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Author Topic: 1D Mandelbrot  (Read 5077 times)
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« on: November 23, 2009, 11:14:12 PM »

Maybe it would be an interesting experience to try and find a one dimensional representation of the Mandebrot Set?

I tried z[n]=z[n-1]+c but that seems to produce no complexity at all. All points from c=-2 to about c=0.25 are inside the set and all others are outside the set.

Any ideas?
« Last Edit: November 23, 2009, 11:20:41 PM by choose » Logged
cKleinhuis
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« Reply #1 on: November 23, 2009, 11:16:00 PM »

lol, congratulations, you found the 1 d mandelbrot ... cheesy  afro
what did you expect ?!?!?
you can use another line through the set ... cheesy
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« Reply #2 on: November 24, 2009, 01:00:32 AM »

Any ideas?

Instead of using a straight line use a strange attractor ?
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« Reply #3 on: November 24, 2009, 01:16:59 AM »


I tried z[n]=z[n-1]+c but that seems to produce no complexity at all. All points from c=-2 to about c=0.25 are inside the set and all others are outside the set.

Any ideas?
For the real z[n+1]=z[n]*z[n]+c, if you plot the C values from -2 to 0 int the X axis and the the iteration values in the Y axis, you will get a bifurcation diagram: http://en.wikipedia.org/wiki/Bifurcation_diagram
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« Reply #4 on: November 25, 2009, 12:13:45 PM »

I've thought about a 1D Mandelbrot set too. It was in desperation trying to find the 3D version. I thought if I can't succeed go up, what happens when you go down.

The problem seems to be that you can't properly rotate using only 1 dimension. in 2D you can, and in 3D, you can rotate in any of the 3 planes. But 1D lacks this ability.

Interesting concept though - we'd expect to see a dot-dash kind of pattern appear, but with lots of patterns of dots and dashes etc.
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« Reply #5 on: November 25, 2009, 01:15:25 PM »

>> we'd expect to see a dot-dash kind of pattern appear, but with lots of patterns of dots and dashes etc.
I would expect to see a line, which is what you see if you use 1d numbers (the reals). Mandelbrot is (almost certainly) connected in 2d, so a connected version in 1d could only be a single line segment.
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« Reply #6 on: November 25, 2009, 01:32:24 PM »

Wouldn't the line have some on/off structure to at least give the appearence of dots and dashes (though obviously some very short/long dots/dashes sometimes).
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« Reply #7 on: November 25, 2009, 01:57:41 PM »

the reason why it gives a line on the x axis is because it is connected i believe ....  alien
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mrrgu
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« Reply #8 on: January 06, 2010, 12:59:30 PM »

With power 1 it is not a fractal anymore...no folding or stretching.
You can go close to 1 though and less than 1.

Less than 1 inverts the fractal..what usually is the inside guts is outside and vice versa..
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« Reply #9 on: January 06, 2010, 07:17:48 PM »

mrrgu:
I think the question is not for power one but for dimension one which already was correctly answered by Aexion but nobody seemed to pay attention lol.

However I wonder what happens if you represent the Mset in Bifurcation-style... Maybe you get a somewhat-3D sheet-furcation-thing smiley
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mrrgu
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« Reply #10 on: January 06, 2010, 08:52:44 PM »

I just looke at the formula and it had power one... but you are right though  wink

mrrgu:
I think the question is not for power one but for dimension one which already was correctly answered by Aexion but nobody seemed to pay attention lol.

However I wonder what happens if you represent the Mset in Bifurcation-style... Maybe you get a somewhat-3D sheet-furcation-thing smiley
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« Reply #11 on: January 06, 2010, 10:46:07 PM »

However I wonder what happens if you represent the Mset in Bifurcation-style... Maybe you get a somewhat-3D sheet-furcation-thing smiley

This is the logistic formula, not the Mandelbrot formula, but Mandelbrot would probably look somewhat similar:
   
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« Reply #12 on: January 06, 2010, 11:46:55 PM »

@I think the question is not for power one but for dimension one which already was correctly answered by Aexion but nobody seemed to pay attention lol.
Actually, I don't see how the logistic map is the same as a 1d mandelbrot, since r*x*(1-x) isn't  x^2 + c

But a 1d mandelbrot is surely simply the set of points at i=0 on the mandelbrot. i.e. a line segment.
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« Reply #13 on: January 07, 2010, 10:40:26 PM »

it's not exactly the same but it's of the same kind.

you could do those bifurcation map for the Mset aswell.

If you coordinate-transform the logistic bifucration, you can get it to exactly match the Mset and its special positions like the biggest Minibrot.
Afaik, that Minibrot is the first bigger lake of stability after the beginning chaos...

Basically do the Mset recursion with x▓+a and do the same as you'd do for the logistic map with a is what in the logistic map would be r.
The results are equal.

As you see, the picture of the "complex logistic map" above, it features a lot of mandelbrot-ish bulbs smiley
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« Reply #14 on: January 08, 2010, 02:48:11 AM »

(I'm pretty sure the OP meant to write the formula with a second power - he even said it went from -2 to 0.25)

All these things such as the bifurcation diagrams produce 2D (or 3D) objects. I think the idea of using a strange attractor sounds cool, but not very mathematically "pure". Maybe taking the segment from (0,i) to (0,-1)? In other words, z[n+1]=z[n]*z[n] + i*c? This is reasonable, and can produce interesting segments. I'm sure that in places where it intersects (for instance) a spiral, you might get something interesting; increasingly rapid dashes, maybe something like   --------    ----  -- - - -- ---- -------- ? If you just want one curve (in 2D), maybe something like plotting y = the greatest imaginary component so that a number with the x-coordinate as its real component belongs to the Mandelbrot set. Even more interesting, the absolute value of the difference between two nearby pixels. This would produce something akin to a pin plot, with high pins where represent places where a branch extends over another.
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