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

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

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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.

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.

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russ_watters

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It depends on whether you mean ideal fractals or anything which exhibits fractal behvaiour?Originally posted by Loren Booda

Is it possible to show for fractalsin generalthat a one-to-one mapping exists which replicates precisely the fractal between different scales?

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.

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climbhi

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

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

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.Originally posted by Loren Booda

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

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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.

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