Logo by dainbramage - 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. October 02, 2022, 11:55:46 AM


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: What is the largest positive real value inside the complex Mandelbrot Set ?  (Read 5127 times)
0 Members and 1 Guest are viewing this topic.
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« 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 smiley
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://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.
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
bib
Global Moderator
Fractal Senior
******
Posts: 2070


At the borders...


100008697663777 @bib993
WWW
« Reply #1 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)
Logged

Between order and disorder reigns a delicious moment. (Paul ValÚry)
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« Reply #2 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.
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« Reply #3 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:



Full version:

http://makinmagic.deviantart.com/art/Another-4-Mandelbrot-IFS-123936424

And here's a zoom:



Full version:

http://makinmagic.deviantart.com/art/IFS-Brots-123936915
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
cKleinhuis
Administrator
Fractal Senior
*******
Posts: 7044


formerly known as 'Trifox'


WWW
« Reply #4 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 smiley ,
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 cheesy

i will try that if i have some spare time  afro

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 ...  cry at least for me ...

@david appealing images !
« Last Edit: May 28, 2009, 01:46:19 AM by Trifox » Logged

---

divide and conquer - iterate and rule - chaos is No random!
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« Reply #5 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 cheesy

i will try that if i have some spare time  afro

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).
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
Duncan C
Fractal Fanatic
****
Posts: 348



WWW
« Reply #6 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 cheesy

i will try that if i have some spare time  afro

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
Logged

Regards,

Duncan C
Timeroot
Fractal Fertilizer
*****
Posts: 362


The pwnge.


WWW
« Reply #7 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.
Logged

Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
Duncan C
Fractal Fanatic
****
Posts: 348



WWW
« Reply #8 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
Logged

Regards,

Duncan C
Timeroot
Fractal Fertilizer
*****
Posts: 362


The pwnge.


WWW
« Reply #9 on: February 20, 2010, 06:04:58 PM »

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
*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...
Logged

Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
Pages: [1]   Go Down
  Print  
 
Jump to:  


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