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Author Topic: Optimizations for bulbs other than main cardioid and period-1 bulb?  (Read 1960 times)
0 Members and 1 Guest are viewing this topic.
Posts: 35

« on: May 23, 2013, 11:36:24 PM »

I know that there are equations to determine if a point is in the main cardioid or period-1 bulb that can be used to optimize code, but has anybody tried to find equations for the period-2 bulbs, period-3 bulbs, etc.? It seems like this could make Mandelbrot programs much more efficient.

Just for reference, the bailout conditions for the cardioid and period-1 are (according to wikipedia) :

(x + 1)^2 + y^2 < 1/16
q*(q + x - 1/4) < 1/4*y^2

where  q = (x - 1/4)^2 + y^2
« Last Edit: May 23, 2013, 11:44:02 PM by Levi » Logged

Math isn't the solution, math is the question.
M Benesi
Fractal Schemer
Posts: 1075

« Reply #1 on: May 24, 2013, 08:37:18 AM »

  The other buds aren't perfect circles... at least with the standard mapping.  Wonder if altering the pixels mapping would make the buds perfect circles at some point?

  I mapped the pixel from   pixel^1.25  to pixel ^3  in the following video.  Can map higher z^n as well- just map pixels to X / (n-1), with X being the type of mandy you want to force map.

<a href="http://www.youtube.com/v/28Yb8IS0CxQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/28Yb8IS0CxQ&rel=1&fs=1&hd=1</a>

  Wondering about the eccentricity of the period 3 bulb when X=2?  (It's shaped like a z^3 Mandelbrot).  Doubt it's worth pursuing. 
« Last Edit: May 24, 2013, 08:43:44 AM by M Benesi » Logged

Posts: 35

« Reply #2 on: May 24, 2013, 06:37:57 PM »

They're not perfect circles, but are they some kind of ellipsoid or other well-defined shape?

Even equations that don't remove the entire bulb, but just a large percent of their area, could be good.

I know there are equations for the points at which every single bulb connects to the main cardioid, which is a good start...http://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_and_period_bulbs

Additionally, the center of each bulb can be found by solving  Q(n+1)(c) = (Q(n)(c))^2 + c  for Q(n)(c) = 0 and n is any natural number (in the wiki article under hyperbolic components).
« Last Edit: May 24, 2013, 06:45:56 PM by Levi » Logged

Math isn't the solution, math is the question.
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