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Author Topic: Variety of the Mandelbrot set  (Read 3335 times)
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twinbee
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« on: November 02, 2007, 02:37:58 PM »

How much variety is in the Mandelbrot fractal? I know this sounds like a vague question, but it's not easy to put such an open question concisely without losing much of its power.

As a few examples:

Does the Mandelbrot contain other fractals as subsets of itself? For example, is it theoretically possible you could find approximations to the Sierpinski carpet or the Pythagoras tree if you zoomed into the Mandelbrot far enough?

If you zoomed in far enough, could you see simple shapes appearing like multiples of triangles or squares? Is this theoretically possible? Could a hypothetical fractal even more beautiful than the Mandelbrot possibly do this as well as contain the amazing detail style of the Mandelbrot too?

Also, do any fascinating fractals have the Mandelbrot as a subset?

Finally, I know I've asked this before elsewhere, but is it theoretically possible that a 3D dimensional version of the Mandelbrot exists (and no, the Mandelbrot mountain or 4D quaternionic 'smoothed' versions don't count smiley ).

Love to hear answers and musings to these thoughts!!
« Last Edit: November 02, 2007, 02:45:59 PM by twinbee » Logged
gandreas
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« Reply #1 on: November 02, 2007, 04:34:09 PM »

How much variety is in the Mandelbrot fractal? I know this sounds like a vague question, but it's not easy to put such an open question concisely without losing much of its power.

As a few examples:

Does the Mandelbrot contain other fractals as subsets of itself? For example, is it theoretically possible you could find approximations to the Sierpinski carpet or the Pythagoras tree if you zoomed into the Mandelbrot far enough?
You'll find things similar to a Julia set in an MSet (and vice-versa), due to the fact that the two can be considered 2D slices through a 4D set (where one 2D "axis" contains the Z0 value, and the other is C).
Quote
If you zoomed in far enough, could you see simple shapes appearing like multiples of triangles or squares? Is this theoretically possible? Could a hypothetical fractal even more beautiful than the Mandelbrot possibly do this as well as contain the amazing detail style of the Mandelbrot too?
Due to several things, you are unlikely to find things involving squares or triangles.  First of all, the whole thing is "self-similar" (or "magnification independent") meaning that it look similar at any magnification - if you suddenly started getting Koch snowflakes, for example, it wouldn't be self-similar.  Secondly, complex math results in doing things like rotating and "smushing" angularly, which causes curves to appear.
Quote
Finally, I know I've asked this before elsewhere, but is it theoretically possible that a 3D dimensional version of the Mandelbrot exists (and no, the Mandelbrot mountain or 4D quaternionic 'smoothed' versions don't count smiley ).

No, the MSet that everybody knows and loves is constructed using complex math (which is 2D).  Complex math is a subset of quaternion math (as well as split-complex math, for what it's worth), but due to weird group theory issues, each "larger" math space has 2^n components (so there can be 2D, 4D, 8D, 16D, though each higher one loose useful properties - 4D quaternions are non-commutative (A x B is not necessarily equal to B x A), 8D octonions are non-associative (A x (B x C) is not necessarily equal to (A x B) x C), and 16D sedenions looses the ability to do division (there are sedenions A and B, neither of them zero, but A x B = 0 = B x A)).

One can formulate an arbitrary 3D set with specific operations for things like multiplication and addition, but they lack the "elegance" that complex, quaternions, octonions and sedenions.


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twinbee
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« Reply #2 on: November 02, 2007, 09:02:28 PM »

Thanks for your response!...

Quote
Due to several things, you are unlikely to find things involving squares or triangles.  First of all, the whole thing is "self-similar" (or "magnification independent") meaning that it look similar at any magnification - if you suddenly started getting Koch snowflakes, for example, it wouldn't be self-similar.  Secondly, complex math results in doing things like rotating and "smushing" angularly, which causes curves to appear.

But it isn't strictly self-similar - only quasi self-similar. Could the 'quasi' bit allow for surprises including approximations to simple shapes? Given the correct conditions, could the curves possibly cancel each other out at certain portions, and zoomed in far enough?

Try this for size at about 150 iterations:
+0.1767, +0.5818 .... to.... +0.1787, +0.5803
It just begins to seem that the area is reminiscent of the classic H-Fractal, itself a type of canopy fractal. Could one get a better approximation still deeper into the Mandelbrot theoretically? With fewer wrong off-shoots, and straighter er... 'lines'?


Quote
No, the MSet that everybody knows and loves is constructed using complex math (which is 2D).
......
One can formulate an arbitrary 3D set with specific operations for things like multiplication and addition, but they lack the "elegance" that complex, quaternions, octonions and sedenions.

Hehe, how did you guess I was trying to create my own 3D complex numbers ;-) Seriously, I know that there isn't a true equivalent to complex numbers in 3 dimensions, but isn't it possible that the beauty and complexity of the Mandelbrot is derived from math which is part of a more general theory? In other words, is it possible one may define special calculations in 3D numbers which are analogous to their complex counterparts, whereby a 3D Mandelbrot somewhat akin to this artists impression would be allowed:

http://www.renderosity.com/mod/gallery/index.php?image_id=1308487&member
« Last Edit: November 03, 2007, 03:37:29 AM by twinbee » Logged
gandreas
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« Reply #3 on: November 06, 2007, 06:31:59 PM »

Just a few thoughts about trying to make the "Mandelbulb" 3D object:

If you imagine such a beast, and do a slice through it, along two axis you should probably get the standard 2D MSet.  So if you're 3D coordinate system is (X,Y,Z) (with X being the "core" - up and down in the sketch you linked), this implies that:
(X,Y,0) ^ 2 ~= (X + i * Y) ^ 2
(X,0,Z) ^ 2 ~= (X + i * Z) ^ 2

i.e., the Y and Z axis are similar to the i component in complex numbers, and that when one is zero, you've got complex multiplication with the other two.

Now if you have a cross section where X = 0, you'll probably end up with something that looks like a rotated (by 45 degrees) version of the z = z ^ 5 + C.  This gives us:
(0,Y,Z) ^ 2 ~= (sqrt(2),sqrt(2)) * (Y + i * Z) ^ 5 ~= (sqrt(2),sqrt(2)) * (Z + i * Y) ^ 5.

If needed, you can negate any or all of the sqrt(2) parts, since rotation by 45 = 135 = 225 = 315
Expand all of those, and hopefully you'll get a series of consistent equations that allows you to solve the general (X,Y,Z)^2.  I'm not positive that such a thing is possible, or that you'll get useful results for the general case (i.e., we've only fixed the shape along 3 perpendicular slices, and the rest could bulge out or smear smooth).


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FractalMonster
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« Reply #4 on: December 21, 2007, 01:36:42 AM »

Check out the very deep zoom sequence from page 4 and forwards in,

http://klippan.seths.se/lists/Diverse/Blomkalsfort.pdf

The end motive is uploaded in my deviant gallery,

http://fractalmonster.deviantart.com/art/Cauliflowerfort-40969732

Regards,
/Ingvar
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Duncan C
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« Reply #5 on: December 24, 2007, 07:49:59 PM »

Check out the very deep zoom sequence from page 4 and forwards in,

http://klippan.seths.se/lists/Diverse/Blomkalsfort.pdf

The end motive is uploaded in my deviant gallery,

http://fractalmonster.deviantart.com/art/Cauliflowerfort-40969732

Regards,
/Ingvar


Ingvar,

What applicaiton did you use to create those zooms? My application can't zoom as far as yours. My images start to fall apart due to floating point errors at page 11. (Plot width of about 4.0E-14. By 1.0E-14, the images are very badly degraded.)

The app also seems to do a pretty good job of creating color tables that show detail at a wide variety of magnifications.


Duncan C
« Last Edit: December 25, 2007, 12:01:37 AM by Duncan C » Logged

Regards,

Duncan C
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« Reply #6 on: December 29, 2007, 01:26:22 AM »

What applicaiton did you use to create those zooms? My application can't zoom as far as yours. My images start to fall apart due to floating point errors at page 11. (Plot width of about 4.0E-14. By 1.0E-14, the images are very badly degraded.)

The app also seems to do a pretty good job of creating color tables that show detail at a wide variety of magnifications.


Duncan C
[/quote]

Hi Duncan,

fractal eXtreme,

http://www.cygnus-software.com/

A very "easy-to-handle"  application with very fast routines.
It's also easy to create zoom-animations smiley However it has
no filter routines wink

Sorry for the late answer. Spend allmost all my cyber time
time at deviantART..

Regards
/Ingvar
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fractalwizz
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« Reply #7 on: August 25, 2008, 12:12:40 AM »

I have found a bit of variety in the M-set
I have figured out where to go to find a julia set(sort of) made out of another julia set
It's still rendering, but I have a picture of what I mean.
animation coming soon
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fractalwizz
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« Reply #8 on: September 13, 2008, 10:14:41 PM »

Here is the picture I rendered.
took about an hour but yeah.



* pictrenderforforum.jpg (130.63 KB, 644x497 - viewed 577 times.)
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Duncan C
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« Reply #9 on: September 13, 2008, 11:11:47 PM »

Here is the picture I rendered.
took about an hour but yeah.



FractalMonster,

Cool image. What are the "specs"? (Plot type, coordinates, magnification)

Duncan C
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Regards,

Duncan C
fractalwizz
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« Reply #10 on: September 15, 2008, 12:08:18 AM »

Duncan C.
Are you talking to me, or FractalMonster?
I just wasn't sure which.
If you were talking to me, the magnification is 2.086e+093
the plot type is mandelbrot set
The software is Fractal xTreme
the coordinates are
Real : -1.810,049,670,432,060,661,108,693,467,312,281,658,259,602,262,637,211,563,550,239,824,916,665,985,881,723,793,306,160,501,257,436
Imaginary :
+0.001,109,840,839,711,142,896,358,434,960,832,824,387,483,153,282,000,241,881,695,036,678,716,360,398,116,254,630,248,731,560
« Last Edit: September 15, 2008, 12:20:39 AM by fractalwizz, Reason: additional info about image » Logged

Duncan C
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« Reply #11 on: September 15, 2008, 12:47:23 AM »

Duncan C.
Are you talking to me, or FractalMonster?
I just wasn't sure which.
If you were talking to me, the magnification is 2.086e+093
the plot type is mandelbrot set
The software is Fractal xTreme
the coordinates are
Real : -1.810,049,670,432,060,661,108,693,467,312,281,658,259,602,262,637,211,563,550,239,824,916,665,985,881,723,793,306,160,501,257,436
Imaginary :
+0.001,109,840,839,711,142,896,358,434,960,832,824,387,483,153,282,000,241,881,695,036,678,716,360,398,116,254,630,248,731,560

fractalwizz,

Sorry, my mistake. I was indeed talking to you. Thanks for the coordinate info. That is a really deep zoom!

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Regards,

Duncan C
fractalwizz
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« Reply #12 on: September 15, 2008, 05:50:22 PM »

I will have a deep zoom video to this and beyond as soon as I get the time. Check back for it. cheesy
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