Mandelbrot "c" formula: a little idea

[Count Fractula]

Hello "Mandelbrothers" and/or "Mandelsisters"! (copyrighted ). Let me share this little idea.


As you know, the Mandelbrot formula is:

My formula (I call it Mandelbrot formula) is:

As you can see, my formula keeps the original .

However, becomes a Mandelbrot formula, too.



(And I guess the concept could also be translated to the equation for the Julia set. "The equation

for the Julia set is the same as the Mandelbrot set, except that z=pixel and c is set as a constant in the

beginning- i.e. c is the "Julia Seed."[www.tonkoppens.nl/Tutorial01/test.html]).


Have you already attempted this? Or have you got any comments on this?

I'm ready to iterate my explanations, if you want to. I'll wait for your answer many days...



(By the way, although I have visited these forums many times, this is my 1st post. When I first saw

the animations, I couldn't believe my eyes!).

[makc]

where's your image?

[bib]

Looks interesting. Sorry I'm not a math expert, but I think the symbol <=> means "is equivalent to" and take place between 2 equations, not 2 expressions. For fractal iterations, in general we write z<-z²+c, so can you please clarify the iteration rule for dummies like me?

z0=c
z1=c²+?

Thanks

[makc]

maybe
z0 = c0
z1 = z0^2 + c1, c1 = c0^2 + d;
z2 = z1^2 + c2, c2 = c1^2 + d;
...
?

[bib]

Yes, that should be it. Who will do the image first?

[makc]

I just did quick test, it seems to kill off details for any value of d



late edit:

I keep messing with the idea of variable c (not this formula any more) but whatever I do, I only get less details:



it's magic...

[makc]

btw d=0 gives perfect circle EDIT d = const, like 0.1 -0.5 i, give julia-like sets. but again with little details

[Count Fractula]

Thanks for the quick replies makc and bib! That's very kind of you.

Dear bib:
About the symbol <=>. I meant the kind of arrow which has the upper part pointing to the left and the lower part pointing to the right. You can see an example of that arrow at http://webweevers.com/fractals.htm
where it says "The simple formula first used by Benoit Mandelbrot", at mid page.
Sorry about that, I don't know how to type the exact symbol.
And as regards the iteration rule, well...my formula is a product of my infinite ignorance.

Dear makc:
Thanks for the time you've spent on the concept. I really appreciate it.

I am conscious that my post looks as if I know what I am talking about...but I don't.
It looks like maths, but it's ignorance. Please, forget about that "formula".
Instead of writing that formula, I should have stated the concept in plain language and let the experts do their thing.
My strategy didn't work completely.



1º) Breaking Rules intelligently (gradually, chaotically, fractally, etc)
So let's state the concept in plain language, and roughly.
The fractals I see here and there are somewhat imprisoned.
They obey a formula and are not allowed to break it.
Now, what if you allow them to break the iteration rule(s)?
They could break it gradually or even chaotically.
Enthropy comes to my mind.

Is there any part of the formulas that can be less "constant"?
Is there room for doing any of this?


2º) Fractal Interference
The second concept is related to 1º, in that it is an attempt at breaking the rules intelligently.
As you probably know, waves can create interference.
Interference is everywhere...think of the colours in soap bubbles.
But you can also think of tuning a musical instrument such as a guitar.
When two frequencies are very close to each other they produce "beats", a pulsating ondulation.
For example, 200 Hz + 202 Hz will produce 2 beats per second (because 202 - 200 equals 2).
Sometimes these beats are extremely pleasant and soothing but in certain contexts they can become irritating.

Is there any part of iteration formulas that can make use of the phenomenon of interference?
Is there room for doing any of this?


3º) Fractal love
This concept is of a more weird nature but well...I love weird!
Is it possible to a fractal to make love?
Is it possible to construct a community of fractals that date and have mathematical intercourse, so to speak?
Many of you have drawn faces in fractals.
So the idea is latent somehow.
Imagine a community of these little mathematical organisms living together and interchanging their codes, like interchanging DNA.
I have seen something like this done for making certain algorithms evolve.
Studying evolution, for example, they create 100 simple software brains and a computer allows to mate only those which fulfill certain characteristics.
After a certain number of offspring, the brains show a type of evolution that would have been too difficult for programmers to work out.
Yes, I know, too much computing power. But at least we could get the first glimpses.

Is there room for fractal lovers?



Thanks for your time and patience.

[makc]

Now, what if you allow them to break the iteration rule(s)?
They could break it gradually or even chaotically.

oh I did that here - drag the sliders to mess with corresponding iteration

[makc]

Imagine a community of these little mathematical organisms living together and interchanging their codes, like interchanging DNA.

sounds like you need to install this.

[Count Fractula]

Great job makc!
That's just bright.
It's the 1st time a programmer allows me to interact with the Mandelbrot set in such an efficient way.
I'm really thankful.

Your Flash creation will keep me up tonight...moving all those sliders!
However, for the sake of coherence, I must tell you that it's not what I would really like to see...and it's not only because of the number of colours used or the beauty of the graphics.
Every transformation is symmetric.
No matter which slider you move, you get a symmetric result.
Is it possible to make the Mandelbrot set to behave wildly?
Let's call it Chaosbrot.
Are chaosbrots possible?

In many natural objects, there is some kind of obvious symmetry.
But it's never perfect, it is naturally wild.
Our first geometries deal with circles, and squares, and triangles, etc.
All idealized forms.
These new fractal concepts get closer to the real objects.
You can easily recognize organic patterns, and structures.
But, somehow, symmetry, and idealization are still there.
It's true that anyone can take a picture of a fractal and edit it until you can barely recognize it.
But an infinitude of the results and possibilities would not be the same.

So again, do you think "chaosbrots" are possible?
Do you think there's a chance of breaking symmetries (or rules) in an intelligent way?


(Note: I will also see what electricsheep.org is all about, it sounds quite similar to what I had in mind but it seems even better! I hope it's not about "naked" sheeps kissing each other!  ).

[makc]

Every transformation is symmetric.
No matter which slider you move, you get a symmetric result.
Is it possible to make the Mandelbrot set to behave wildly?

yes it is possible. the applet I posted was made long before your request, so it's just a coincidence that it does somewhat close to what you had in mind. if you had enough time to learn very basic math and programming, you could easily tweak it to your taste.

[Count Fractula]

Great, I'm glad it is possible.
In a way, you have already proved it with your "random-like" transformations applet.
And I've been looking at other works that also point in that direction.

By the way, you mentioned my "request."
I'm not making a request, not at all.
I'm Count Fractula, not Count Dracula...so I'm not trying to vampirize nobody.
Although I admit it may have looked like that.
As I wrote at the very beginning, I want to share the concept of a less symmetrical approach to these visualizations or sets.

Looking at the other applets you've created at wonderfl.net, I was very happy to find an interactive Pythagoras Tree...gorgeous! (http://wonderfl.net/c/tS3h).
There's a 3D one there too, but my computer doesn't seem to cope well with it (and it doesn't look so appealing as the 2D version).
That's what I meant by killing symmetry in an intelligent way...I don't know the technical terms for these kind of transformations, sorry.

So what I've learned is that my little ideas are not only little, they also lack originality.
However, my knowledge has achieved a little more depth.

Now I have to leave my Dear makc, there's a young lady in her chamber with a long neck ready to be sucked...ahhhhh those young and fractal vein structures..."Veins":

<a href="http://www.youtube.com/v/KjEWZP0Z6jY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/KjEWZP0Z6jY&rel=1&fs=1&hd=1</a>

See you

[miner49er]

re: "Fractal Love" - as insane as this particular idea sounds I, too have thought something sort of similar!
Perhaps not quite what this guy is thinking though (but probably just as insane). I envisaged some sort of analysis of what makes an image attractive to us and have the algorithm create 'random' images. I further thought it that should be possible to teach the algorithm what is pleasing to use by rating images somehow. Perhaps, a range of thumbnails could be presented to the user and one is chosen as being 'the nicest' :-)
I do realise this is more difficult than is seems as the 'learning' algorithm would have to 'see' the picture...

yes, ridiculous.