Title: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: David Makin on May 27, 2009, 12:38:51 PM Hi all, although I'm into fractals I've never really studied the Mandy itself in detail so I don't know the answer to this :)
I want to know as I'd like to do a 4 Mandelbrot IFS where the "butt-ends" of the 4 Mandelbrots are next to each other instead of using the pointed ends - this is what that produces: (http://fc00.deviantart.com/fs44/f/2009/145/8/e/Mandelbrot_IFS_by_MakinMagic.jpg) http://makinmagic.deviantart.com/art/Mandelbrot-IFS-123729985 Actually ideally I'd like to know the imaginary value of the maximum real points as well. Obviously I can get estimates easily but I just wondered if anyone knew the exact value, or a simple way of computing it. Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: bib on May 27, 2009, 02:21:47 PM Are you asking for the pure real number inside the M-Set (in this case it seems experimentally obvious to be 0.25), or the real part of the complex number that is at the far east ? (in this case, it seems to be around 0.4711853)
Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: David Makin on May 27, 2009, 04:43:56 PM Are you asking for the pure real number inside the M-Set (in this case it seems experimentally obvious to be 0.25), or the real part of the complex number that is at the far east ? (in this case, it seems to be around 0.4711853) Yes I meant the largest positive real part of any complex number in the Mset - ideally I'd also like to know the imaginary value for the two points concerned as well. I just wondered if there was a reasonably simple mathematical way of getting the exact value, failing that I'll just get an approximate value by inspection. Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: David Makin on May 28, 2009, 01:32:52 AM In the end I just translated the 4 Brots each by 1 unit in the appropriate direction:
(http://th02.deviantart.com/fs45/300W/f/2009/147/c/c/Another_4_Mandelbrot_IFS_by_MakinMagic.jpg) Full version: http://makinmagic.deviantart.com/art/Another-4-Mandelbrot-IFS-123936424 (http://makinmagic.deviantart.com/art/Another-4-Mandelbrot-IFS-123936424) And here's a zoom: (http://th02.deviantart.com/fs44/300W/f/2009/147/4/f/IFS_Brots_by_MakinMagic.jpg) Full version: http://makinmagic.deviantart.com/art/IFS-Brots-123936915 (http://makinmagic.deviantart.com/art/IFS-Brots-123936915) Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: cKleinhuis on May 28, 2009, 01:39:34 AM hmm, clearly those points do exist,
but i could only think of a method to determine the lowest real part :) , by using the fact that the mandelbrot set has no longer extend than the first arm, and that the black area ( the mandelbrot set ) is connected hey david, one method for finding the point - for sure it is not a rational number - is to simply newton iterate it ... just choose one point INSIDE the set e.g. (0,0) and one point OUTSIDE the set ( e.g. -2,0) and then start an NEWTON iteration or more clearly a binary search, meaning calculate the horizontal midpoint of these two (-0.5,0) check if it is inside the set ( here a VERY large iteration count could be of help ) and repeat with the point a weekend :D i will try that if i have some spare time O0 this method could also be used to determine the lower and upper parts ( if you know the exact midpoint of the bulbs ) i think but the biggest extend is a bit harder ... :'( at least for me ... @david appealing images ! Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: David Makin on May 28, 2009, 01:46:20 AM hey david, one method for finding the point - for sure it is not a rational number - is to simply newton iterate it ... just choose one point INSIDE the set e.g. (0,0) and one point OUTSIDE the set ( e.g. 2,0) and then start an NEWTON iteration or more clearly a binary search, meaning calculate the horizontal midpoint of these two (0.5,0) check if it is inside the set ( here a VERY large iteration count could be of help ) and repeat with the point a weekend :D i will try that if i have some spare time O0 Actually I think that would just get you (0.25,0) - I want the complex points that have the largest real value - as blb said they're around 0.4711853. A search algorithm in 2D could work but it's not worth the effort for what I wanted - as I said I was just wondering if there was a "simple" way (e.g. based on the special Fractal properties of the Mset). Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: Duncan C on February 20, 2010, 03:28:33 AM hey david, one method for finding the point - for sure it is not a rational number - is to simply newton iterate it ... just choose one point INSIDE the set e.g. (0,0) and one point OUTSIDE the set ( e.g. 2,0) and then start an NEWTON iteration or more clearly a binary search, meaning calculate the horizontal midpoint of these two (0.5,0) check if it is inside the set ( here a VERY large iteration count could be of help ) and repeat with the point a weekend :D i will try that if i have some spare time O0 Actually I think that would just get you (0.25,0) - I want the complex points that have the largest real value - as blb said they're around 0.4711853. A search algorithm in 2D could work but it's not worth the effort for what I wanted - as I said I was just wondering if there was a "simple" way (e.g. based on the special Fractal properties of the Mset). I did a little research, and can't find a mathematical answer to the question. The OTHER end of the Mandelbrot set is at exactly -2. As to the largest real value of a Mandelbrot point: .25 is the cusp of the main cardioid. The "cheeks" of the cardioid extend further to the right, and there are small tendrils that extend slightly beyond the "cheeks." It's pretty easy to "zero in" on the largest real value visually. I found the same value that you mention. I got 0.471185334933396 when I zoomed in to a magnification of around 5E-15. I could not find any pure mathematical way to find this value. Regards, Duncan C Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: Timeroot on February 20, 2010, 05:37:10 PM It's actually not a transcendental value, but it is impossible to write it in closed form. Since it is the tip of an antenna, it has a certain polynomial corresponding to it. Considering that the bulb it is attached to is (I think) period-4, the polynomial would be of order 32, which can then quickly be reduced to two possible 16th order polynomials. You could divide out some roots from the period 2 bulb and period 1 bulb, but it still would be very insoluble, I think. There is the fact that the antennas curve, so the tip might not be the most real of them all. In that case, perhaps the only (mathematical) way to do it would be looking through the antenna for a pattern, plugging that pattern of antennas into a formula for a polynomial, then using a root-finding method to calculate the root.
Btw, I think the root - if the tip really was the most real - would be of the polynomial ((((c^2+c)^2+c)^2+c)^2+c)^2+c=c^2+c. Subtract c, take the square root, and for the two branches you'll get totally different results. I don't how many iterations the antenna take before becoming periodic. If it is more than 2, then it will be an ever larger polynomial. Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: Duncan C on February 20, 2010, 05:48:27 PM It's actually not a transcendental value, but it is impossible to write it in closed form. Since it is the tip of an antenna, it has a certain polynomial corresponding to it. Considering that the bulb it is attached to is (I think) period-4, the polynomial would be of order 32, which can then quickly be reduced to two possible 16th order polynomials. You could divide out some roots from the period 2 bulb and period 1 bulb, but it still would be very insoluble, I think. There is the fact that the antennas curve, so the tip might not be the most real of them all. In that case, perhaps the only (mathematical) way to do it would be looking through the antenna for a pattern, plugging that pattern of antennas into a formula for a polynomial, then using a root-finding method to calculate the root. Btw, I think the root - if the tip really was the most real - would be of the polynomial ((((c^2+c)^2+c)^2+c)^2+c)^2+c=c^2+c. Subtract c, take the square root, and for the two branches you'll get totally different results. I don't how many iterations the antenna take before becoming periodic. If it is more than 2, then it will be an ever larger polynomial. Did you say you're in 8th grade?!? You are head-and-sholders ahead of my son, who's in 9th grade, and quite bright. In fact you are head-and-sholders ahead of most adults. To say that I'm impressed would be a gross understatement. Duncan C Title: Re: What is the largest positive real value inside the complex Mandelbrot Set ? Post by: Timeroot on February 20, 2010, 06:04:58 PM Did you say you're in 8th grade?!? *blush* :embarrass: Thanks...math is really my passion, and I pursue it as much as I can. Plus, I've learned a lot about these things just from going through Beauty of Fractals...You are head-and-sholders ahead of my son, who's in 9th grade, and quite bright. In fact you are head-and-sholders ahead of most adults. To say that I'm impressed would be a gross understatement. Duncan C |