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Author Topic: what is fractal dimension of a circle ?!  (Read 10214 times)
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cKleinhuis
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« on: October 22, 2012, 05:32:56 AM »

i am just wondering, what is the result when box counting a circle ?!?! huh?
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cKleinhuis
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« Reply #1 on: October 22, 2012, 05:36:06 AM »

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 ?!'
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Tglad
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« Reply #2 on: October 22, 2012, 06:38:28 AM »

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.
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kram1032
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« Reply #3 on: October 22, 2012, 09:12:23 AM »

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.
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taurus
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« Reply #4 on: October 22, 2012, 09:36:16 AM »

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.
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Tglad
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« Reply #5 on: October 22, 2012, 10:45:32 AM »

Quote
I can't see the need to name it "fractal dimension", even in fractalforums.

Why not?   smiley
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.
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hobold
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« Reply #6 on: October 22, 2012, 11:07:12 AM »

Why isn't a line a fractal? it is self similar.
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.
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taurus
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« Reply #7 on: October 22, 2012, 12:02:38 PM »

It doesn't matter much...
O K !!
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taurus
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« Reply #8 on: October 22, 2012, 01:12:18 PM »

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... grin
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cKleinhuis
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« Reply #9 on: October 22, 2012, 02:03:04 PM »

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!
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Syntopia
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« Reply #10 on: October 22, 2012, 04:49:24 PM »

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.

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



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Tglad
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« Reply #11 on: October 23, 2012, 01:28:38 AM »

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.
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hobold
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« Reply #12 on: October 23, 2012, 03:38:32 AM »

What about this curve - it is self-similar and smooth. Is it fractal? :-)
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. smiley
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kram1032
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« Reply #13 on: October 23, 2012, 09:17:41 AM »

a curve isn't self-similar
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.
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taurus
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« Reply #14 on: October 23, 2012, 10:16:50 AM »

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

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




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