Welcome to Fractal Forums

So..
I've spent the last 1 1/2 years thinking about and researching fractals.
We know the formulas that generate fractals. We have the means to calculate "simple"fractals, which is at the core of this forum.
We know they are all around in nature.. (in my view: literally everything came into existence through recursion, resulting in a fractal universe)
But one question remains.

Why are these forms?
Why are there spirals, nets, snakes, those fundamental forms in the m-set?
Why can the recursion of the simplest formula z²+c generate all these endless forms? How does this actually happen??
I mean, I love it!
But whats up with this?!  :hmh:

mind to share your personal view?

haha, right, there is an obvious connection between the fractals and the real world that is yet unknown (mandelbrot scratched the surface, describing a method to describe the world) i personally think there is much into it, but we are far far far away from the deep understanding of it

one thing should have become clear by comparing the shapes that occur in the mandelbrot and in the mandelbox and in most of the other fractals, for once we have to be happy to have taken that step

as tglad is always pointing out that the fundamental transformations are translate,scale,rotate,fold, inversion it is perhaps not much, but i for myself am getting enlighted by that, lol, the most fractals like trees or sierpinski triangles that lay in the mandelbox are a hint that these transformations are fundamental

what it actually means for real live, e.g. the distribution of genetic codes or the way trees are formed is yet to be found.

one thing that i found really interesting recently was the 3d variant of the sierpinski triangle that looked like the brocolli

so, real world is 3d, but the underlying methods using obviously repetive stuff to create and we are far away of understanding how the world is build up, but its the water drop that forms the stone over ages ;) so we just have to continue and play around and one beautiful mind some day gets enlighted and gets the final spirit to find a meaningful connection

:D my five cents

These kind of why questions are not really keeping me busy, since the answer is always some kind of circle. Like in the ultimate question about life and universe. Every single case uses some other case, which needs to be investigated even further. Like the following;

Why does life exists?
Because the local earth enviroment is just right.

Why is it just right?
Because it's warm enough for water to vaporize and cool enough to liquify and even freeze (and the pressure is just right)

Q1 Why is it warm enough?
Q2 Why is the pressure just right?
Q3 Why does life need water?
Q4 Why does earth hold water?
A1 The speed and direction of earth relative to the gravity field of our sun was just right, and the internal movement of earth brings friction and heat aswell.
A2 This is due to the total mass of the earth together with the properties of our atmosphere.
A3 Has something to do with the chemical reaction in which light energy from the sun is converted to usable energy for the organism.
A4 Probably some comet(s) have brought it in.

Q1.1 Why was the speed and direction of the earth the way it was?
Q1.2 Why is the gravity field of our sun the way it is?
Q1.3 Why is there internal movement in the earth?
Q1.4 Why does movement create heat?

Q2.1 Why does the earth have mass (and why is it the amount of mass it is)?
Q2.2 Why does this mass influence the speed and direction of the earth?
Q2.3 What are the properties of our atmosphere, and why are they?

Q3.1 Why does this chemical reaction specifically needs water?
Q3.2 Why does life need (sun)light?

Q4.1 Why are there comets?
Q4.2 Why do comets crash into earth?
Q4.3 Why do comets have water?

For every real life event or results of an event, there exists multiple connected events that causes, shaped, steered or otherwise influenced the last event. This shows the bigger fractal-like structure that all why-questions will follow. Start with one, and soon you'll have infinite.

We see the universe as a collection of discrete chunks of (things, persons, documents, stories, events) in order to be able to think about them. But all those boundaries we see, they don't really exist. Everything is connected, nothing stands apart. You have more non-human living cells in your body than your "own" cells. Every few year, all your cells have been regenerated. We only exists because some star had enough mass not to burn out, but to collapse and explode. Why do stars explode when there's enough mass? Why are stars born in the first place? Because our universal constants allow for it (at this moment in universe) and perhaps some quantum flunctuations. Why are the universal constants the way they are? Because of some fundamental law. Why is that law? Well, it's just fundamental. Finally you will get answers like this. You can NOT divide any smaller (aka "stop asking why"). It's just infinite (aka "stop asking why"). It was a singularity (aka .... ). People don't like to say they do NOT know. Our brains hates emptyness aswell, so the automatic thing to do is to make up some story (set it apart, create beginning, plot, end. finish. stop). But nothing really ends, since there is no beginning to point to anyway. Just keep calm while continuing.

We can always answer How something comes to be...
but we can never answer Why.

How = straight line segment, beginning middle and end.
Why = circle, never ending.

z²+c = How  to approx Mandelbrot set, but why does this complex shape exist? and why such an elegantly simple formula?

Only "The One" can answer why.

This is something I have known felt since I was about 10 yo when I started asking "how" because "why" would, more often than not, get the answer "That's just the way it is" which never satisfied my curiosity at all.

How? we know.
Why? we try to understand.

 :beer:

By the way, read https://medium.com/starts-with-a-bang/numbers-game-a5546123a213 ... it overlaps/intersects with this subject. Also some kind of quote comes to mind, stated in the context of quantum physics. Something like "you never understand it, you just get used to it". This very much resonated with me, which could be the same as for dick ulus (drop the why, focus on the how).

The mapping is conformal so shapes of (small) things stay the same but they exist at every scale.
Consequently there are a limited number of fractal forms that can exist...
a curve becomes a logarithmic spiral
a junction (like a Y shape) becomes a logarithmic tree
a loop becomes a 'sponge' / apollonian gasket type thing (bubbles / nets)
a single blob becomes a 'cluster' of blobs
etc

Certain fractal shapes don't crop up in the mandelbrot set, such as clusters, I think that's because the mandelbrot only emulates connected julias and clusters are disjoint.

Hard to answer your question fully, but I guess what I'm saying is that very basic non-scale-symmetric shapes give these commonly seen forms when made to be scale symmetric.
I made an attempt to classify these forms for 2d and 3d here: https://sites.google.com/site/simplextable/ (https://sites.google.com/site/simplextable/) by ignoring the exact shape and looking more at its connectivity, a bit like topology does.

This video helps me to understand how the shapes of Julia spirals arise.

https://www.youtube.com/watch?v=R1gpm7WsNhg (https://www.youtube.com/watch?v=R1gpm7WsNhg)

Each Julia set can be visualized by repeatedly iterating a transformation on a circle. Each transformation folds the circle onto one half of itself, and then duplicates it onto the other half, so it's a 1:2 mapping (this is actually the inverse of the map used to render Julia sets in the typical escape-time fashion). Each iteration introduces small distortions. These distortions stack on top of each other to create valleys, and eventually spirals. Due to the 1:2 nature of the map, these spirals are duplicated all over the main shape in different sizes.

It does nothing to help me understand how "shape stacking" works, that is, how Julia sets can combine features from two or more other Julia sets to make more varied sets. It's amazing that such a  simple transform can give rise to such complex sets.

hu, why havent we collected the link to this fantastic description ?
http://www.karlsims.com/julia.html

this is another video, for the julia sets the c translate is what creates the different shapes, this uses a c of -7,-1 seed
just the translation is what creates all the different shapes, there are different rules that create different shapes, i have a
julia visualisation as well in preparation, but since the seed is influencing everything heavily i am yet unaware wich seeds
to use

but some insights:

- a seed of 0.0 creates just a circle
- one interesting thing about this: at every-zeropoint of a mandelbrot^n formula is a center of a minibrot and produces julias with n repetitions, by repetitions i mean the whole translation undergoes n different locations before i starts again
- i hope to finish my own dot visualisation about that


https://www.youtube.com/watch?v=-V8HnG9XB2g

Hmmm, getting a bit off topic here, sorry about that, but looking at this way to produce a julia shape... would it be possible to do the same thing in 3D starting with a sphere... you beautiful minds of mathematics... ?

To not hijack this thead I have moved my question here:
http://www.fractalforums.com/the-3d-mandelbulb/transformation-of-a-sphere-to-make-3d-julia-possible-to-do/

Please answer in the new thread!  O0

Great links Johan, and good question on making those 3D. While I consider myself to have a beautiful mins it is certainly not of mathematics (but wish it was). I think that some of the answers for these shapes come from answers as simple as gravity and efficiency. Gravity certainly has a lot to do with the forms of the universe, and here on earth, evolution has found these forms to be extremely efficient for growth, protection, and the process of procreation.

@youhn : the more completely we understand How things work and their interrelationships, the closer we are to knowing Why.

There is nobody on this planet that can answer your question.

We live in a "shut up and just calculate world "

As you can see from quantum physics they have no idea why, they don't get deep. It drove Einstien mental trying to understand all the complete weird behaviour in quantum physics.

If anyone says they understand it they are lying, because as the saying goes if you truly understand you will be shocked and amazed, plus completely baffled.

Having said that the only person was Mandelbrot himself, and get this, FOR  A LONG TIME people thought he was a complete and utter nutcase.

Now for me it was the biggest revelation of my life to look at a cloud , mountain and plants and finally see fractals as iterative functions.   Thank you Mandelbrot you made my life so much better.

We clearly live in a completely fractal universe - that's clearly a computer simulation. There is plenty of compelling proof of this:

http://listverse.com/2013/12/02/10-reasons-life-may-be-a-computer-simulation/


This is the only video on the planet that tries to explain what a Mandelbrot actually is:

http://youtu.be/ce0lms78nt4

So it's obvious then right ? We live in a holographic clearly computer simiuation, hence nature is obviously fractal in nature.  It's computer code.

So then it's just trying to understand the code and see how that code creates it. Like the video above that shows how that mandel brot creates itself.

Unfortunately though, you can count on one hand the number of people on this planet that truly understand how these interative maths actually creates these wonderful fractals.

For example I can't look at a cloud now or mountain without thinking of fractal noise code or code that creates fractal landscapes.   That's a solved problem to me.

There isn't much about that shows other things like how other fractal equations creates forms.

You just have to learn to cut code and draw these fractals and get first hand experience, how the iterations over time create complexity.

I've had a few major complications in my life, so sorry for the looong delay.

Thank you very much Tglad!
This cleared up a lot.
From that point of view, one might call the mandelbrot set a multifractal, as most of the basic fractal patterns can be found in it.. :)
nice!

I also love your shape classifications and think that this is outstanding work you did there!

I'm pretty sure there is, but if he told me, I probably wouldn't believe him..  at least if he doesn't try it in a scientific way...

Word.
what a trailblazer he was...
I have a serious problem using our human terms "computer & simulation" in this context.
they seem to degrade the awesome beauty of the cosmos (though I'm sure that's not what you're trying to do. ;))
I too firmly believe that the universe is the result of an ongoing recursive process.
But I think that math and calculation is just a very usefull language to express what is happening. but not more. Content is king, not the language that we use to describe it... ;)

And consciousness! How can the one thing that is fundamentally central for YOU be completely "ignored" by science?
Math, a simulation without consciousness, without an observer is nothing. it's only virtual.
(I recommend a youtube search for Peter Russel - primacy of consciousness, a real eye opener in that area)

Btw, have you heard of Konrad Zuse and his idea of calculating space?
http://en.wikipedia.org/wiki/Calculating_Space

doing that certainly helps. but it's not the only way. I've come from the other direction, the strictly visual. as you described how it happend for you, I just started to recognize these patterns one day. and since then discovered them in many more places, were I think they are still hidden for most.

the really hard part is to find a language besides deep math, that helps interested people to see and discover it on their own, without having to work hard and think about it for years..