Welcome to Fractal Forums

fractals that are 3d 4d 5d 6d and more is there a stopping point to them or dues it go on and so on


is there a true fractal stopping point ??

are fractals just the starting point of it all ??

I would like to hear what you think ?


this has ben making me nuts trying to get a grip of a stopping point for fractals ??

There are only 4 normed division algebras (https://en.wikipedia.org/wiki/Composition_algebra) of the 8 fundamental loop spaces (https://en.wikipedia.org/wiki/Bott_periodicity_theorem), which are used to define fractal iterations. Only a smaller subset of those, the complex numbers, are conformal.

What lycium states is the full truth, although encoded in mathematical terminology.

In plain(er) English, it might sound like this:
Most of the fractals we know were discovered/designed not as sculpted structures, but as mathematical formulas. The meaning of these formulas depends on the idea of what a number is, how numbers behave. The ultimate imagery, the shapes, the structure of a fractal then emerges from the interaction between numbers within the formula.

Now here come the constraints:
For "numbers" to "work" for a given formula, they need to have certain qualities. For example, you probably want addition and multiplication to be clearly defined and free of contradictions; also + and * should interact in specific ways. In that sense, numbers from spaces of different dimensionality, can be very different beasts. A perhaps surprising fact of the matter is that numbers cannot have all desired qualities for any arbitrary dimensionality. In other words, there is only a small list of high dimensional spaces where numbers behave the way that is required for interesting fractals to emerge.

It gets worse:
Since people began studying fractals, we found out that there are more constraints, geometrical in nature, that have a big influence on how varied and how detailed fractals can be in a given space of numbers.

Concluding:
Fractals do exist in every dimensionality, but they might be limited to being "simple" fractals, strictly self-similar, unvarying, uninteresting. Only in a select few spaces we know fractals that are much more varied in structure, approaching the richness of the Mandelbrot set.

*bump*

That's sad, I would have liked to see a real 3D or 4D equivalent of the Mandelbrot set.

Still, I wonder how 4D and higher-dimensional fractals could look like.

4D: The development of e.g. a fractal tree in time.

5D: The development of said tree under different circumstances. Different weather (to stay with the metaphor), different events shaping the tree. (Of course, this may mean way more than one additional dimension.)

6D: The development if the laws of science were different?