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Author Topic: if the Mandelbrot set is infinite...  (Read 8935 times)
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teamfresh
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« on: March 08, 2010, 11:50:00 AM »

I have been given a theoretical question:

if the Mandelbrot set is infinitely large, does that mean it contains all possible shapes? Or does it repeat to often to include all possible forms? In other words, if you set your color palette to use all colors, and I gave you any image, and infinite amount of time, would you be able to find that image somewhere in the set?

I would value your opinions on this one!
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Sockratease
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« Reply #1 on: March 08, 2010, 12:01:11 PM »

I have to say no.

Infinite is not all inclusive.  The set of counting numbers is infinite, but you'd never find your name in it because it contains no letters!

There are different types of and magnitudes of Infinity.

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kram1032
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« Reply #2 on: March 08, 2010, 12:53:03 PM »

I guess, the natural numbers together with names are a rather bad example because you could interpret them as a binary and then as a string again. Your name will be somewhere in them that way.
However I agree with you: You wont see every possible shape if you just go deep enough into the Mset. Some of the structures change with certain rule, that already are known. All the other structures look a lot like having rules which just aren't entirely described yet.
If you want a simpler example: Take time (in the classical newtonian way). Time is measured by splitting a periodic process into smaller units or counting how many times that peridoic process happens.
While this process might be infinite, it does include only itself.

For a simpler math example, just take the sine and cosine or something like that.
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hobold
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« Reply #3 on: March 08, 2010, 02:18:16 PM »

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.
« Last Edit: March 08, 2010, 02:20:08 PM by hobold » Logged
kram1032
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« Reply #4 on: March 08, 2010, 02:35:00 PM »

That again is based on which number model you use smiley

Some weird concepts allow stuff like numbers, bigger than the infinity you get by just counting to infinite.
Then, there are different kinds of infinities.

One important thing is also, that the Mset isn't quite infinite in its expansion. It's infinite in its details. The Mset is all about the infinitely small rather than the infinitely big.
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visual.bermarte
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« Reply #5 on: March 08, 2010, 02:45:11 PM »

In this case a subset of natural numbers (even or odd numbers) has the same cardinality of the set of natural numbers.
Cantor's Aleph null is the first and smallest infinite cardinal.
http://en.wikipedia.org/wiki/Cardinal_number

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bib
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« Reply #6 on: March 08, 2010, 02:51:50 PM »

Or, in simpler terms: when going to infinity, things might no longer add up.
They might in fact even "substract". I read an article about infinity and the special notations for very very big numbers (eg. where the number of levels using an exponent notation is himself hugely big : 10^10^10^....h...u...g...e...^10) and they gave an example of a rule and a sequence that grows extremely rapidly (e.g 1, 4, 1456767, 1E50, 1E22345...) but that eventually converges to 0 even if each terms is bigger than the previous one ! But don't ask me for the details, it's still sounds impossible to me...!
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hobold
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« Reply #7 on: March 08, 2010, 04:07:56 PM »

As several of you noted, it is still quite possible to reason about infinity or even infinities. But I didn't want to overwhelm the original poster, or worse, scare him away.

No matter how much you learn about infinity, be it a baby step or a giant leap, it's still only an infinitesimal fraction of the whole ... smiley
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teamfresh
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« Reply #8 on: March 08, 2010, 11:47:51 PM »

wow thanks for the response guys! I will pass these answers on to the person that asked =) 
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Tglad
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« Reply #9 on: March 09, 2010, 12:16:37 AM »

My guess is yes.
If you ever study artificial neural networks you will know that a network 3 layers deep (and n nodes wide) is capable of approximating any function at all, to arbitrary precision depending on the number n. The mandelbrot set might have a similar property.
The question is whether the mandelbrot can approximate any shape to arbitrary precision, depending on the zoom level.
It doesn't have to exactly match (e.g. be an exact filled square), it just has to average the right colours under each pixel for a given resolution of the shape. A higher res shape can use a different (deeper) part of the set.
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KRAFTWERK
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« Reply #10 on: April 12, 2010, 11:12:43 AM »

Found this thread wich I remembered this weekend when I found a man (we can call him Daniel ; ) looking out at me from a cave in the mandelbulb.

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

Off course everything depends on iteration count and light setup and all other variables, but anyway, here he is  afro

J


* Maninthebulb.jpg (78.58 KB, 355x305 - viewed 2689 times.)
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Sockratease
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« Reply #11 on: April 12, 2010, 11:45:06 AM »

Found this thread wich I remembered this weekend when I found a man (we can call him Daniel ; ) looking out at me from a cave in the mandelbulb.

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

Off course everything depends on iteration count and light setup and all other variables, but anyway, here he is  afro

J

I found this guy in my first exploration of the Mandelbulb.


Bigger : http://www.sockrateaze.com/stuff/1b1.png

Further refinement of the coloring algorithm brought out even more life-like detail...


Bigger : http://www.sockrateaze.com/stuff/1b.png
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Krumel
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« Reply #12 on: June 08, 2010, 07:39:53 PM »

I would say no.
There are many sets which have infinity things in it, but not everthing.
Simple example:
There are infinite numbers betwen 0 and 1, but you'll never find 2.
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kram1032
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« Reply #13 on: June 08, 2010, 09:55:15 PM »

What you're talking about is set theory smiley
The Integers are an infinite set of lower cardinality than the reals.
The intervall  0...1 over the reals also is an infinite set.
Hmm... Which set has more elements? xD
All integers or the reals from 0 to 1?
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trafassel
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« Reply #14 on: June 09, 2010, 12:10:28 AM »

You must also bear in mind, that the M-Set
has some mathematical properties.

Example: because it is connectet, you
will never found a picture of the letter i
in the M-Set. 
   
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