GlytchMeister wrote:Sgt. Howard wrote:... fractals AREN'T three dimensional...?
Ehhh... As best as I understand it:
Well, the classic
Mandelbrot Set isn't really 3-D. It's just a rediculously complicated 2-D image.
Well, strictly speaking, it's a kind of 2.5-D, the added .5th dimension being "roughness"... Sorta. Look, it's spacey-wasey, ok?
A three dimensional fractal would be like a sphere with a bunch of smaller spheres budding off of it, and each one of those has a bunch of smaller spheres, and then
those spheres have
more spheres...
But it's not exactly 3-D, no is it? More like 3.5-D... Or something. Maybe 3.7? Or 3.55? Because it's rough... But along three different axis!
[insert twilight zone theme song here]
Well, yeah, fractals are so named because they have a dimensionality expressed as a fraction. But a plane set doesn't suddenly jump out of 2D, just like a linear set doesn't jump out of 1D. But a curve, now …
For a fractal curve, this dimension might be log(3)/log(2), or if you like, log₂ (3). Not that log₂ (3)D isn't still a mouthful.
But the boundary of the Mandelbrot set, this fraction turns out to be integral, equal to 2. Yup, the circumference of the Mandelbrot set has the same dimensionality as the set itself. That's kinda like saying, if you could draw a Mandelbrot set on a sheet of paper, both its area
and its circumference could be measured in square inches!
There is really nothing special about a plane figure being 2D. Most of them are. But for a circumference to be 2D – that's notable.
