what is fractal dimension of a circle ?!

[Syntopia]

ok. but why does a math professor (Kenneth Falconer) claim, that a fractal's hausdorff dimension "strictly exceeds" its topological dimension, while the menger-sponge (and others) contradicts this claim. I doubt that this is a simple mistake - it is too obvious.

Or is my assumption for the Menger-sponge wrong?
topological dim=3 > 2.727=hausdorff dim

I don't think Falconer is properly quoted. Looking at the book that Wikipedia references ("Fractal Geometry: Mathematical Foundations and Applications" - try Google), he says in the introduction:

"Usually, the ‘fractal dimension’ of F (defined in some way) is greater than its topological dimension."

And

"In his original essay, Mandelbrot defined a fractal to be a set with Hausdorff
dimension strictly greater than its topological dimension. (The topological
dimension of a set is always an integer and is 0 if it is totally disconnected, 1 if
each point has arbitrarily small neighbourhoods with boundary of dimension 0,
and so on.) This definition proved to be unsatisfactory in that it excluded a number
of sets that clearly ought to be regarded as fractals."

Nowhere is the phrase "strictly exceeds" used.

[taurus]

Thanks a lot Syntopia for investigating. 
You are very cautious with your rating for that phrase. Wikipedia has taken Falconer's words completely out of context, and turned around the sense.
Seems to be the old Wikipedia problem... good for a quick glance, but don't take too serious.

[kram1032]

The topological dimension of the menger-sponge is 2. 2.7 is strictly greater than 2.

The topological dimension deals with surface, not volume. It's the numbers of parameters you need to fully describe a given surface.
The involved functions in case of something like the Menger sponge would be pretty crazy, jumping around like mad, but technically you only need 2 variables to describe its entire surface. - Simply because it's a surface.

Similarly, a sphere (the surface of a ball) has a topological dimension of 2, namely, where r is a constant:

r cos phi sin theta
r sin phi sin theta
r cos theta

(top of my head, maybe the sin theta and cos theta should be exactly opposite)

Since r is fixed, phi and theta fully describe every point of the sphere. You can't get away with less variables than those two. (Unless you somehow obscurely use something like a hilbert-curve to parametrize it with only one variable, but I'm really not sure wether that's possible...)

[Syntopia]

The topological dimension of the menger-sponge is 2. 2.7 is strictly greater than 2.

The topological dimension deals with surface, not volume. It's the numbers of parameters you need to fully describe a given surface.
The involved functions in case of something like the Menger sponge would be pretty crazy, jumping around like mad, but technically you only need 2 variables to describe its entire surface. - Simply because it's a surface.

The topological dimension of the Menger sponge is actually 1. It is more like a curve than a surface:
http://en.wikipedia.org/wiki/Menger_sponge

[taurus]

Still investigating, but as I see, it seems to be easy to claim this and that and finding reliable sources is next to impossible for a layman.

@kram1032 you seem to describe the euklidean dimension, not the topological http://www.mathe-seiten.de/fraktale.pdf and i found no signs, that two parameters are really enough.
@syntopia 1 seems the solution (ok wikipedia says that) as Menger showed that the Lebesgue covering dimension of the sponge is equal to the related curve. But I still need to find out how the Lebesgue covering dimension relates to topology (just for the interrest).

Regardless of all that Wikipedia didn't quote Falconer correctly.

[Syntopia]

But I still need to find out how the Lebesgue covering dimension relates to topology (just for the interrest).

Regardless of all that Wikipedia didn't quote Falconer correctly.

As I understand it, the Lebesgue covering dimension *is* the topological dimension - at least according to Wikipedia :-) http://en.wikipedia.org/wiki/Lebesgue_covering_dimension). It is tricky stuff - I really didn't expect the topological dimension of the Menger to be 1 - I expected 3.

Now, where are the examples of fractals, where the Hausdorff dimension is less than the topological dimension? Falconer's quotes suggests that these exists, although they are "unusual".

[taurus]

As I understand it, the Lebesgue covering dimension *is* the topological dimension - at least according to Wikipedia :-) http://en.wikipedia.org/wiki/Lebesgue_covering_dimension). It is tricky stuff - I really didn't expect the topological dimension of the Menger to be 1 - I expected 3.

Indeed tricky stuff...

I mined this from the net:

The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to refer to Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.

Now we only have to decide, whether the Menger sponge is a "sufficiently nice" space. But at this point I reach some limits in understanding english terms and I found no sufficiently nice translation...

A mathematical object is colloquially called nice or sufficiently nice if it satisfies hypotheses or properties, sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informal antonym for pathological. For example, one might conjecture that a differential operator ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interesting topological invariant should be computable "for nice spaces X."

So far I din't find one of the expected fractals with a greater topological than hausdorff dimension, but i didn't try so hard. It might also be possible, that Mandelbrot's initial definition of "fractal" is actually correct...

[kram1032]

Ah, I think, I know why the Mengersponge has topological dimension of 1:
It *probably* has a representation similar to the Sierpinsky Sponge, by means of the 3D-Sierpinsky Arrowhead Curve:


Namely probably some sort of 3D version of the Sierpinsky Curve:

http://en.wikipedia.org/wiki/Sierpi%C5%84ski_curve

Because you can do this, you technically only need one parameter, so that's the topological dimension.

@taurus66:
The one I called "geometric" is the euclidean dimension.
My "topological" dimension is correct. The Menger Sponge still is a "three-dimensional" object in the sense of the euclidean dimension.

What was incorrect of me, was that the topological dimension has to be one less than the euclidean dimension. This is obviously not the case if you think of (topologically) one-dimensional curves that, to not be distorted by projections, may or may not require representations in Rⁿ, n>2, like for instance a helix that cannot exist in R² without distortion.

However, the Hausdorff dimension still is strictly greater than the topological dimension and smaller than the euclidean dimension.

Of course, my definition that you do need n parameters for n-dimensional topology becomes a bit weird at n=0 for a point cloud. However, it still seems accurate, since in that case, you simply can't desribe the set with a parameter. You have to explicitly list the points in some way, or give an algorithm that finds every point. - This algorithm might again use parameters but those are not of the same "kind" as usual geometric parameters.

My definitions are also failing as the euclidean dimension goes towards infinity. But so do most intuitively useful definitions, if not all...

The linked Lebesgue covering dimension is of course more general but for "usual" surface-sets that can also be reasonably ploted in eucildean geometry (and I guess, also in general geometry of varying curvature), my simplified ways of saying it should be equivalent.

I guess, I also exclude the case where you consider the volume, which should require one more parameter (for the before-mentioned sphere , if you do not fix r and vary it along intervals, you obviously obtain the respective ball or, if you go 0<a<=r<=b, a ball with a smaller ball taken out of it, with dimension (e.g. parameter-count) 3: r, phi, theta)
That *is* a gross simplification, but it's not difficult to rectify.
The topological dimension is essentially, for non obscure cases, the minimum number of parameters, you need to fully describe a set. This now holds for the volume case too.

"sufficiently nice" is "hinreichend nett", as in not a bastard topology. IMHO, the Menger Sponge is sufficiently nice, since it can be broken down in the above-mentioned curve that ultimately is made up of line segments.
Though I can not guarantee that for all fractals. Likely, there are some monster-toplogies that are not only monsters but also bastards.

[kram1032]

I just found this:
http://en.wikipedia.org/wiki/Dimensionality#More_dimensions
Can anyone else appreciate the subtly punny naming of that subcategory?