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Author Topic: what is fractal dimension of a circle ?!  (Read 10213 times)
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cKleinhuis
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« Reply #15 on: October 23, 2012, 12:51:07 PM »

@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 wink
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taurus
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« Reply #16 on: October 23, 2012, 02:18:54 PM »


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 wink


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
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hobold
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« Reply #17 on: October 23, 2012, 05:41:10 PM »

Also, I'm not sure, don't quote me on that, but the Lorentz-Attractor seems fairly smooth to me.
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.
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« Reply #18 on: October 24, 2012, 12:04:24 AM »

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. 
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« Reply #19 on: October 24, 2012, 12:27:25 AM »

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.

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cKleinhuis
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« Reply #20 on: October 24, 2012, 04:32:26 AM »

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 . . . smiley
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kram1032
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« Reply #21 on: October 24, 2012, 09:22:50 AM »

Guys, you should read the very top of the wiki page you just linked.
Quote from: Wikipedia
According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension.[1]
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.
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hobold
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« Reply #22 on: October 24, 2012, 12:36:08 PM »

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.
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taurus
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« Reply #23 on: October 24, 2012, 12:48:58 PM »

Guys, you should read the very top of the wiki page you just linked.
Quote from: Wikipedia
According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension.

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!
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David Makin
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« Reply #24 on: October 24, 2012, 01:18:57 PM »

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".
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David Makin
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« Reply #25 on: October 24, 2012, 01:22:16 PM »

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

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 wink
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Syntopia
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« Reply #26 on: October 24, 2012, 01:47:28 PM »

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?

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).
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taurus
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« Reply #27 on: October 24, 2012, 02:13:57 PM »

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 head batting

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?
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hobold
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« Reply #28 on: October 24, 2012, 03:43:32 PM »

Mathematical jargon:

"exceeds" often means ">=",
"strictly exceeds" always means ">" (explicitly excluding the case of being equal).
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taurus
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« Reply #29 on: October 24, 2012, 04:09:39 PM »

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

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