Mandelbox article

[JosLeys]

See http://images.math.cnrs.fr/Mandelbox.html, the site of the CNRS is France.
(Centre National de la Recherche Scientifique)

[KRAFTWERK]

Cool!

But not a word about the mandelbulb?

[bib]

Excellent article, merci

@Kraftwerk, of course Jos Leys wrote an article about Mandelbulb : http://images.math.cnrs.fr/Mandelbulb.html

[KRAFTWERK]

Of course.. I think I even remember it now... 
Thanks for the link bib!

[Tglad]

C'est formidable
j'aime tresment votre explanation, mais il y a un part qui est peut etre un peu faux, vous ecrit "Pour un bon dessin du Mandelbox, cette valeur de distance correspond à un nombre d’itérations qui est normalement inférieur à 20, ce qui explique pourquoi les dessins ne souffrent pas de la transformation en poussière"
vraiment, je pense que en 3d il ne jamais pas transform en poussiere, je n'est pas sur mais voici le discussion- http://www.fractalforums.com/3d-fractal-generation/the-mandelbox-is-tenuous/.
Pardon pour mon francais qui est er.. pas asses bien!

OK, that taxed my brain, phew!  great article, really.

[ker2x]

C'est formidable
j'aime tresment votre explanation, mais il y a un part qui est peut etre un peu faux, vous ecrit "Pour un bon dessin du Mandelbox, cette valeur de distance correspond à un nombre d’itérations qui est normalement inférieur à 20, ce qui explique pourquoi les dessins ne souffrent pas de la transformation en poussière"
vraiment, je pense que en 3d il ne jamais pas transform en poussiere, je n'est pas sur mais voici le discussion- http://www.fractalforums.com/3d-fractal-generation/the-mandelbox-is-tenuous/.
Pardon pour mon francais qui est er.. pas asses bien!

OK, that taxed my brain, phew!  great article, really.

Félicitation ! Votre français est compréhensible

J'aime vraiment votre explication, mais il y a une partie qui est inexacte, vous écrivez "Pour un bon dessin du Mandelbox, cette valeur de distance correspond à un nombre d’itérations qui est normalement inférieur à 20, ce qui explique pourquoi les dessins ne souffrent pas de la transformation en poussière".
Vraiment, je pense qu'en 3D il n'y a jamais de transformation en poussière, je ne suis pas sur mais voici la discussion http://www.fractalforums.com/3d-fractal-generation/the-mandelbox-is-tenuous/.
Pardon pour mon français qui ... Mmm... n'est pas assez bon!

I think i got it right

[JosLeys]

To Tom :
I remember the thread on the 'tenuous' Mandelbox very well...I started it!
I still think that at higher iterations it turns to Cantor dust..

A professor from the Paris Polytechnic commented on the article, he said : "Au dela de l’intérêt esthétique très évident, il serait utile de savoir ce qui a conduit Tom Lowe à ce produit de transformations (repliement, inversion « seuillée » et homothétie). Est-ce purement expérimental et un peu par hasard, ou ce processus a-t-il des racines mathématiques (voir physiques) profondes comme c’est le cas en ce qui concerne le Z^2+C de Benoit Mandelbrot ?

In English (for the sake of the other readers, as I now know your French is ok!) : 'Over and above the aesthetic interest, it would be useful to know what drove Tom Lowe to this sequence of transformations. Is it purely experimental, and more or less by accident, or does this process have deep mathematical (or physics) roots regarding Mandelbrot's Z^2+C ?

Over to you Tom..

(BTW: my native language is Dutch!)

[knighty]

Cool!

[Tglad]

"I remember the thread on the 'tenuous' Mandelbox very well...I started it!"
Oh, ha ha, I forgot who it was Well the thread was very interesting, I think that it is true that the disappearing into dust act happens on menger carpets, sierpinski triangles and probably any 2d fractal with dimension <2. It is a problem of how to render. A menger carpet has dimension about 1.9, if you try to render it like you render a filled area, i.e. blacken each pixel in proportion to the percent of the pixel that is in the set, then the fractal will disappear at higher iterations/accuracy. If you try to render it like you render a line, i.e. blacken the pixel if it is within half a pixel width of a point in the set, then the fractal won't disappear but it would overemphasise the set. Maybe we need some rendering method inbetween that designed for areas and that designed for lines! Never-the-less, I think it isn't so much of a problem in 3d, I know my renders often used 400 iterations.

To the professor: this Mandelbox shape isn't a fundamental object like a dodecahedron or a mandelbrot, but it isn't accidental either. The box fold is mimicking three properties of the mandelbrot's Z^2 transformation, it multi-covers the space (n points map to 1 point), it doesn't introduce stretch and it is continuous. In 3d it is proven that the only transformations that don't stretch are the mobius transforms (+ reflection), so the only possible way to multi-cover the space while remaining continuous is to fold the space. There are an infinity of ways to fold the space but the box fold seems to be the only fold that preserves the symmetry of the object you are folding around. The scale by 2 allows some points to escape, but 2 is quite arbitrary. Lastly, the 'ball fold' was added because (apart from rotation) it adds the final mobius transform into the mix, and just looks more interesting than without it
I won't even try to translate that all into Dutch

ker2x, thanks for fixing my French that is actually helpful. Would love to practice more, maybe I should move to France.