मण्डलबेथ (maṇḍalabeth) 3D analog of the mandelbrot set

[bethchen]

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!

[bethchen]

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])

[jehovajah]

Hiya Betchen.

Although i would easily go out on a date with you becaue you are my kind of gal  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. Very fractal 

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

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

[DarkBeam]

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

[jehovajah]

@ 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.

[jehovajah]

Some amateur mathematicians don’t like the Man. d. alaBeth formulae because they are
“too complicated”. If you view the summation of conjugates over the entire symmetry
group, then that’s the forest, very simple & elegant. If you view each individual group
element as a vector-matrix conjugate, thats a tree, very simple & elegant. But if you focus
on each individual component of the vector-matrix, then it gets really messy, because you’re
focussing on the leaves, and missing both the forest & the trees!
So the amateur mathematicians & fractal artists at Fractal Forums [sic] are still not satis-
fied that they have found the “real” 3D Mandelbrot set. And thus, the holy grail continues...
[17] Fractal Forums [sic]
http://fractalforums.com/

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.

[DarkBeam]

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

[jehovajah]

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.