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.

 

"To see a World in a Grain of Sand"
https://www.poetryloverspage.com/poets/blake/to_see_world.html
Why? Because reality is fractal

 

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 ?

 

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.

 

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.

 

Life is a fractal.

 

I think the whole universe is a fractal and it gets very complicated very quickly. Fractal math is way beyond me. It's probably why they have not figured out the dark energy/matter thing yet.

 

I look for fractals in nature.
Queen Ann's Lace is my favorite example.

 

https://theconversation.com/explainer-what-are-fractals-10865
https://www.treehugger.com/amazing-fractals-found-in-nature-4868776
https://aiartists.org/fractal-art-generators
https://xaos-project.github.io/

 

I look for fractals in nature.
Queen Ann's Lace is my favorite example.

View attachment 244082

I just mowed some of those that had escaped into our lawn!

Bob

 

The root of Queen Ann's Lace is technically edible, if I recall correctly - it's a type of wild carrot (or related to it).

 

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.

Well bear in mind there is no process/algorithm to determine that a point iteration will ever "escape", it's a bit like the halting problem, there's no way to prove that any pair of points when iterated will "escape" that's why they place some limit, if it has not escaped after say 1 million iterations then they assume it never will, but they don't know, they don't know whether it might escape if iterated just a few more times, this is the essence of chaos - unpredictability.

You can do this with any software that draws the Mandelbrot set, set the iteration limit to 1,000,000 then run it again with the limit 1,000,001 and you find some additional points that now escape but did not escape with the limit of 1,000,000.

Note that the "set" part of Mandelbrot set is all those points that never "escape" but since we can never really prove if a point definitely will escape the members of the set are unknown.

 

Well bear in mind there is no process/algorithm to determine that a point iteration will ever "escape", it's a bit like the halting problem, there's no way to prove that any pair of points when iterated will "escape" that's why they place some limit, if it has not escaped after say 1 million iterations then they assume it never will, but they don't know, they don't know whether it might escape if iterated just a few more times, this is the essence of chaos - unpredictability.

You can do this with any software that draws the Mandelbrot set, set the iteration limit to 1,000,000 then run it again with the limit 1,000,001 and you find some additional points that now escape but did not escape with the limit of 1,000,000.

Note that the "set" part of Mandelbrot set is all those points that never "escape" but since we can never really prove if a point definitely will escape the members of the set are unknown.

Right. This, I already knew (I frequently use the Mandelbrot set as a kind of crude "benchmark" of processing power/speed, especially multi-threading). The thought/idea was less of a technical/mathematical question and more of an "artistic choice" question.
I have noticed that the closer you "zoom" into sections of it, the more iterations it takes to properly generate the patterns, which is likely because of the short scales at which the math is being done and the slow divergence/convergence of the iterations.

 

 

Romanesco broccoli is another great example.

 

Romanesco broccoli is another great example.

View attachment 244107

fractals are deeply embedded into the makeup of nature. Just like your broccoli,fractals demonstrate how simple geometric shapes can form the incredibly
complex, elaborate, and seemingly chaotic patterns just like our day-to-day life.Fractals, in this way, represent to me a type of rhythm to life.
Modern physics and mathematics prove that time is an illusion of our limited consciousness.In reality, there is only
infinite potentiality and within that, from our point of view
, there is the probability that certain groups of energy,
quanta, will be in relation to other groups of quantaThis
means that everything and every relationship possible exists. And only if every possibility of all
relationships exist
can free will be possible. This also allows for the potential for the connectivity
of consciousness through the complete
space of the Universe and thus the possibility that we can live forever in one stream of totally connected consciousness.
This is the flow which appears to move through the infinity
of potential universes or connections. But it is us that's moving through space and thus gives us the illusion of time.
We are Fractals surveying small parts of potentiality from within potentiality.

That's my philosophy.

 

See my post #6.

 

Life is a fractal.

I read intently all your posts.

 

I recall one lunch-break many years ago I wrote a WordPerfect macro to generate a simple fractal, but omitted to put any limit on the number of iterations in its calculation. The resulting file created when the macro ran grew large enough to crash the office's server after about half an hour! The IT department weren't too happy.

 

That's interesting, as is the title of the video with the word "analog".

I was thinking yesterday that in nature there are no calculations, nature does not do calculations or iterate (we humans do) so I was thinking about an analog circuit I could build with some opamps that implements the Mandelbrot transform function and then employ feedback to have the circuit's output represent the result of continued iteration.

The circuit would have two inputs and two outputs.

We can do this kind of thing already, for example we can calculate derivatives with opamps very easily indeed yet with a computer we need to write code to do rather involved calculations.

I have no idea what would emerge from this, and no idea how long it would take for the output to "settle" and so on, but the idea was on my mind.

I suppose supplying the coordinates of some points would cause the system to oscillate, never settle.

As that video shows, doing repeated calculations is not necessary, if fractals exist in nature and natures does not do calculations then this is clearly possible.