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 Author Topic: Is this allowed?  (Read 687 times) Description: 0 Members and 1 Guest are viewing this topic.
Chillheimer
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Fractal Schemer

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Just another fractal being floating by..

 « on: August 13, 2016, 12:15:05 PM »

I'm in a little email discussion with someone and pointed out that I think that the depiction of the axes in the attached image is false.
Am I right? Or is it ok to move the axes around as long as you don't put any fixed values as reference points? I thought the x axis always has to go through the middle of the cardiod.

And a second question that came up: Is it in any way appropriate to call the Mandelbrot set an IFS?
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TheRedshiftRider
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 « Reply #1 on: August 13, 2016, 12:33:36 PM »

I think it is allowed to move around an axis as long as the values stay the same. In this example the imaginary axis has been move by a value of two but the axis does not indicate values, as long as the values that make up the set do not change with 2i it is okay. In this case it can still be considered correct I think.

Not sure about the IFS. I don't think it is correct, you can not call an escape-time fractal an IFS fractal, eveb though there is an iterated process going on.
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cKleinhuis
Fractal Senior

Posts: 7044

formerly known as 'Trifox'

 « Reply #2 on: August 13, 2016, 04:35:40 PM »

i second the notion for the axis, since the mandelbrot is 90 degree rotated, and the axis show no numbers it is totally valid to display it as such

i tend to call mandelbrots iterated functions systems as well, the term depicts a collection of function not just one single function which is repeatedly applied,although one single function would make a special case for such a system, regarding to the (german) wikipedia definition https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem

the only rule that needs to be valid is that any function combined with another function from the collection of functions (one called after another) is a function of that system as well, which would hold true for the mandelbrot -> my five pence would lead to that mandelbrot is a iterated function system following the definition....

one could argue that the resulting image creation methods differ strongly, but that is always an interpretation, both methods could as well just plot dots in a graph.... so representation of values is just from the makers point of view, the underlying mechanic of using formulas repeatedly holds true though

perhaps a true mathematician can shed light on the issue
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Chillheimer
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Just another fractal being floating by..

 « Reply #3 on: August 13, 2016, 04:47:06 PM »

thank you guys. I guess I'll have to admit that I was wrong.
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Sockratease
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 « Reply #4 on: August 13, 2016, 05:43:04 PM »

thank you guys. I guess I'll have to admit that I was wrong.

I'm not so sure you're wrong.

From the most elementary point of view, it shows an intersection of the 2 axes.  Where else do they intersect besides 0,0?

It's OK to shift axes, I suppose - but how do you explain moving the origin? I would think showing them shifted should omit the intersection.

I must be missing something...
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valera_rozuvan
Safarist

Posts: 80

a test in time

 « Reply #5 on: August 13, 2016, 06:10:00 PM »

And a second question that came up: Is it in any way appropriate to call the Mandelbrot set an IFS?

Many online resources mix several things together:

From Wikipedia on the Mandelbrot set:

"The Mandelbrot set is the set of complex numbers $c$ for which the function $f_{c}(z)=z^{2}+c$ does not diverge when iterated from $z=0$, i.e., for which the sequence $f_{c}(0)$, $f_{c}(f_{c}(0))$, etc., remains bounded in absolute value."

From Wikipedia on Iterated Function Systems:

"In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting constructions are always self-similar. ... The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński gasket, also called the Sierpiński triangle. The functions are normally contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar fractal nature."

The important thing to note, is that in IFS, "the fractal is made up of the union of several copies of itself." In the case of the Mandelbrot set, the iterated formula only produces the answer to the question: is the current point in the set? So, the original method that produces the Mandelbrot set does not use an Iterated Function System.

Now, an interesting question. Is it possible to build an IFS that will produce the Mandelbrot fractal? So far this has not been done. Is it possible? I don't know. This question has already been discussed over at mathoverflow.net: Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?
 « Last Edit: August 13, 2016, 06:16:24 PM by valera_rozuvan » Logged

TheRedshiftRider
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 « Reply #6 on: August 13, 2016, 06:32:25 PM »

I'm not so sure you're wrong.

From the most elementary point of view, it shows an intersection of the 2 axes.  Where else do they intersect besides 0,0?

It's OK to shift axes, I suppose - but how do you explain moving the origin? I would think showing them shifted should omit the intersection.

I must be missing something...
That is pretty simple I think. You just move the set by 2+2i after all calculations have been done. It would be kind of strange. All calculations would be -2+-2i relative to the axes. Although I must say it is strange to do that.
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Sockratease
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Posts: 3181

 « Reply #7 on: August 14, 2016, 10:36:03 AM »

That is pretty simple I think. You just move the set by 2+2i after all calculations have been done. It would be kind of strange. All calculations would be -2+-2i relative to the axes. Although I must say it is strange to do that.

Wow.  Me dummy...

I haven't done any mathematical analysis of this sort since college, decades ago.  Can't believe I missed that

My work has me analyzing some large datasets, but it's all so narrowly focused that the stuff I have been away from for so long has been all but forgotten...  Until it get pointed out to me!  Then it comes back.

Thanks, Red!
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cKleinhuis
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formerly known as 'Trifox'

 « Reply #8 on: August 14, 2016, 12:46:16 PM »

sock, the axis did not show the positions, so they just indicated the direction, and hence the mandelbrot on the picture is rotated by 90 degree it fits
you could put numbers on it, if the cross center would be 0,0 then the fractal formula is added 2,2 like red explained, otherwise the cross center is -2, -2, or something to indicate its line, and yes the axisses cross all the time which is what we call a coordinate
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TheRedshiftRider
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 « Reply #9 on: August 14, 2016, 04:20:54 PM »

Wow.  Me dummy...

I haven't done any mathematical analysis of this sort since college, decades ago.  Can't believe I missed that

My work has me analyzing some large datasets, but it's all so narrowly focused that the stuff I have been away from for so long has been all but forgotten...  Until it get pointed out to me!  Then it comes back.

Thanks, Red!
sock, the axis did not show the positions, so they just indicated the direction, and hence the mandelbrot on the picture is rotated by 90 degree it fits
you could put numbers on it, if the cross center would be 0,0 then the fractal formula is added 2,2 like red explained, otherwise the cross center is -2, -2, or something to indicate its line, and yes the axisses cross all the time which is what we call a coordinate

Good to see I am helping out.

In fact I was just being silly, I was a bit tired when I wrote it.
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