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Hi -
I am an artist not a mathemetician
but I have a particular interest in higher dimensional geometry.

It seems to me that a lot of the morphing in some of the
incredible 3D fractal fly throughs being created
is actually revealing the structure of the fractal in other dimesions.

For instance - a Mandelbox may have a particular parameter where values
less than 1 or greater than 2 give no form at all
but movement between the values of 1 and 2 reveals an incredible variety of different structures -
in which case we could say that this parameter is a dimension in which the fractal extends 1 unit
and each value for this parameter represents a different cross section of this particular dimension.

Therefore if a fractal is defined by 20 different parameters
it can be conceptualised as having extension in  20 different algorithmic dimensions
which are displayed by our software as variations in the fractals 3 dimensional form
as one changes the value of the parameter.

When a video maker tweaks the parameters to produce morphing
it is the equivalent of moving the higher dimensional shape in a different direction
through our 3D space.

I would love to know if this idea has already been extensively discussed
and whether it is commonly conceptualised this way
or even if the idea has been dismissed as misleading or irrelevant ?

To me it seems that the ultimate extension of this idea is that all 3D fractals
are cross sections of the one infinite dimensional shape that contains all possible fractal forms.
The more varieties of Mandelboxes and Bulbs etc that we see
the more we get the impression of what this section of the Uberfractal structure is like.

Please comment and illuminate this poor finite mind that
is trying to understand these incomprehensible things  ;D

...or are all fractals the inifinte parameterization of the same 1 dimensional fractal? The god fractal!
I agree with your premise that varying a parameter is like adding a dimension. Beyond that I can only add my own confusion.

Imo, the answer is yes for the question 1 (there are many dimensions)... one formula i wrote proves it. also you can do it replacing in a formula three parameters with C components (you have to rewrite the formula to do this)

for q2... in theory yes but in practice no, you are flying too high  ;)

Ah, the inverse example.. the Julibrot fractal is a generalization of Julia and Mandelbrot 2D fractals in 4D space :)

Thanks for your replies.

The more I watch the fly through videos of morphing Mandelbox and Mandelbulbs etc
the more intuitively obvious it is to me that we are observing the 3D cross sections
of multidimensional structures rotating and moving through 3D space.

Compare moving through the human body with a EMR machine
and the way bone structures with suddenly appear apparently out of nowhere
as we move from one level of depth to another.

I am still surprised that there is not more interest or discussion
of the multidimensional nature of these algorithmic forms.

@eiffie I agree. As I understand it, all possible states of all universes can be fully defined as single points within an infinite dimensional hilbert space. (Im sure thats not rigorously expressed but I hope you get the drift). Likewise, all possible fractals are points within a infinite dimensional algorithmic space.

I have been thinking about this a lot, and also discussed it here before. One (bad) example: http://www.fractalforums.com/3d-fractal-generation/folding-boxes-in-higher-dimentions/msg25574/
Can't find any more examples at the moment, but check out the philosophy corner here: http://www.fractalforums.com/philosophy/

The original mandelbulb is only 3D, but the mandelbox is actually folded through the fourth dimension and I have always imagined it "taking something with it back" from there...  O0

But to "take it down to earth a bit", it is only the mathematic fourth dimension, it is not real world objects... but I like the concept.  :beer:

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