How Do Fractals Influence Measurements in Non-Differentiable Environments?

[eljose79]

assuming its coastline is given by a (fractal) nowhere differentiable but continuous function...so the expresion for distance integral ds**2=1+(y´)**2 is not valid?..in fact..how do you meassure a fractal distance, surface or volume?.

-Can you assign a metric to define distances if curves are fractals?...could we construct with that metric integrals or derivativess?.

 

[arcnets]

AFAIK, a fractal has the following property:

The fractal is made of n parts which, enlarged by a factor s (scale), equal the original.
We know this from normal geometry:
Length of line (1-dimensional): n = s
Area of figure (2-dim): n = s^2
Volume of body (3-dim): n = s^3

We could generalize n = s^d, where d is the 'dimension',
yielding d = ln(n)/ln(s).

Now take e.g. Koch's fractal which has n = 4, s = 3.
You get d ~ 1.262
which is not an integer, but a 'fraction'.
That's why it's called a fractal . Having 'fractal dimension'.

From this it follows that the volume (length, area,...) of Koch's fractal cannot be measured in meters, nor m^2, ... Instead, the appropriate unit to use would be m^1.262... which makes no sense really.

Forget England's coast, that's a bad example. It doesn't have the property required from a fractal.

 

[russ_watters]

You probably need to define a "depth" for the fractals. A limit to how far you can zoom in. I've never studied it though.

 

[Brad_Ad23]

When dealing with fractals in real life, it is often wise to use a band limit (the 'depth') to limit how fine the fractal gets. A pure fractal coastline by definition has infinite length. However a real one would have some other lenght, given by the number of smallest sized increments that exist along the perimiter of the coastline, which in turn are determined by the exact specifications of your band limit.

 

[Loren Booda]

An interesting point comes up. With Koch's fractal, Serpinski's triangle or the Cantor set, an exact scale factor may easily be specified. Is it possible to show for fractals in general that a one-to-one mapping exists which replicates precisely the fractal between different scales?

 

[sir-pinski]

Originally posted by Loren Booda
Is it possible to show for fractals in general that a one-to-one mapping exists which replicates precisely the fractal between different scales?

It depends on whether you mean ideal fractals or anything which exhibits fractal behvaiour?

As far as the original question ... what is the length of Englands coastline? This was the basis for the beginnings of fractal geometry. A guy called richardson examined the coastlines for different countries at different scales and plotted a graph of:

ln(coastline length) against ln(measuring stick)

See here for a richardson plot

It turns out that the UK has a fractal dimension of about 1.24. However as has been pointed out already the fractal behaviour of real world fractals (pseudo-fractals) only exists over a certain range of scales. What you usually find is that the linear relationship in the richardson plot would approach limits at the two measuring extremes i.e. measuring stick ~ size of the county or ~size of the smallest elements e.g. atoms. This is quite well documented (I have a paper which addresses this if anybody want's it - just let me know). As far as the length goes ... the length probably can be measured ... one just needs to have a measuring stick approx same size as the smallest elements. But there is probably not much point in doing that as it doesn't really tell you anything of value.

Anyhow that's enough of my ranting I hope this helps.

 

[bogdan]

http://math.bu.edu/DYSYS/chaos-game/node6.html

 

[Loren Booda]

Also interesting that (pseudo)fractals should be based eventually on linear quantum mechanics. Is there a theory that shows how this can happen? Do all physical "fractals" exhibit (pseudo)scaling dependent in some regard on Planck's constant? Is it possible for (pseudo)linear behavior to be based on fractal origins?

 

[climbhi]

Originally posted by bogdan
http://math.bu.edu/DYSYS/chaos-game/node6.html

That link did a good job at explaining the fractal dimension aspect of it. Thanks.

 

[sir-pinski]

Originally posted by Loren Booda
Also interesting that (pseudo)fractals should be based eventually on linear quantum mechanics. Is there a theory that shows how this can happen?

I'm not sure what you mean by this. Could you clarify?

Originally posted by Loren Booda
Do all physical "fractals" exhibit (pseudo)scaling dependent in some regard on Planck's constant?

Depends on the physical fractal. If you look at dendritic patterns such as diffusion limited aggregates then no. The scaling exponents are related to the process involved. So for example if you take the Koch curve then the fractal dimension is defined according to the motif you replace line segments with. This is why it can be calculated analyticaly. I suppose really planks constant comes in when you have a pseudo fractal extending all the way down to the atomic scale e.g. fracture surfaces. Typically though most real world fractal behaviour is limited to a range of scales and doesn't always extend down to the atomic scale.

 

[Loren Booda]

sir-pinski:

Can fractals arise from a linear quantal medium? I. e., how can (pseudo?)fractals generate up from or down to the linear Schroedinger equation scale? As you mentioned, all physical "fractals" may actually be pseudofractals.