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Author Topic: मण्डलबेथ (maṇḍalabeth) 3D analog of the mandelbrot set  (Read 6750 times)
Description: new 3D fractal with various symmetry groups
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bethchen
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« on: December 13, 2010, 02:34:53 PM »

नमस्ते! (i bow to you!)
please feel free to check out the मण्डलबेथ (maṇḍalabeth), new 3D fractal with various symmetry groups.
http://www-personal.umich.edu/~bethchen/mandalabeth/

actually, it's a 3D analog of the mandelbrot set!
the standard 2D generating function (over the standard unit circle) gives you the usual 2D mandelbrot set.
the 3D generating function (over a bouquet of rings) is the sum of 2D generating functions (over each ring),
where the function is conjugated by the isometry (between the ring & standard unit circle) & its inverse.
if the power–1 is a multiple of the natural symmetry of the bouquet of rings,
then the 3D fractal will have the same symmetry group as the bouquet of rings.

i invite you to explore these new worlds, and have fun drawing lots of pretty pictures!
bethchen@umich.edu
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marius
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« Reply #1 on: January 10, 2011, 06:54:18 AM »

नमस्ते! (i bow to you!)
please feel free to check out the मण्डलबेथ (maṇḍalabeth), new 3D fractal with various symmetry groups.
http://www-personal.umich.edu/~bethchen/mandalabeth/

actually, it's a 3D analog of the mandelbrot set!
the standard 2D generating function (over the standard unit circle) gives you the usual 2D mandelbrot set.
the 3D generating function (over a bouquet of rings) is the sum of 2D generating functions (over each ring),
where the function is conjugated by the isometry (between the ring & standard unit circle) & its inverse.
if the power–1 is a multiple of the natural symmetry of the bouquet of rings,
then the 3D fractal will have the same symmetry group as the bouquet of rings.

i invite you to explore these new worlds, and have fun drawing lots of pretty pictures!
bethchen@umich.edu

Interesting work, deserves more attention from this forum. huh?

Here's a little animation of one of your formulae, rough & uncolored but it should get the idea across.
<a href="http://www.youtube.com/v/sPxgboG6z5k&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/sPxgboG6z5k&rel=1&fs=1&hd=1</a>

* mandalabeth-1522.zip (5.93 KB - downloaded 168 times.)
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knighty
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« Reply #2 on: January 17, 2011, 09:12:52 PM »

Quote
Interesting work, deserves more attention from this forum.
Indeed! grin
Looks like the problem is "language barrier": Not easy for everyone (incuding me of course) to understand mathematical language and formalism. grin. Personally; it took me a whiiiiile to understand and the reference to borromean rings didn't help because there are no ones in the formula, just planes arranged in symmetric fashion. Nevertheless, it's a beautiful concept. the formula gives interresting overall shapes but it seems that it stretches more and more as one zooms in.
Some silly suggestions (would try them if enough time*will  cry):
- Rotating the "rings" on their planes.
- Different exponent for each "ring".
- Average instead of the sum.
- Some kind of product instead of the sum.
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visual.bermarte
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« Reply #3 on: January 18, 2011, 08:11:00 PM »

test made with Fragmentarium  smiley


« Last Edit: January 18, 2011, 09:06:16 PM by visual » Logged
knighty
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« Reply #4 on: January 20, 2011, 08:57:46 PM »

Quote
test made with Fragmentarium
Syntopia have released the binaries for windows. It's awesome. dancing chilli
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marius
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« Reply #5 on: January 20, 2011, 10:55:05 PM »

Quote
test made with Fragmentarium
Syntopia have released the binaries for windows. It's awesome. dancing chilli

Fragmentarium looks pretty interesting. Kinda the setup i have been pushing boxplorer towards, except more organized  grin
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jehovajah
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« Reply #6 on: January 26, 2011, 11:15:39 AM »

These are definitely more appealing. I have contacted Betchen so i hope she stops by and shares some more of her insight.

Did you guys catch the formulas in the appendix? Seriously good for exploration! tease
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
bethchen
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« Reply #7 on: February 01, 2011, 07:37:03 PM »

did someone call my name? my ears are burning!

here is a very beautiful picture of M1034, made by terry gintz (http://mysticfractal.com/)
we expect to release "μυστική मण्डलबेथ" (mystic maṇḍalabeth) 3D very soon! and also the 4D version later

if you would like to include the मण्डलबेथ (maṇḍalabeth) 3D & 4D fractals in your fractal program,
please email me, and ill be happy to send you the source code!

bethchen@umich.edu


* M1034.terry.gintz.jpg (153.68 KB, 800x600 - viewed 389 times.)
« Last Edit: May 03, 2014, 12:03:30 AM by jehovajah » Logged
Tglad
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« Reply #8 on: February 02, 2011, 02:12:13 AM »

"actually, it's a 3D analog of the mandelbrot set!"

So what properties of the mandelbrot set does it preserve in 3d? It must preserve some properties if it is a 3d analog.

- universal (not affected by applying transforms inside each iteration)?
- conformal (stretchless)?
- box counting dimension = hausdorff dimension = correlation dimension = topological dimension?
- fully connected?
- contains mini versions of itself?
- unique, simple formula?

Exploring is fun, but searching for an object that provably doesn't exist will probably get boring after a while. Just sayin'.
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jehovajah
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« Reply #9 on: February 02, 2011, 01:05:02 PM »

did someone call my name? my ears are burning!

here is a very beautiful picture of M1034, made by terry gintz (http://mysticfractal.com/)
we expect to release "μυστική मण्डलबेथ" (mystic maṇḍalabeth) 3D very soon! and also the 4D version later

if you would like to include the मण्डलबेथ (maṇḍalabeth) 3D & 4D fractals in your fractal program,
please email me, and ill be happy to send you the source code!

bethchen@umich.edu

Hiya Betchen. Welcome to the forum!

If you got time can you explain your degree 2 formula a bit?

Nice render, and very suggestive. Terry's work is always quick, intuitive and right! Hope, Jesse Or Bib can get into theses to see if there is any interesting places down there!
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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« Reply #10 on: February 02, 2011, 01:33:01 PM »

Quote
Interesting work, deserves more attention from this forum.
Indeed! grin
Looks like the problem is "language barrier": Not easy for everyone (incuding me of course) to understand mathematical language and formalism. grin. Personally; it took me a whiiiiile to understand and the reference to borromean rings didn't help because there are no ones in the formula, just planes arranged in symmetric fashion. Nevertheless, it's a beautiful concept. the formula gives interresting overall shapes but it seems that it stretches more and more as one zooms in.
Some silly suggestions (would try them if enough time*will  cry):
- Rotating the "rings" on their planes.
- Different exponent for each "ring".
- Average instead of the sum.
- Some kind of product instead of the sum.

Precisely right, Knightly. which is why it is important to take the opportunity to ask questions of the author.

One things for sure this is not the mandelbrot as we know it, because it is actually based on the nature of the underlying complex magnitudes directly. We use √-1 as a square root,giving lip service in a sense to it being a Fourth root . Its underlying nature is therefore a fourth root of unity on the spherical surface which produces bands or contiguous circles or as we call it the lathed mandelbrot.

We actually get the mandelbulb by symmetry breaking! This is a very deep concept in modern physics of the black hole etc.

So your "wacky" suggestions are on the money.
« Last Edit: February 03, 2011, 07:23:47 AM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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« Reply #11 on: February 02, 2011, 01:42:55 PM »

One other thing i think may be of interest is the hyperbolic plane.

I have only recently learned about it , but wonder if all the "stretchiness" we are seeing is due to it being revealed just below the spherical surface?

If it is then we will have to switch to hyperbolic sinh and cosh to reveal its wonders!
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
bethchen
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« Reply #12 on: February 02, 2011, 04:48:52 PM »

when you take 2D cross sectional slices of the maṇḍalabeth fractals, you get fractals that resemble 2D mandelbrot sets

compare the following:
maṇḍalabeth M114,M112,M116 = 2D mandelbrot sets with 4,2,6 fold symmetry
maṇḍalabeth M322, 2D slices normal to 4,2,3 fold symmetry axes
maṇḍalabeth M432, 2D slices normal to 4,2,3 fold symmetry axes
maṇḍalabeth M622, 2D slices normal to 4,2,3 fold symmetry axes


* mandalabeth.M114.M112.M116.png (78.59 KB, 480x720 - viewed 376 times.)

* mandalabeth.slice.M322.png (71.29 KB, 480x720 - viewed 361 times.)

* mandalabeth.slice.M432.png (63.47 KB, 480x720 - viewed 383 times.)

* mandalabeth.slice.M622.png (74.1 KB, 480x720 - viewed 390 times.)
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bethchen
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« Reply #13 on: February 02, 2011, 04:52:58 PM »

compare the following:
maṇḍalabeth M112,M11A,M116 = 2D mandelbrot sets with 2,10,6 fold symmetry
maṇḍalabeth M652, 2D slices normal to 2,5,3 fold symmetry axes
maṇḍalabeth M1032, 2D slices normal to 2,5,3 fold symmetry axes
maṇḍalabeth M1522, 2D slices normal to 2,5,3 fold symmetry axes


* mandalabeth.M112.M11A.M116.png (74.08 KB, 480x720 - viewed 392 times.)

* mandalabeth.slice.M652.png (54.23 KB, 480x720 - viewed 373 times.)

* mandalabeth.slice.M1032.png (57.64 KB, 480x720 - viewed 409 times.)

* mandalabeth.slice.M1522.png (97.38 KB, 480x720 - viewed 402 times.)
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jehovajah
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« Reply #14 on: February 03, 2011, 07:34:00 AM »

The last 4 images show the distribution of the hyperbolic plane(s) around the interior of the sphere ! very nice!

I think the uniqueness of the mandelbrot it is what is being shattered here! We are so used to the image in 2d that we actually do not realise that it is not a unique set, but a foundarional one which appears in all other convex symmetries of the circle and spherical surface.
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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