Is this art really a fractal?

[David Makin]

My definition of "fractal" is as loose as it can get - *everything* is fractal

[Tglad]

I think the topological dimension of cantor dust is 0, it is made from 0 dimensional points/dust. Its fractal dimension is about 0.63.
The Mandelbrot set is the set of interior points. They fill an area, and just like a disk, if you bring it twice as close it fills 4* as many pixels. Its topological dimension is 2, and its fractal dimension is 2. So the 'set' isn't fractal by Mandelbrot's 1975 definition.
The border of the set is a line, which is 1 dimensional, but if you bring it twice as close it fills 4* the pixels to draw the border, so its fractal dimension is 2, larger than its topological dimension, so a fractal. This is why Wikipedia is careful to say "the boundary of which forms a fractal."

But that's being fussy, everyone knows it as a fractal

[hobold]

Warning, unfounded philosophical babbling ahead!

I regard the Mandelbrot set as an example of a battle between two conflicting forces. One force tries to attract the orbits, another force tries to repel them. The areas held by either force, i.e. the inside and the outside of the set, are fairly boring, ordinary 2 dimensional planar sets. But the front line between them, that's the place where chaos reigns.

Two forces aren't really enough for chaos to happen (that would just be z^2), but some small third force is needed that disturbs the two great armies (the + c part). But that doesn't change the general notion that chaos always happens - if it does happen at all - at some border between conflicting forces.

[Sockratease]

Ah, the age old questions concerning...   What is "art"?  and  What is a "fractal"?      

Well, I can't help with What is a "fractal"?  Too many definitions out there...

But I can babble with the best of them about What is "art"?

There is no criteria which can be used as a blanket statement to define whether anything is "Art" or not.

Art is an Intent and an Act of Creating Something. Not a finished item.

It is a Process, not a material object.

It is a Thing To Do, not a Thing.

It is a Verb, not a Noun!

Sometimes the finished product is not the point at all, but merely the by-product of the Art.

[Timeroot]

Hobold, did you steal that from somewhere within the pages of "The Beauty of Fractals"? 

Socratease, that's a very nice way of describing things. I think the same might be said of cooking, to some extent: Sure, everyone might be enjoying the finished product, but the part where your heart is really in it is while you're stirring through the batter. 

[hobold]

There is certainly a danger that this idea infected me when I read "The Beauty of Fractals" back in my school days. But I have no conscious memory of such a quote in that book. The book did profoundly impress me, though, and was a big part of the reason why I chose to study mathematics, and do that at the University of Bremen. Funnily enough, I then proceeded to never meet Professors Peitgen or Richter during all of my education.

I don't regret the path I have taken, though. And I did get to meet some some of the " fractal celebrities" eventually.

[Timeroot]

Whom have you gotten to meet?

I think that, especially in the sections about Newton fractals and the Volterra-Lotka equation, it talked about how in some sense there are several attractors, each pulling on the point. The Mandelbrot set is somewhat like a plot of all the different ways that the small attractor would win; it's complement is all the ways it would lose. In the Volterra-Lotka dynamics, you have the predator vs. the prey, but there is also some strange force trying to make them battle for eternity, and one trying to make them live in harmony... or something like that. It's a great book.