Title: Roots of real polynomial x²+x Post by: stigomaster 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 :D Title: Re: Roots of real polynomial x²+x Post by: makc on February 01, 2010, 08:17:11 PM attached essential part
Title: Re: Roots of real polynomial x²+x Post by: makc 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
Title: Re: Roots of real polynomial x²+x Post by: lkmitch 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 :D 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. Title: Re: Roots of real polynomial x²+x Post by: makc 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?
Title: Re: Roots of real polynomial x²+x Post by: makc 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 (http://wonderfl.net/code/73215e17e9514ed7af449d2273a9130647ab78ad), except that there are now "islands". I am going to try one with random coefficients now...Title: Re: Roots of real polynomial x²+x Post by: makc on February 01, 2010, 11:46:15 PM I am going to try one with random coefficients now... This turned out (http://wonderfl.net/code/46a3f7720334837ad71f6affc40be87324250e2a) to be much more fun than typing stuff in Wolfram Alpha :)Title: Re: Roots of real polynomial x²+x Post by: makc 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.
Title: Re: Roots of real polynomial x²+x Post by: kram1032 on February 02, 2010, 02:52:58 PM really a nice little program :D
Title: Re: Roots of real polynomial x²+x Post by: Timeroot 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. ;D
Code: Code: ChangingSignsMand{Title: Re: Roots of real polynomial x²+x Post by: jehovajah 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. Title: Re: Roots of real polynomial x²+x Post by: Timeroot 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??? ??? ??? ???
Title: Re: Roots of real polynomial x²+x Post by: makc on February 08, 2010, 11:57:09 AM your typical random() code (http://en.wikipedia.org/wiki/Xorshift) doesn't involve much computations either.
Title: Re: Roots of real polynomial x²+x Post by: Timeroot on February 09, 2010, 12:43:58 AM Okay fine, but the amount of chaos it produces is quite surprising I find. |