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i am just wondering, what is the result when box counting a circle ?!?! ???

ok, perhaps i can give the answer myself ...

the circumfence of a circle is simply calculated through the radius and the pi number, hence it is limited it is not fractal ... but the question remains, what dimension has a circle then ?!?! 2 ? or something less than 2 ?!'

A circle has fractal dimension 1, it is a smooth curve
A disk has fractal dimension 2, as it is an area. (A disk is a filled in circle)

Similarly for a sphere, lots of people think a sphere is solid, but it is the name of the surface, so has dimension 2, the solid (sometimes called a ball) has dimension 3.

Basically, for a curved object, you do the same you'd do for an easily overlap-free subdividable flat one (e.g. a square or cube). At first, the potential overlap gives you a lot of error, but as you go to smaller and smaller substeps, the error reduces and in the limit, you get the exact measure.

A circle really isn't a problem in this. Just use the Yardstick-, Boxcount- or Circlecount measure and you'll arrive at accurate representations.
Generally, line-objects don't pose any problem. It does get a bit problematic, however, if you choose curves surfaces, like a sphere. There, you can't find good macroscoping measures as the curvature guarantees some kind of undesired overlap. Luckily, it turns out that the error produced that way is of higher order than the accuracy of the measure at any given scale. So even here, as you go towards infinitely small scales, you'll have an accurate result.

In doing so, you'll see 1 for the circle, 2 for the sphere, 2 for the disk or 3 for the ball. They are all too smooth to give irregular results.

I guess, there is no need to quantify a fractal dimension, 'cause none of those objects is fractal. What Tglad describes, is the topological dimension, which is equal to fractal dimension for euklidean objects. But I can't see the need to name it "fractal dimension", even in fractalforums.

Why not?   :)
Why isn't a line a fractal? it is self similar. Same for an area. That's why a Cesaro fractal can go from a square outline (fractal dimension 1) to a filled box (fractal dimension 2) and anything in between.

OK, a curve isn't self-similar, but it still has an exact box counting dimension.
The way I see it, Euclidean shapes are just special cases of fractal geometry. Just like a square is a special case of a quadrilateral... it doesn't stop it being a quadrilateral. 2 is still a real number, even though it is also an integer, etc.

It is also differentiable, when parameterized with arc length. It is even _possible_ to parameterize a line with arc length, which is generally not true for fractal curves.

Of course, we could debate forever whether smoothness should win over self similarity with regards to a line being called a fractal. :-) It doesn't matter much, though, because the set of all smooth self similar curves probably doesn't contain more than a single element, and is thus rather boring as a set.

:ok:

just a side note: In Germany, we don't know about the distinction between ball and sphere. We have one word for both - Kugel - with a volume (3-dim) and a surface (2-dim). Ball (same word as english) means the physical object, you can throw or kick around... ;D

hey thank you for the discussion, thank you for clarifying the surface and volume/area distinction, i wanted to prepare for questions like, hey, why is a circle not a fractal ? that could occur when i show slices of the quaternion rotated mandelbrot, but the most important thing is that we actually know the circumfence of the circle very well, this can be understood very well by the people, thank you for the discussion!

What about this curve - it is self-similar and smooth. Is it fractal? :-)

If you look at the line getting smaller, the line length scales by perhaps 0.8 each wave, call the scale k, the length of the line is l + kl + kkl + kkkl + ... which is a geometric series that sums to a finite number. So the line has Haussdorff dimension 1. I think the same can be said of a logarithmic spiral.

The whole curve is smooth, but only a single point is self similar. Try to zoom in on another point to see what I mean. In the case of a straight line or a fractal, either all points are self similar, or at least a dense subset of points is self similar.  (Where "dense subset" means that within within any contiguous piece of the curve, that is larger than a single point, there is at least one self similar point.)

A nice "counter"-example nonetheless! It proves that I have to work harder on the clarity of my expression. :)

Be careful with such generic words. A line is a curve, the Hilbert- and Peano-CURVES are curves. The Kochcurve is a curve...
What you meant to say is, that not all curves are self-similar.
Self-similarity is a kind of scale-varying symmetry. Many curves have that, property.
A very simple example would be x sin(1/x).

Note, a line isn't just self-similar, it's completely scalefree.
I'm sure that, if you relax that down to mere self-similarity (e.g. only a finite number of fixed scales where the curve is the same) or even pseudo-self-similarity (self-similarity to the first order, like the MSet has), you'll obtain a lot of smooth curves that fulfill that property.

Also, I'm not sure, don't quote me on that, but the Lorentz-Attractor seems fairly smooth to me.

x sin(1/x) isn't smooth at the 0-point but it is everywhere else.

cKleinhuis, mathematically speaking, it's Kugel and Sphäre for ball and sphere respectively. Or alternatively Hohlkugel and Vollkugel.

Note, however, that fractals do not need to have a fractal Hausdroff-dimension, as clearly given by the before-mentioned Peano- and Hilbert-curves aswell as the MSet, all of which have a Hausdorff-dimension of 2.

Maybe on a pure academic level (I doubt that), but in formula collections, there is no such distinction!

@kram sure, this is the reason why i left the hausdorff dimension of the border outside, but they understood that the coastline of britain and mandelbrots border have something in common and i said as well that opposed to the coastline of england the mandelbrot set is infinite between each part ... at least i hope so ....

couldnt it be said that anything is fractal which has a higher dimension as the topological dimension !?
so, when a line has a hausdorff dimension above 1 it is fractal ?! any counter examples please ;)

Maybe, but there are also fractals with a lower fractal dimension, than the topological one.
Menger sponge: 2.727
Sierpinski carpet (antenna): 1.8928
Sierpinski triangle: 1.585

The Lorentz attractor is made of smooth fiber (or fibers?), but those fibers are distributed fractally in space. You can think of it as an extreme example of the whipped cream effect.

Wiki has a nice page on fractal dimensions: http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

It seems that many fractals with lower fractal dimension than topological are made by removing parts, and many fractals with higher fractal dimension than the topological are made by folding or twisting the objects (or by adding parts).

Then it should be possible to make a fractal with topological dimension equal to the fractal dimension by doing both folding and reductions: for instance by creating a koch surface, but at each step remove parts of each face. 

Interesting: on the Wikipedia-site the Mandelbulb is listed as having Hausdorff dimension 3, which a link to FractalForums for the derivation: http://www.fractalforums.com/theory/hausdorff-dimension-of-the-mandelbulb/15/

I know next to nothing about Hausdorff dimensions, but as I read it, the argumentation is that the Mandelbulb has a cross-section, which is the Power-8 (2D) Mandelbrot boundary (which is proven to have Hausdorf dimension 2). Thus in a neighbourhood around the cross-section we must have a 2+1 dimensions (<- I'm not sure about this last part - it combines reasoning from Euclidean dimension intuition with fractal dimensions).

What bothers me is this:

Look at the Menger: It has fractal dimension 2.7268. But each of its faces is a Sierpinski carpet with dimension 1.8928. Using the same logic the Menger should have dimension 2.8929 - but it does not.

I dont like the link back to frsctalforums at this point as well because it is a too simple desvription

ad far as i understand it is is although it has whipped cream the crispy parts make up for it . . . :)

Guys, you should read the very top of the wiki page you just linked.
Note: The topological dimension is essentially the dimension of the "surface" of an object.
Then there's the, I think it's called, geometric dimension, which is essentially the dimension of the lowest-integer-dimensional euclidean space it can lie in.

For example, a 2-Sphere has topological dimension 2 (the dimension of its surface), however, its geometric dimension is 3, since it has to be put into euclidean space.

The Hausdorff dimension always lies between those two dimensions. So an object with Hausdorff-dimension, say, 0.5, will be something between a set of disconnected points (0-Dimensional) and a set of lines (1-Dimensional) It will also have a topological dimension of 0 and a geometric dimension of 1.

A line-section also has topological dimension 0 (the "surface" of a line-segment is the two end points which are both 0-dimensional) and a geometric dimension 1. The Hausdorff-dimension in this case is 1 too.

The possible range of Hausdorff dimensions for a particular fractal construction is not limited to the interval between adjacent integers. For example fractals made of discrete points (i.e. clouds or dust) can be line-filling (dimension 1), plane filling (dimension 2), space filling (dimension 3), and so on.

It starts to get clearer, some way down in wikipedia article I learned, that the SURFACE of the menger sponge has a dimension of 2.727 not the cube itself. so the rule of exceeding hausdoff dimension is not broken. But I still ask myself what dimension has the sponge? no dimension, because the volume is zero?

Interresting also the fractal dimension of brain and lung surfaces. Our lung has a fractal surface of almost three dimensions (2.97)
Surprisingly high!

Personally I would redefine "Fractal" as anything that can be created (or simulated) using fractal techniques - of course that means *everything* - all other geometries are thus special sub-sets of fractals rather than the other way around.
To me this makes more sense than viewing fractals as "special" and other more restrictive geometries as "general".

Given how efficient evolution is and the fact that the greater the surface area of the lungs the better I don't really think it's that surprising - I'd expect fish gills to have a similar dimension ;)

I think the Wikipedia page says that both the surface and volume fractal dimension is 2.727 (see note at the right of the Mender).

for sure, I should read till the end :headbatting:

But what about this definition, that for a fractal the hausdorff dimension "strictly exceeds" topological dimension? I understand "strictly exceeds" as "must be higher". Do I misunderstand?

Mathematical jargon:

"exceeds" often means ">=",
"strictly exceeds" always means ">" (explicitly excluding the case of being equal).

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.

Thanks a lot Syntopia for investigating.  :ok:
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.

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...)

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

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 (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.

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".

Indeed tricky stuff...

I mined this from the net:
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... :dink:

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...

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:
(http://upload.wikimedia.org/wikipedia/commons/3/3a/Sierpinski_arrowhead_3d_stage_5.png)

Namely probably some sort of 3D version of the Sierpinsky Curve:
(http://upload.wikimedia.org/wikipedia/commons/0/04/Sierpinski-Curve-3.png)
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.

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