Negative Exponent Mandelbrot
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Negmandel, what is it?
In short, it's when you use a negative exponent with the standard mandelbrot formula.
The Generalized "Mandelbrot Fractal" Formula
When the standard mandelbrot-set formula:
- z->z^2 + c
Is "generalized" to allow any exponent, you get the generalized "Mandelbrot Fractal" formula:
- z->z^n + c
Multibrot - Positive Integer Powers
For positive integer powers greater than 2.0, the mandelbrot formula gives the so-called 'MultiBrot' sets.
Negmandel - Negative integer powers
Negative integer powers give a shape like a curved star, with the number of points one greater than the power, so a power of -4 gives a 5 pointed star. The iterations no longer bail out so all points of a negmandel are Inside.
Fractional Powers
When the power is not an integer, the solution becomes "multi-valued", because part of the formula has an infinite number of solutions to it. This can be visualized like a piece of spiral pasta sticking out of the page (see the Wikipedia article about Complex Logarithm)
This has the visual result of introducing infinitely fine 'cuts' into the fractal. This artifact probably occurs because two or more overlapping solutions are not tracked at once in the rendering of the fractal. It remains to be seen whether one could improve this situation in a usable fashion.
So it seems that the cuts are because of an artifact of the rendering, but they can still look pretty cool!
Negative fractional powers also have the interesting property that some parts inside the lake get extremely dense with detail.
Examples
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Copyright © 2008 by Dan Wills
upr
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tea
Copyright © 2008 by Dan Wills
upr
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Thin orbit traps Negmandel
The contrast has been enhanced here a bit from the original render.
Copyright © 2008 by Dan Wills
upr
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