Logo by HPDZ - 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, 09:39:55 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: Inside  (Read 1420 times)
0 Members and 1 Guest are viewing this topic.
reesej2
Guest
« on: April 22, 2010, 11:32:59 PM »

I've been thinking about the complexity inherent in the structure outside the Mandelbrot, and noticing that the interior doesn't seem to have nearly that detail no matter what means are used to color it. I'd be interested to know if anyone knows of any strategies that get that kind of detail...

Pickover stalks and similar start to get close, but it's still a very simple pattern. Distance estimation does practically nothing, and escape time is of course trivial. Any ideas?
Logged
teamfresh
Fractal Lover
**
Posts: 246


nothing is everything


fractalco
WWW
« Reply #1 on: April 23, 2010, 04:05:50 AM »

this really should be looked into!
Logged

hobold
Fractal Bachius
*
Posts: 573


« Reply #2 on: April 23, 2010, 03:55:09 PM »

Well, the points inside the Mandelbrot set certainly go through interesting and varied orbits. Even those in the main cardioid, that ultimately converge. Furthermore, the orbits vary smoothly (except at the individual points where different cardioids/disks touch each other).

The hard part lies in "distilling" some invariant simple numbers from those orbits, that can be mapped to a colour palette, but capture enough information from the orbit to be actually interesting.
Logged
lkmitch
Fractal Lover
**
Posts: 238



« Reply #3 on: April 23, 2010, 05:59:05 PM »

I think the key is to look at inside points differently.  For interior points (as opposed to boundary points), you may wish to begin by plotting the orbits of some target points to get a feel for the types of orbits.  Generally, the closer a point is to the center of its respective disk or cardioid, the more quickly its orbit will converge to a limit cycle (thus, potentially less interesting graphically).  I like to think about questions about the orbits that can be answered graphically.  For example, how often (percent of iterations) is the real part positive?  How close does the orbit come to a particular point or shape?  If the orbit is chaotic, how chaotic is it?  If I connect the iterates, what kind of pattern will it make?  If the orbit is ultimately chaotic with period n, what happens if I draw lines between every (n-1) or (n+1) points?

True, you won't find the same type of detailed structure with inside points, but there is structure there nonetheless.
Logged
reesej2
Guest
« Reply #4 on: April 23, 2010, 09:10:19 PM »

Oh, yes, certainly there IS structure there--that can be seen by simply plotting Pickover stalks or preperiod iterations. The thing is that the structures produced that way are not nearly as rich as the exterior, which seems strange, since the paths of the interior points are at least as varied as those of the exterior.

I think connecting the iterations has been done, I read a paper on it a while ago--you tend to get a sort of blur, not very exciting. But the idea of connecting points at intervals based on the period is interesting...

Also, on the notion of "how chaotic is it", perhaps plotting the Lyapunov exponent would be relevant? And maybe the Lyapunov exponent of the nth-iterate map, for various n?
Logged
Pages: [1]   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
Inside Out UltraFractal Gallery bib 0 843 Last post June 09, 2009, 10:46:25 PM
by bib
Orbits inside Images Showcase (Rate My Fractal) « 1 2 » aluminumstudios 16 3784 Last post March 15, 2010, 02:19:50 AM
by aluminumstudios
From the Inside Mandelbulb3D Gallery lenord 0 574 Last post September 28, 2011, 05:45:36 AM
by lenord
Inside Kaliset Mandelbulb3D Gallery Kali 0 536 Last post November 05, 2011, 09:38:29 AM
by Kali
Inside the Inside Mandelbulber Gallery mclarekin 0 782 Last post October 22, 2015, 10:04:59 AM
by mclarekin

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 0.178 seconds with 24 queries. (Pretty URLs adds 0.01s, 2q)