Welcome to Fractal Forums

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. :hmh:

Here's a little animation of one of your formulae, rough & uncolored but it should get the idea across.
http://www.youtube.com/watch?v=sPxgboG6z5k

Indeed! ;D
Looks like the problem is "language barrier": Not easy for everyone (incuding me of course) to understand mathematical language and formalism. ;D. 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  :'():
- Rotating the "rings" on their planes.
- Different exponent for each "ring".
- Average instead of the sum.
- Some kind of product instead of the sum.

test made with Fragmentarium  :)
(http://fc03.deviantart.net/fs70/f/2011/018/7/a/fragmentarium_mandalabeth_by_bermarte-d37hlth.jpg)

(http://fc05.deviantart.net/fs71/f/2011/018/1/c/fragmentarium_mandalabeth_box_by_bermarte-d37hqzm.jpg)

Syntopia have released the binaries (http://syntopia.github.com/Fragmentarium/get.html) for windows. It's awesome. :chilli:

Fragmentarium looks pretty interesting. Kinda the setup i have been pushing boxplorer towards, except more organized  ;D

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:

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

"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'.

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!

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.

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!

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

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

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.

if you view each term as a matrix equation inverse(B)*A*B is very simple & elegant! in fact, it is merely the change of basis formula. on the other hand, if you look at the individual components, then this equation will seem very messy indeed!

so basically, the maṇḍalabeth construction is really quite elegant (from a higher mathematical perspective)... the entire summation (over the symmetry group) is the forest. each matrix term inverse(B)*A*B is a tree. each individual component term is a leaf. so if youre complaining about wanting a "simpler" formula, that means youre focussing too much on the leaves, and missing both the forest & the trees!

there is another construction which is simpler: take the cartesian product of 2 mandelbrot sets.
you get a 2-parameter family of 4D fractals (with m & n fold symmetry) (generated by [m,n])

[x,y,z,a] –> [x0,y0,z0,a0] + [real((x+i*y)^(m+1)),imag((x+i*y)^(m+1)),real((z+i*a)^(n+1)),imag((z+i*a)^(n+1))]

technically these 4D fractals are not 4D मण्डलबेथ maṇḍalabeth fractals, however they are closely related to 4D πολύχωροι (singular πολύχωρος).
the cartesion product of 2 generating rings (with m & n control points) (with m & n fold symmetry) is a torus/duoprism (with m*n control points) (generated by [m,n])

Hiya Betchen.

Although i would easily go out on a date with you :embarrass: becaue you are my kind of gal  ;D i think you have just lost a lot of people with the technicality of your explanation!

If you use a washing machine, or cook with a pot and pan i am sure you can explain what you have just said in those terms, seriously. Much of modern design incoporates forms and surfaces higher mathematics is only just beginning to explore.

I love your analogy about the trees and the leaves, though. ;D Very fractal  :dink:

About the date: i am married so i can't come on my own  :embarrass:  ;D

Of course, i do want to know what poluchoroi are, so put me out of my misery! :love:

Hi Betchen and all!

I have opened Betchen's site and *surprise*! there are no longer explicit formulas in the PDF. (like x'=x^3+y^3, y'=x*y ...) Yes they vanished :-\ :-\ :-\

Is somebody able to find them?

Luca :-X :'(

@ Luca. Sorry,I thought i had downloaded the original pdf but it seems not. Maybe try asking her if she has the original, and if it is not subject to copyright.

@Betchen. Hope you are well and still interested in the topic of symmetries. I have not quite got there yet, but my research into Grassmann may give us something to discuss in that regard.

The real issue is that we want to engage in what we are creating, and to do that we need to understand. Professional mathematicians are notorious for disengaging with the public whom they serve. At all levels in this forum there is a genuine willingness to learn and explore and engage in experimental mathematics for aesthetic reasons. All any one asks for is a good and patient teacher.

Okay no problems, I already solved my problem a long ago. Don't say this around  :laugh:

Found this on Brtchrns site?
http://www-personal.umich.edu/~bethchen/mandalabeth/mandalabeth.3D.source.rtf

Got the figures I saw in the back of one of her PDFs

Could be fun exploring.