Logo by Cyclops - Contribute your own Logo!

END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

it was a great time but no longer maintainable by c.Kleinhuis contact him for any data retrieval,
thanks and see you perhaps in 10 years again

this forum will stay online for reference
News: Visit us on facebook
 
*
Welcome, Guest. Please login or register. March 28, 2024, 04:07:23 PM


Login with username, password and session length


The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!


Pages: [1]   Go Down
  Print  
Share this topic on DiggShare this topic on FacebookShare this topic on GoogleShare this topic on RedditShare this topic on StumbleUponShare this topic on Twitter
Author Topic: The Christmas Tree 3D Mandelbrot Set  (Read 7436 times)
Description: The Christmas Tree 3D Mandelbrot Set
0 Members and 1 Guest are viewing this topic.
bugman
Conqueror
*******
Posts: 122



WWW
« 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 3403 times.)

* Mandelbrot-Christmas.jpg (98.9 KB, 563x282 - viewed 2340 times.)
« Last Edit: December 18, 2009, 08:58:36 AM by bugman » Logged
bugman
Conqueror
*******
Posts: 122



WWW
« 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 2431 times.)
Logged
bugman
Conqueror
*******
Posts: 122



WWW
« 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 1925 times.)
Logged
BradC
Safarist
******
Posts: 85



« 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
Conqueror
*******
Posts: 122



WWW
« 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
Pages: [1]   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
Christmas Tree + misc Mandelbrot images Images Showcase (Rate My Fractal) twinbee 4 1942 Last post May 19, 2008, 10:53:13 PM
by fractalwizz
Fractal Christmas tree X-Mas 2008 Fractal Chrimbo (completed) Duncan C 0 3112 Last post December 17, 2008, 04:20:02 AM
by Duncan C
Fractal Christmas tree 1 X-Mas 2008 Fractal Chrimbo (completed) AGUS 0 3142 Last post December 20, 2008, 07:47:08 AM
by AGUS
Christmas Tree X-Mas 2008 Fractal Chrimbo (completed) Cosine 0 2881 Last post December 22, 2008, 07:01:16 AM
by Cosine
Mandelbrot Tree stump. Images Showcase (Rate My Fractal) Caleidoscope 2 882 Last post June 01, 2016, 12:30:48 PM
by Caleidoscope

Powered by MySQL Powered by PHP Powered by SMF 1.1.21 | SMF © 2015, Simple Machines

Valid XHTML 1.0! Valid CSS! Dilber MC Theme by HarzeM
Page created in 0.171 seconds with 26 queries. (Pretty URLs adds 0.013s, 2q)