Thursday, November 12, 2009

Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal

Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal

TIMESTAMP 08/11/2009. The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years with its cascading patterns and hypnotically colourful detail. It's known as a 'fractal' - a type of shape that yields (sometimes elaborate) detail forever, no matter how far you 'zoom' into it (think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc.).

It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal. For the impatient, you can skip to the nice pics, but the below makes an interesting read (with a little math as well for the curious).
Some said it couldn't be done - that there wasn't a true analogue to a complex field in three dimensions (which is true), and so there could be no 3D Mandelbulb. But does the essence of the 2D Mandelbrot purely rely on this complex field, or is there something else more fundamental to its form? Eventually, I also started to think that this was turning out to be a Loch Ness hunt. But there was still something at the back of my mind saying if this detail can be found by (essentially) going round and across a circle for the standard 2D Mandelbrot, why can't the same thing be done for a sphere to make a 3D version?

Our story continues with mathematician - Paul Nylander. His idea was to adjust the squaring part of the formula to a higher power, as is sometimes done with the 2D Mandelbrot to produce snowflake type results. Surely this can't work? After all, we'd expect to find sumptuous detail in the standard power 2 (square or quadratic) form, and if it's not really there, then why should higher powers work?

But maths can behave in odd ways, and intuition plays tricks on you sometimes. This is what he found (also see forum thread, and the full size pic at the 'Hypercomplex Fractals' page of his site):

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