghd]

There were helix'. It makes me wonder if the universe is a deterministic system or not. I'm pretty sure that all the patterns that arise naturally would appear due to topological transformations of certain number sequences. It's interesting that we have complex numbers though. If we think of them in terms of natural processes, the fact that they even exist is a wonder.

[ThePhysicsfreek]

My interest is in the projection of the MS in the quaternions in a 10 dimensional space/time. You have an interesting look. Really liked your imaging.

[DocFarnoff]

When I was introduced to the Mandelbrot by a TV special hosted by Arthur C. Clarke, my mind was completely blown! My life was changed.Everything had new meaning. I am a dunce when it comes to geometry but it look likes the Mandelbrot is a model of infinite dimensions not just four. One symbol representing a creator down to the smallest sub atomic particle and every variable in between!

[TheGodlyFoods]

AMAZING!



[funkyjj900]

look its Andromeda

[AluminumStudios]

@killthetaco Our intuitive understanding is based on the fact that we only EXPERIENCE 3 spacial dimensions in our physical lives (there may be other dimensions according to string theory or depending on how you choose to define time, but despite that, we can't experience them.) Therefore, our brains really only process what front-back, up-down, left-right means. Trying to imagine another direction that is perpendicular to all o them is difficult to impossible.