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Author Topic: The Christmas Tree 3D Mandelbrot Set  (Read 4013 times)
Description: The Christmas Tree 3D Mandelbrot Set
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bugman
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« on: December 17, 2009, 11:15:10 PM »

I was thinking it might be worth exploring different kinds of 3D Mandelbrot sets using power formulas that travel the same distance around the sphere no matter what the angle. Other people have already thought of this idea, but I came up with the following power formula and figured I'd give it a try. As it turns out, this formula is identical to the 3D slice at y = 0 of my 4D Hopfbrot formula (the one that looks like a Christmas Tree):
http://www.fractalforums.com/theory/3d-mandelbrot-formula-based-on-the-hopf-map/

As it turns out, this method is very similar to the cosine method. Here is the power formula:


* Christmas.gif (3.96 KB, 485x240 - viewed 1681 times.)

* Mandelbrot-Christmas.jpg (98.9 KB, 563x282 - viewed 878 times.)
« Last Edit: December 18, 2009, 08:58:36 AM by bugman » Logged
bugman
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« Reply #1 on: December 17, 2009, 11:15:32 PM »

We can also define a multiplication operator which is commutative but not associative (same as the Mandelbulb multiplication operator):


* Multiplication.gif (1.03 KB, 386x56 - viewed 1196 times.)
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bugman
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« Reply #2 on: December 17, 2009, 11:15:45 PM »

We can generalize the above power formula to define other power formulas that travel the same distance around the sphere no matter what the angle. So in general we can define phi = n*f(atan2(z,y)) where the function f can be almost anything. For example, if I use the sine function for my function f, then I get the following quadratic and 8th order Mandelbrot sets:


* Mandelbrot-Sine.jpg (93.99 KB, 563x282 - viewed 820 times.)
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BradC
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« Reply #3 on: December 18, 2009, 02:41:40 AM »

I'm having trouble picturing how this is different from the "cosine method" (formula #3) on your summary page (http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/?action=dlattach;attach=527;image), with the x-axis here playing the role of the z-axis there. The formulas do look different I think, but I can't picture what the difference is geometrically. Aren't they both using spherical coordinates, but with the elevation angles measured as angle away from a pole rather than angle away from the equator as in the usual formula? In the picture for formula #3, I can almost imagine a Christmas tree pointing downward, maybe.

Edit: Try starting with formula #3 from the summary page and make the replacements \{x\rightarrow y,y\rightarrow z,z\rightarrow x\}, and I think it becomes the formula given on this page.
« Last Edit: December 18, 2009, 04:28:01 AM by BradC » Logged
bugman
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« Reply #4 on: December 18, 2009, 04:44:01 AM »

Wow, you're right. The formulas become equivilent if in my formula you define phi = pi/2 - n*atan2(y, z). But the renderings look distinctly different, so apparently that makes a difference.
« Last Edit: December 18, 2009, 05:19:15 AM by bugman » Logged
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