3d Fractals - AGAIN

[CuriousCanuck]

HA.. the moderator and I just posted at the same time so my question just snuck in before the post was closed... ANYWAY..
Here it is again:

I was reading on a physics site that a 3 dimensional fractal has a Zero volume. How can this be? And while we are on the subject, what is meant when it is said that a 3D fractal really is not 3d, it is more like 2.5d.

Thanks :)

[ultrafilter]

The notion of dimension is based on the idea of self-similarity. Suppose you've got a square. Cut the sides in half, and you get a square one-quarter the size of the original. With a cube, you'd get a cube one-eighth the size of the original. The equation that defines the dimension of a shape is 2x = n, where n is the number of smaller objects you get when you cut the sides in half.

Consider the Sierpinski sieve (http://mathworld.wolfram.com/SierpinskiSieve.html). Cut the sides in half, and you get an object one-third the size of the original. So its dimension is the number x which satisfies 2x = 3. x is a little over 1.5. Same idea for fractals of higher dimension--the sieve is just one of the easiest to understand.

As for the zero volume, fractals are the limit of a sequence of shapes. Each shape has a surface area and a volume, and the surface area and volume of the limiting shape is the limit of the respective sequences of surface area and perimeter. For most of the ones I've seen, the surface area diverges and the volume goes to zero.

Do some reading at the site I linked to. It explains things very well.

[CuriousCanuck]

Thanks for the link.. alot of informative "stuff" there. Now explain to me what a Klein bottle shaped fractal might look like, including volume and dimension ;)

(kidding)

(though if you can do it... I would be impressed)

(then again, I do imagine the math involved would be very similar)

(ok... enough parentheses)

[ultrafilter]

The Klein bottle is a 4D object, so I'm not going to get very far on that.

[Urban Ranger]

Originally posted by ultrafilter
As for the zero volume, fractals are the limit of a sequence of shapes. Each shape has a surface area and a volume, and the surface area and volume of the limiting shape is the limit of the respective sequences of surface area and perimeter. For most of the ones I've seen, the surface area diverges and the volume goes to zero.

That doesn't work that way. If the surface area "diverages" (I take that to mean it goes to infinity) volume isn't zero. It can either be finite or infinite.

Take an equilateral triangle. Divide each side into three equal sections. Erase the middle section on each side. Construct an equilateral triangle (without the bottom line) in each. Now you get a snow flake of sorts.

Keep repeating the procedure. You'll end up with a figure with a finite area but an infinite perimeter.

[CuriousCanuck]

But theoretically it should be possible to calculate a 4D fractal's properties shouldn't it?

[ultrafilter]

Originally posted by Urban Ranger


That doesn't work that way. If the surface area "diverages" (I take that to mean it goes to infinity) volume isn't zero. It can either be finite or infinite.

Volume can damn well be zero with a divergent surface area. In the Sierpinski sieve, the perimeter is infinite, but the area is zero. A similar thing happens with some fractals of dimension greater than two.

Take an equilateral triangle. Divide each side into three equal sections. Erase the middle section on each side. Construct an equilateral triangle (without the bottom line) in each. Now you get a snow flake of sorts.

Keep repeating the procedure. You'll end up with a figure with a finite area but an infinite perimeter.

Yep. And that example proves nothing about fractals in general.

[ultrafilter]

Originally posted by CuriousCanuck
But theoretically it should be possible to calculate a 4D fractal's properties shouldn't it?

Yes, but it's a bit hard to describe the shape. :)

Seriously, to really accurately describe the properties of higher-dimensional shapes, you need to deal with some pretty advanced math. I don't know a lot of it, and what little I do know is not helpful enough to make it worth typing. There are some other people who can explain it, though; let's hope they show up.

[ultrafilter]

Actually, Urban Ranger, I shouldn't be so snarky. You are correct that some fractals can have finite non-zero area/volume/etc., with infinite perimeter/surface area/etc., which was not entirely clear from my first post. But that doesn't mean that no fractals have that zero-infinity situation I described.

[Achernar]

I've never heard of a 3-D fractal with an infinite volume. Is there some trick to it, or does it just go off to infinity in some direction?

[ultrafilter]

Infinite surface area, not volume. That would be tricky unless it couldn't be bounded by any subset of R3.

[Achernar]

That's exactly what I said. Either there's a trick, or it's unbounded, right? Is there such a trick? Anyway, I was referring to part of Urban Ranger's statement:

"If the surface area "diverages" (I take that to mean it goes to infinity) volume isn't zero. It can either be finite or infinite."

[ultrafilter]

In order for the volume of a shape to be infinite, the shape must be larger than any subset of R3 with finite volume. No trick.

[Roger43]

If a 3-D fractal had infinite volume wouldn't it contain the Universe? And wouldn't heaven and hell be in there too? :)

[Chronos]

If a 3-D fractal had infinite volume wouldn't it contain the Universe? And wouldn't heaven and hell be in there too?Not necessarily. Even an ordinary shape can have infinite volume without containing the entire Universe. Picture, for instance, a really long pipe. Infinitely long, in fact. But this pipe only has a one inch diameter. What's the volume? Infinite. But it's still not much, compared to the entire Universe.

[Roger43]

I can see the infinite volume concept. But, is a pipe a 3-D fractal when only one of the three dimensions is allowed to become unbounded?

Just now, as I slice my Swiss cheese, I think the answer may have come to me. If I extended the cheese's dimensions infinitely in three orthogonal directions, reproducing its holes, I would have an infinite volume of cheese, and the cheese would have an infinite volume of holes.

Now, if I were to increase the hole-dimensions linearly with cheese block size, eventually one hole would be large enough to contain the universe, as we know it, which is finite.

Of course the universe, most likely, infinite, is a sub-particle of a neutrino, which, in turn, is a sub-particle of a neutrino, which, in turn, is ... ad infinitum.

And where does the God Particle (if it exists) fit into this?

[Monty]

Arent' fractals those geometric objects whose dimensions are the fractions between whole numbers? Doesn't that rule out the Klein Bottle (a 4D object) as a fractal?

[ultrafilter]

Originally posted by Monty
Arent' fractals those geometric objects whose dimensions are the fractions between whole numbers? Doesn't that rule out the Klein Bottle (a 4D object) as a fractal?

Strictly speaking, sure. But you could cut out a similar shape from the middle of the Klein bottle, etc., etc., etc., and get a fractal that resembles the Klein bottle.

[Achernar]

I always thought of manifolds like the Klein Bottle or the circle as fractals, although dismally uninteresting ones. Is there some minimum structure a fractal has to have to be a fractal? Doe the dimension have to be non-integral?

[ultrafilter]

Actually, [b]Achernar[/b may be right. Fractals may only need self-similarity. However, in common parlance, "fractal" is always used to refer to a shape of non-integral dimension.

[Achernar]

Also, this has nothing to do with fractals, but I think the Klein bottle is technically considered to be a 2-D object. It's a 2-D manifold anyway. What people mean when they say that it's a 4-D object is that to get a good idea of its structure, you can't embed it in three dimensions. This is analagous to not being able to embed a 1-D knot in two dimensions. The next-highest space you could embed it in, then, would be 4-D space.

However, I'm wondering now if it would be possible to embed it 3.5-D space. Hmmm....

[Jabba]

A fractal (http://www.weihenstephan.de/ane/dimensions/dimensions.html) is a shape whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension.

[Jabba]

The link I gave doesn't work now. It did when I checked it the first time.
This site (http://www.math.okstate.edu/mathdept/dynamics/lecnotes/lecnotes.html) gives some information.
So does this one (http://www.math.umn.edu/~swenson/augie_website/chaos/toc.htm).

[Achernar]

From a page in your first link (http://www.weihenstephan.de/ane/dimensions/subsection3_3_5.html), Jabba:Mandelbrot offers the following definition of a fractal: A fractal is a set for which the Hausdorff-Besicovitch dimension exceeds its topological dimension. This definition, although correct and precise, is too restrictive, since it excludes many fractals that are useful in physics.

An alternative definition is given by Mandelbrot: A fractal is a shape made of parts similar to the whole. This definition uses the concept of self-similarity. A set is called strictly self-similar if it can be broken into arbitrary small pieces, each of which is a small replica of the entire set.Of course I was referring to Mandelbrot's second definition. :D