Who likes fractals?

Thread Starter

Motanache

Joined Mar 2, 2015
537
For me, fractals are the essence of the universe.
I like Benoît Mandelbrot.

I'm waiting for others who want to talk about fractals. For now, anything about them.
 

andrewmm

Joined Feb 25, 2011
1,464
For me, fractals are the essence of the universe.
I like Benoît Mandelbrot.

I'm waiting for others who want to talk about fractals. For now, anything about them.
Used to see these as demos oh how fast a processor was
how do you convert a number to those lovely plots

is it just real and imaginary,

how do they assign colours ?
 

Delta Prime

Joined Nov 15, 2019
994
Hello there!

:)
Fractal spacetime,dimensional flow, dimensional reduction,electroweak scale,QCD scale,holographic principle, holographic bound.The mechanism underlying the generation of mass scales in field theory remains elusive.
The concept of fractal spacetime having minimal deviations from
four-dimensionality (the so-called minimal fractal manifold defined through ɛ = 4 - D, with ɛ <<1) can naturally account for the onset of these scales.A counterintuitive outcome of this analysis
is the deep link between the minimal fractal manifold and the holographic principle.One of the many unsettled questions raised by field theory revolves around the vast hierarchy of scales in nature,a large numerical disparity exists between the Planck scale,electroweak scale,the hadronization scale of Quantum Chromodynamics and the
cosmological constant scale expressed as energy density in (3+1 dimensions). Now I need to take a couple Tylenol.
o_O
 

ZCochran98

Joined Jul 24, 2018
141
Used to see these as demos oh how fast a processor was
how do you convert a number to those lovely plots

is it just real and imaginary,

how do they assign colours ?
For things like Mandelbrot, the colors are assigned by the number of iterations it takes for iterated calculations to "escape" - i.e.: how long it takes for, given \(z_{i+1} = f(z_i)\) (initialized to some initial complex number in the plane), \(\left|z_i\right|^2 > r^2\) for some "radius" \(r\). If, after a certain number of maximum iterations it does not "escape," that point (the pixel corresponding to the starting point) is colored black. The more iterations you let a fractal run through, the sharper your edges will be. The points that do not "escape" and are colored black are considered part of the Mandelbrot set. The Julia sets are a subtype of the Mandelbrot set (indeed; you can form the Mandelbrot set from all the Julia sets).

The way I color them, I use the HSV colorscheme for the iteration number. For instance, if the maximum number of iterations permitted is 1000, and a point "escapes" after 500 iterations, then I color its point corresponding to H = 0.5 (S = 1, V = 1), though, in my "BASIC" days, I would just cycle through the BASIC integer color codes (1-15, I think).

For things like the Newton fractal, the color corresponds to which root the initial point leads to.

I would be curious to see what the fractals would look like if you iterated them until the points "escaped," and instead of coloring based on iteration count, coloring them based off of the argument (angle) or the modulus (magnitude) of the "final" \(z_i\) after the last iteration.
 
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