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 Author Topic: if the Mandelbrot set is infinite...  (Read 8937 times) Description: 0 Members and 1 Guest are viewing this topic.
Fractal Molossus

Posts: 703

 « Reply #15 on: June 09, 2010, 01:42:07 AM »

"you will never found a picture of the letter i"
Usually you colour each pixel black if the point under the centre of each pixel is inside the set, so the pic below shows how you can get the letter i because you will never be rendering with infinite resolution.
Since anti-aliasing just operates at a higher (but finite) resolution it should be clear that you can also get an anti-aliased i with the correct shades of grey, even when the real fractal underneath is quite different in shape.
It isn't a big extension to see that you can get any iteration count that you like under each pixel, and so, if the iteration colour spectrum is complete, you should be able to find any colour picture in the mandelbrot set, e.g. the one below
 anything.gif (3.51 KB, 363x140 - viewed 556 times.)  george.jpg (3.4 KB, 96x123 - viewed 526 times.) « Last Edit: June 09, 2010, 01:44:17 AM by Tglad » Logged
lkmitch
Fractal Lover

Posts: 238

 « Reply #16 on: June 09, 2010, 04:27:57 PM »

It isn't a big extension to see that you can get any iteration count that you like under each pixel, and so, if the iteration colour spectrum is complete, you should be able to find any colour picture in the mandelbrot set, e.g. the one below

While it's true that some images of the Mandelbrot set resemble some other things, I'm pretty sure that one will never find a region that looks like that photo of Mr. Bush.  There's a difference between infinite (which the Mandelbrot set is) and all-containing (which it ain't).  The latter implies the former, but not vice-versa.
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kram1032
Fractal Senior

Posts: 1863

 « Reply #17 on: June 09, 2010, 11:06:30 PM »

Also it's really a matter of how you colour the set. If you use the often-seen blue-yellow gradient, there is no possibility that something like tan could ever happen.
The Mset is not natively colours. They only are used to clarify the shape.
If you want a photo inside the Mset, tile it over the set as done a million times already
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trafassel
Fractal Bachius

Posts: 531

 « Reply #18 on: June 10, 2010, 07:36:57 PM »

The black and white Version of the hypothesis can defined
as follow:

Let BWPicture be a set of pairs of integer points and
minx, maxx, miny, maxy natural numbers with
minx < maxx and miny < maxy.

The open question is: exists for each set BWPicture
a transfomation T
(as combination of rotations, translations and scale
operations) and a natural number n, such that
for each integer pair (x,y) with minx<x<maxx and
miny<y<maxy: Mandelbrot(T(x,y),n)=1 if and only if
(x,y) is in BWPicture?

I think, this question is hard enought. Forget the colors.

scaling)

I open question
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teamfresh
Fractal Lover

Posts: 246

nothing is everything

 « Reply #19 on: June 11, 2010, 12:03:56 AM »

yeah i like t-glads argument as well, I also like this...

The mandelbulb might be a better place to look for all shapes possible?

still if you could find your name that wold be sooo cool!
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Timeroot
Fractal Fertilizer

Posts: 362

The pwnge.

 « Reply #20 on: June 11, 2010, 03:34:11 AM »

The concept of "infinity" is counter-intuitive in many ways. In particular, "infinity" is not a number that you can simply use in computation. For example, take a subset of all integers, say, all numbers from zero to one thousand. Half the numbers in that subset are even, and the other half are odd numbers.

The split between even and odd numbers remains the same, even for larger subsets, say, all numbers from zero to one million. In fact, the subsets may be arbitrarily large (ignoring the case when there is one more number of one kind than the other kind). And in all these subsets, the number of even integers is smaller than the size of the whole subset.

This changes drastically when you look at the whole set of all integers (i.e. no upper limit). Then there are infinitely many numbers in the set, with half of them being even, and the other half being odd.

But there are infinitely many even numbers! This can be seen with the mapping
f(x) := x -> 2*x
which establishes a one-to-one correspondence between all integers and the even integers.

The even numbers are now a subset of the positive integers, but that subset is as large as the whole set, despite the lack of odd numbers.

Or, in simpler terms: when going to infinity, things might no longer add up. :-) To come back to your original question: a set being infinitely large does not imply that it contains everything. Half of infinity is still infinity.

You're working with Cardinal numbers here, one of several types of infinity. Ordinal numbers behave like a perfectly respectable infinity in many ways, and it's algebra is well defined. One of the weird things is that $\infty = 1 + \infty = \infty - 1$, but non of these are equal to $\infty + 1$. Still, in most respects it makes sense.

I think you would be able to find all shapes - especially if you're just sticking to black & white, because the cardinality of black and white raster images is much less than the cardinality of points in the complex plane. I think the Black and White Mandelbrot hypothesis (BWMH?) wouldn't even need to require any trickiness with changing the iteration, since any Minibrot will stay solid no matter what the iteration count, so if the minibrots were arranged just right we could get them to stay solid under the centers of the pixels.

I would state the BWMH as follows:

Let A and B be two finite positive integers, and let R be a finite set of points with positive, finite integer coordinates, such that each point P in R has an x-coordinate less than or equal to A and a y-coordinate less than or equal to B. Then for each A, B, and R there exist somecomplex Z and some real S such that for any two positive integers U <= A and V <= B, the number Z+(S*U/A)+i*(S*V/B) is in the Mandelbrot set if and only if the point (U,V) is in R.

How hard do you think it would be to get this instated as a new Millennium problem? Maybe we'd have to wait 1,000 years..
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
KRAFTWERK
Global Moderator
Fractal Senior

Posts: 1439

Virtual Surreality

 « Reply #21 on: June 15, 2010, 11:45:57 AM »

I just uploaded this to the gallery.

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=2641

Lots of "faces" hiding in it (didn't see mr Bush though... some examples...

 1.png (154.01 KB, 615x579 - viewed 306 times.)  2.png (58.06 KB, 346x336 - viewed 521 times.)  3.png (110.65 KB, 480x487 - viewed 538 times.)  5.png (85.2 KB, 459x428 - viewed 512 times.) Logged

hobold
Fractal Bachius

Posts: 573

 « Reply #22 on: June 15, 2010, 03:51:53 PM »

Many faces are a trick of the human mind. We're wired to recognize and "read" the faces of other humans very quickly, even under adverse conditions. That's why ASCII emoticons work so well: 8vD

We tend to detect faces even when there are none. It's a systematic bug.
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Rathinagiri
Fractal Fertilizer

Posts: 374

 « Reply #23 on: June 15, 2010, 05:40:36 PM »

We tend to detect faces even when there are none. It's a systematic bug.

Nice words.
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KRAFTWERK
Global Moderator
Fractal Senior

Posts: 1439

Virtual Surreality

 « Reply #24 on: June 16, 2010, 08:54:02 AM »

Many faces are a trick of the human mind. We're wired to recognize and "read" the faces of other humans very quickly, even under adverse conditions. That's why ASCII emoticons work so well: 8vD

We tend to detect faces even when there are none. It's a systematic bug.

You are absolutely right hobold, BUT!
What is a "face" anyway?
What we see has to do with how we interact with this universe.
A bit like the images I posted shows how the computer interacted with the formula given the conditions at that time.
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visual.bermarte
Fractal Fertilizer

Posts: 355

 « Reply #25 on: June 16, 2010, 02:08:41 PM »

Maybe it is possible to add extra rules to the mandelbrot set..
Imagine an intelligent mandelbrot set; only that one could reproduce (for example) a raster copy of this page..This special set would be able to read this text and reproduce it..a super mandelbrot self programming set; a kind of a god.
It would know languages or being able to imitate human beings's language and so on..so this is the real difficulty.
For sure could be obtained an 'i' or maybe even a face but what about a particular sequence of images? what about a simple name, a text?
Will we be happy enough with the result? if it is just 'an estimation', maybe yes..but under the condition to patch a lot of random images together..our mind and imagination will play the trick of finding an output more similar to the input here and there.
This set would organize itself to create the appropriate rules to imitate images ..finally the problem is to find these rules to create a raster images of a string like 'if the Mandelbrot set is infinite...'; so why not creating a way to manipulate symbols starting from an alphabet?
 « Last Edit: June 16, 2010, 02:36:17 PM by visual » Logged
hobold
Fractal Bachius

Posts: 573

 « Reply #26 on: June 16, 2010, 03:36:01 PM »

I didn't expect my offhand remark about face recognition to draw so much attention. And I want to apologize for using terminology that equates humans with computers. I do not believe that reducing brains to computing machinery is the right point of view (neither philosophically nor scientifically). But the jargon of a particular community is the most efficient means of communication within that community, so ...

Anyway, if you want to learn a bit more about this face recognition bias, and lots and lots of other interesting facts about art and visual communication, I heartily recommend "Understanding Comics" by Scott McCloud. This is a highly scientific but highly accessible and entertaining book about comics. It is itself written as a comic, and full of striking examples of everything it talks about. Reading this book will permanently change your view, and IMHO enrich it.
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KRAFTWERK
Global Moderator
Fractal Senior

Posts: 1439

Virtual Surreality

 « Reply #27 on: June 16, 2010, 03:41:03 PM »

I heartily recommend "Understanding Comics" by Scott McCloud.

I will check it out hobold!
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visual.bermarte
Fractal Fertilizer

Posts: 355

 « Reply #28 on: June 16, 2010, 04:20:01 PM »

to find a representation of the string 'Scott McCloud' would be necessary only 8 parameters of one or more formulas.
Every parameter would have infos about position, camera etc.
The sequence of parameters would create the sequence of symbols even without the help of AI.

I like a lot that last image.
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KRAFTWERK
Global Moderator
Fractal Senior

Posts: 1439

Virtual Surreality

 « Reply #29 on: June 16, 2010, 04:32:34 PM »

to find a representation of the string 'Scott McCloud' would be necessary only 8 parameters of one or more formulas.
Every parameter would have infos about position, camera etc.
The sequence of parameters would create the sequence of symbols even without the help of AI.

I like a lot that last image.

You are crazy visual!

The last image? The guy with very big glasses?  :
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