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 Author Topic: Roots of real polynomial x²+x  (Read 2553 times) Description: 0 Members and 1 Guest are viewing this topic.
stigomaster
Guest
 « on: February 01, 2010, 06:54:30 PM »

I was playing with the 1D mandelbrot (the iterative function x->x²+x for real numbers) and discovered something nice at WolframAlpha.
http://www.wolframalpha.com/input/?i=%28%28%28%28x%C2%B2%2Bx%29%C2%B2%2Bx%29%C2%B2%2Bx%29%C2%B2%2Bx%29%C2%B2%2Bx

Check out the plot of zeroes in the complex plane and call me if you see something familiar
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makc
Strange Attractor

Posts: 272

 « Reply #1 on: February 01, 2010, 08:17:11 PM »

attached essential part
 1.gif (2.89 KB, 120x120 - viewed 502 times.) Logged
makc
Strange Attractor

Posts: 272

 « Reply #2 on: February 01, 2010, 10:43:13 PM »

tried to up the number of iterations but that site times out after roots of (((((x²+x)²+x)²+x)²+x)²+x)²+x
 2.gif (3.47 KB, 120x120 - viewed 485 times.) « Last Edit: February 01, 2010, 10:47:31 PM by makc, Reason: broken link any way » Logged
lkmitch
Fractal Lover

Posts: 238

 « Reply #3 on: February 01, 2010, 11:08:05 PM »

I was playing with the 1D mandelbrot (the iterative function x->x²+x for real numbers) and discovered something nice at WolframAlpha.
http://www.wolframalpha.com/input/?i=%28%28%28%28x%C2%B2%2Bx%29%C2%B2%2Bx%29%C2%B2%2Bx%29%C2%B2%2Bx%29%C2%B2%2Bx

Check out the plot of zeroes in the complex plane and call me if you see something familiar

You may already know this, but what you'd get if you could continue the process indefinitely would be a point for the center of each cardioid and each disk, which would approximate the boundary of the Mandelbrot set.
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makc
Strange Attractor

Posts: 272

 « Reply #4 on: February 01, 2010, 11:18:13 PM »

I was entering random garbage into this form over and over until I came up with signs alteration idea. I wonder what's going to happen with traditional Mandelbrot set rendering algos modified this way?
 1.gif (9.49 KB, 550x400 - viewed 380 times.) Logged
makc
Strange Attractor

Posts: 272

 « Reply #5 on: February 01, 2010, 11:30:34 PM »

I wonder what's going to happen with traditional Mandelbrot set rendering algos modified this way?
It appears nothing particularly interesting, except that there are now "islands". I am going to try one with random coefficients now...
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makc
Strange Attractor

Posts: 272

 « Reply #6 on: February 01, 2010, 11:46:15 PM »

I am going to try one with random coefficients now...
This turned out to be much more fun than typing stuff in Wolfram Alpha
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makc
Strange Attractor

Posts: 272

 « Reply #7 on: February 02, 2010, 01:05:32 PM »

added manual coefs control to the code in last post and without page reloading. (shift+)click to zoom (out) btw. not too fast, but it's only flash. could be more fluid in c.
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kram1032
Fractal Senior

Posts: 1863

 « Reply #8 on: February 02, 2010, 02:52:58 PM »

really a nice little program
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Timeroot
Fractal Fertilizer

Posts: 362

The pwnge.

 « Reply #9 on: February 07, 2010, 10:26:37 AM »

I made a quick formula(s) for UF that implement this. It calculates an "angle" to multiply c by each iteration. With angle=0.5, you get ((z^2+c)^2 - c)^2 + c etc... with angle=0.25, you get ((((z^2+c)^2+ic)^2-c)^2-ic etc... The incredible thing is, unless the angle has an imaginary part of exactly zero and a real part not close to a multiple of 2, the mandelbrot set loses basically all its interior. At angle=0.0001, it's a full Mandelbrot set with not visible deformation. At angle=-0.00001 or angle=1.99999, it has no deformation, but it has lost all it's interior. Weird, just weird. In fact, I think this might be a viable inside coloring algorithim for the Mandelbrot set due its lack of deformity and interesting shapes.

Code:
Code:
ChangingSignsMand{
init:
sign=-1^(2*@Angle)
iter=0
z=0
loop:
z=z^@Power + (sign^iter)*#pixel
iter=iter+1
bailout:
|z|<=@Bailout
default:
title="Changing Signs (Mandelbrot)"
complex param Angle
default=(0.05,0.0001)
endparam
complex param Power
default=2
endparam
float param Bailout
default=4
endparam
}

ChangingSignsJulia{
init:
signedC=(-1^(2*@Angle))*@C
iter=0
z=#pixel
loop:
z=z^@Power + signedC
iter=iter+1
bailout:
|z|<=@Bailout
default:
title="Changing Signs (Julia)"
complex param Angle
default=(0.05,0.0001)
endparam
complex param Power
default=2
endparam
float param Bailout
default=4
endparam
}
 « Last Edit: February 07, 2010, 10:28:59 AM by Timeroot » 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.
jehovajah
Global Moderator
Fractal Senior

Posts: 2749

May a trochoid in the void bring you peace

 « Reply #10 on: February 07, 2010, 11:22:25 PM »

If this does become a colouring algorithm i just want to say i was first to respond to your insight to this wholly serendipitous train of events!

Delightful.
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
Timeroot
Fractal Fertilizer

Posts: 362

The pwnge.

 « Reply #11 on: February 08, 2010, 07:51:06 AM »

Well, I made a "coloring algorithim" of that idea, if you can call it that. Basically it just iterates this other formula with a very very small angle (which the user can change) and then colors based on iteration, magnitude, angle, real, or imaginary. But I think I've made an incredible discovery. With any option other than iteration, if produces PURE NOISE. Zooming in to any level doesn't simplify it. It fills the entire MSet. I zoomed in to E11, with only 100 iterations (that's relevant, because more iterations mean more chaos), and there was still pure noise. I may have messed something up in the code, but either way, I see some applications in encoding and security considering how much noise is generated by such little computation... anyone have thoughts???
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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.
makc
Strange Attractor

Posts: 272

 « Reply #12 on: February 08, 2010, 11:57:09 AM »

your typical random() code doesn't involve much computations either.
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Timeroot
Fractal Fertilizer

Posts: 362

The pwnge.

 « Reply #13 on: February 09, 2010, 12:43:58 AM »

Okay fine, but the amount of chaos it produces is quite surprising I find.
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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.
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