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Author Topic: Simplest formula for producing arbitrarily complex fractal?  (Read 2127 times)
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quick yellow whale
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« on: January 15, 2017, 12:19:00 AM »

"arbitrarily complex" means that as you zoom in, the location of where you zoomed affects what you see, making it possible to keep zooming and getting more and more complex images. Simplest would mean uses the fewest number of operations in the formula when all operations that use complex numbers are converted to ones that use regular numbers.

The Mandelbrot set formula is one of the simplest I've seen so far, with this formula where the complex operations have been converted into regular operations:

<br />Z^x_0 = Z^y_0 = 0<br />Z^x_{n+1} = Z^x_n*Z^x_n - Z^y_n*Z^y_n + x;<br />Z^y_{n+1} = 2*Z^x_n*Z^y_n + y;<br />

This uses 4 multiplications, 2 additions, and 1 subtraction.

However, if the 2* is removed, the fractal that is produced seems to still be arbitrarily complex on the right side. Is there an even simpler formula that can produce an arbitrarily complex fractal?
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youhn
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« Reply #1 on: January 15, 2017, 02:38:12 AM »

The complexity only arises because the choice of the 2D project system. Try the formula in it's plain form and you will see it doesn't get much simpler than that:

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Tglad
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« Reply #2 on: January 15, 2017, 07:51:13 AM »

Note that z=z^2+c is a single iteration of a complex number, part of a more complicated escape time algorithm.
But Z=Z^2+C is a single equation (not an assignment) operating on the whole complex plane which defines the same set exactly. Very simple with very complex results.
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Tglad
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« Reply #3 on: January 15, 2017, 07:41:15 PM »

Actually not quite right, the set valued formula only works for Julia sets (and with a little c), the Mandelbrot set is a bit harder to express in this way. Might need the language of vector bundles to write concisely.
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quick yellow whale
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« Reply #4 on: January 15, 2017, 11:50:34 PM »

The complexity only arises because the choice of the 2D project system. Try the formula in it's plain form and you will see it doesn't get much simpler than that:

<Quoted Image Removed>

I'm looking for the simplest "real" formula that produces a complex fractal. My reasoning is that imaginary numbers don't exist in the real world, so computers can only do computations on real numbers, which means any fractal can only be realized using real numbers.
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kram1032
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« Reply #5 on: January 16, 2017, 12:35:08 AM »

computers can only manipulate 0s and 1s. With streams of 0s and 1s you can define real numbers, floats (which are NOT actual real numbers) or complex numbers alike.
Imaginary numbers don't not exist any more or less than real numbers don't exist. They are human-conceived abstractions that turned out to be super useful either way. Furthermore, there are great geometric justifications for complex numbers: Essentially pretty much everything about them follows from the fact that i corresponds to a quarter rotation in the euclidean plane. With that exact meaning, i actually smuggles its way into physics equations. - On one hand you can always avoid writing this i. On the other, lots of stuff becomes super awkward. This also is true here: z \to z^2+c is a lot simpler than \left(x,y\right) \to \left(x^2-y^2+a, 2xy+b\right).
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quick yellow whale
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« Reply #6 on: January 16, 2017, 03:55:22 AM »

computers can only manipulate 0s and 1s. With streams of 0s and 1s you can define real numbers, floats (which are NOT actual real numbers) or complex numbers alike.
Imaginary numbers don't not exist any more or less than real numbers don't exist. They are human-conceived abstractions that turned out to be super useful either way. Furthermore, there are great geometric justifications for complex numbers: Essentially pretty much everything about them follows from the fact that <Quoted Image Removed> corresponds to a quarter rotation in the euclidean plane. With that exact meaning, <Quoted Image Removed> actually smuggles its way into physics equations. - On one hand you can always avoid writing this <Quoted Image Removed>. On the other, lots of stuff becomes super awkward. This also is true here: <Quoted Image Removed> is a lot simpler than <Quoted Image Removed>.

When z \to z^2+c is calculated on a computer, a sequence of operations are performed for the calculation. I'm looking for the simplest sequence of operations that can produce a complex fractal. \left(x,y\right) \to \left(x^2-y^2+a, 2xy+b\right) is just a more precise definition of what the computer does to compute the Mandelbrot fractal than  z \to z^2+c.
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hobold
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« Reply #7 on: January 16, 2017, 06:00:51 AM »

Let me just try to clarify my understanding. The words in this case seem to mean:

"complexity":
visual complexity, i.e. the original poster is interested in the generation algorithm which requires the lowest time to arrive at some interesting resulting image. The result does not necessarily have to have a mathematical structure that is related to the plane of complex numbers.

"formula":
the sequence of fundamental computational operations, i.e. the original poster is not interested in mathematical notation, but in algorithms and concrete programs for concrete computers.

Am I getting that right?
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quick yellow whale
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« Reply #8 on: January 16, 2017, 06:21:21 AM »

Let me just try to clarify my understanding. The words in this case seem to mean:

"complexity":
visual complexity, i.e. the original poster is interested in the generation algorithm which requires the lowest time to arrive at some interesting resulting image. The result does not necessarily have to have a mathematical structure that is related to the plane of complex numbers.

"formula":
the sequence of fundamental computational operations, i.e. the original poster is not interested in mathematical notation, but in algorithms and concrete programs for concrete computers.

Am I getting that right?

For "complexity" I'm not exactly sure how to explain it, but when you zoom into the Mandelbrot set the patterns you see as you zoom deeper are repeats of patterns that you passed before, as explained in this video:

<a href="https://www.youtube.com/v/Ojhgwq6t28Y&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/Ojhgwq6t28Y&rel=1&fs=1&hd=1</a>

It's this kind of "Turing-completeness" that I'm looking for.


For "formula" yes, I am looking for the simplest combination of real operations that can produce a complex fractal.
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youhn
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« Reply #9 on: January 16, 2017, 07:58:41 AM »

I'm looking for the simplest "real" formula that produces a complex fractal. My reasoning is that imaginary numbers don't exist in the real world, so computers can only do computations on real numbers, which means any fractal can only be realized using real numbers.

Do "real" numbers then exists in the real world ... ?  hrmm

Our thinking methods (math, geometry, counting) are just simplified models to make some sense of our perception of the world. Negative numbers and imaginary numbers are just as usefull as real numbers, those names are just convention and nothing more.
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quick yellow whale
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« Reply #10 on: January 16, 2017, 08:36:52 AM »

Do "real" numbers then exists in the real world ... ?  hrmm

Our thinking methods (math, geometry, counting) are just simplified models to make some sense of our perception of the world. Negative numbers and imaginary numbers are just as usefull as real numbers, those names are just convention and nothing more.

The key word is "real". In the real world only real things exist. And if you only count real things, then the numbers you get can only be real.
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hobold
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« Reply #11 on: January 16, 2017, 11:37:20 AM »

Let's just clarify the topics further: a "real number" is one that can efficiently be handled by actually existing computing machinery?

(What I am trying to do here is to straighten out what I perceive to be misunderstandings between various participants in this thread.)

If I am getting this right, then the original question is not quite so specific, but more of an exploration. Let me try to phrase it completely differently (the original poster should correct me if I am totally off):

"Can the computer produce images that are at least as interesting as the Mandelbrot Set with a lower amount of computational work?"

Answers would ideally lead to steps of "improvement", i.e. either a reduced number of computational machine operations, for an equally interesting image; or a "better" image (whatever that may be exactly) with the same number of computational operations.

Such a chain of hypothetical "improvements" would naturally lead to the question of "best" such algorithm, i.e. the one that can no longer be improved or simplified.


I have no clue how realistic that search is, and I readily admit that it is a bit vague and philosophical. Still, some "naive" experimentation could lead to a faster generator of pretty pictures, even if it does not lead to truly new insights.

It seems obvious to me that the Mandelbrot Set is probably quite close to such a lower limit already. But it seems completely not obvious to me that the Mandelbrot Set is that lower limit.
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kram1032
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« Reply #12 on: January 16, 2017, 12:02:05 PM »

I'll stand by my comment though I recognize the spirit of your question.
Self-similarity also is a problem as basic definition of fractal complexity: You know what's self-similar? A cube. You can cut it into arbitrarily many other cubes. By the same token, a square or a line will do. And even a point: No matter how deep you zoom into a point, the image won't change at all!

That being said, if you are looking for visually interesting fractals, as the Mandebrot Set is one, I think that will always require some sort of non-linearity. - Mind you, non-linearity is not a guarantee for interesting stuff, but it's a root cause for that.
And this is why the Mandelbrot Set is so insanely simple: It adds one of the simplest types of non-linearity, a squared term, into the mix, and BAM, visually striking beauty.
No doubt here are other simple kinds of non-linearity to consider, which may result in even simpler formulae, but it certainly will be hard to beat out on the MSet.
One extremely simple thing you could do is to, instead of complex numbers, iterate over dual numbers with a constant \epsilon^2=0:
z \to z^2+c
\left(x,y\right) \to \left(x^2+a,2xy+b\right) - you just saved one multiplication and one addition!
However, while it still produces some interesting patterns, the end result isn't nearly as complex as what the standard definition gives you. Here is the result if you go for a Buddhabrot-Style rendering. Not actually sure what the plain old escape time method will give you.
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quick yellow whale
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« Reply #13 on: January 16, 2017, 04:43:56 PM »

One way I'm thinking about finding such a formula is to make a program that generates formulas from every possible combination of variables {x, y, a, b} and operations {+, -, *, /} using up to three of each variable, with an operation between every two variables, and then create a fractal image from each formula using the escape-time method. Then I would look to see if any of the images produced are as complex as the Mandelbrot fractal.
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trafassel
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trafassel
« Reply #14 on: January 16, 2017, 06:59:18 PM »

Kali is the specialist in creating very simple fractal formulas.

See http://www.fractalforums.com/mandelbrot-and-julia-set/mandelbrot-on-real-numbers-t5375/msg26835/#msg26835
or
http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/

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