Title: Higer dimension modulus or something odd Post by: Alef on February 28, 2012, 05:54:27 PM Modulus of z do not generate many fractals, actualy it is abs function who do.
In most of fractal soft |z| is modulus squared and not mudulus, coz its faster and colouring formulas by tradition are intended to be used with modulus squared bailout. (http://upload.wikimedia.org/wikipedia/en/thumb/a/a9/Theorem.gif/220px-Theorem.gif) Algebraic modulus is cabs function defined as square root of real part and imaginary part squared, pithagorean theorem in complex plane-> z=( (real(z)^2+imag(z)^2)^0.5 ); Lacking pure cabs formulas, lets make it to generate something on complex map. z=z*z; z=( (real(z)^2+imag(z)^2)^0.5 ); z=z+C; (http://www.ljplus.ru/img4/a/s/asdam/1modulus2_m.jpg) Oval mandelbrot and circle julia. Ok, be it odd power what wouldn't be the modulus, as it retains negatives. z=z*z; z=( (real(z)^3+imag(z)^3)^0.333333 ); z=z+C; (http://www.ljplus.ru/img4/a/s/asdam/2modulus3_m.jpg) Something very odd. Lets go back to even powers, something fitting in modulus description (no negatives). z=z*z; z=( (real(z)^4+imag(z)^4)^0.25 ); z=z+C; (http://www.ljplus.ru/img4/a/s/asdam/3modulus4_m.jpg) z=z*z; z=( (real(z)^8+imag(z)^8)^0.125 ); z=z+C; (http://www.ljplus.ru/img4/a/s/asdam/4modulus8_m.jpg) It generated different result, ~ power 2 modulus, and equal to power 2 modulus when real or imaginary part are zero. Pythagoras put squares around the right triangle, here is a 8th dimension version of square. 4th power are boring, lets be an extreme, say 100th power modulus. z=( (real(z)^100+imag(z)^100)^0.01 ); (http://www.ljplus.ru/img4/a/s/asdam/5modulus100th_j.jpg) As power goes to infinity the result aproaches to certain ideal shape, to certain numerical result different from normal pythagorean modulus. 8th power seems to be good enought appropriation. This seems consistent considering cyclic nature of numbers. * * * * * * * * * * * * * * * * * * * * * * * Maybe in a 3D. Mandelbulb triplex uses modulus as part of calculation. So switching it with 8th dimension modulus: r2=(sx^8+sy^8+sz^8)^0.125; I got boxeded mandelbulb, soap box julia with seed 0,0,0, and zoom with different fine details. (http://www.ljplus.ru/img4/a/s/asdam/6ModulusMandelbulb_7.jpg) (http://www.ljplus.ru/img4/a/s/asdam/7Modulus_julia_bulb.jpg) (http://www.ljplus.ru/img4/a/s/asdam/8ModulusMandelbulb_zoomed.jpg) p.s. By Edgar Malinovsky Title: Re: Higer dimension modulus or something odd Post by: Ryan D on July 22, 2012, 02:27:14 PM I will occasionally try my hand at formulas that have been posted here, using my tried & true Fractint to see if I can come up with the correct implementation. The 2D fractals shown at the start of this post didn't look like they would be too thrilling, so I was surprised to find very interesting images coming from the orbit cloud histogram rendering method (the "Buddhabrot" method). Plain fractals don't often make exciting orbit clouds. In fact, it often works the other way - interesting images, like standard Julia fractals, are sometimes pretty boring when viewed as an orbit cloud.
Here's a video animating the power term from 0.2 to 15. The orbit cloud is symmetrical about the X axis for all even integer powers. At the odd integer powers, there is a strange discontinuity when the power fractionally increases - for example, the orbit cloud with a power of 3 is very different from that of 3.00001. As the power increases, more of the orbit cloud remains unchanged at these discontinuities. Ryan (new animation, and all the fussing about the old animation has been removed) http://vimeo.com/moogaloop.swf?clip_id=50303892&server=vimeo.com&fullscreen=1 Title: Re: Higer dimension modulus or something odd Post by: Alef on July 25, 2012, 05:38:20 PM Nice video, like mysterious sea creature. There are no much buddhabrot images, and I had seen just few formulas being generated in buddhabrot method, such as tricorn fractal, and this are like tricorn. Maybe odd powers represent kind of singularities, black holes, or top of the wave function. Formulas were intended to show between normal complex modulus function z=((real(z)^2+imag(z)^2)^0.5 ); and function with different powers. Throught it looks, that this was alredy invented by (renowned M3D formula coder Luka) ;) Youtube kind of are extremely harsh with fractal videos. I certain timeas had downloaded youtube movies and used in my own, and they don't detoriate in quality so much as fractal videos, they are kind of optimised for home videos. Maybe they just throw out extra frames and its more like special qualities of a picture compression, not shure, but folks say, it is size that matters. In fractalforums are pretty featurless video embedding;) Just put as link: http://www.youtube.com/watch?feature=player_embedded&v=kcSmqfLp6ys#! (http://www.youtube.com/watch?feature=player_embedded&v=kcSmqfLp6ys#!) |