Fractals without iteration or feedback?

[tumnus123]

My first question is: Are there any fractals (natural or mathematical) which are not constructed by either iteration or feedback?

If the answer to Q1 is no, then Q2 is: Is any process (natural or mathematical) that involves iteration or feedback therefore necessarily fractal in some sense? 

I understand the potential for logical fallacy here... Thanks in advance for your thoughts!

Cheers,
Jason

[hobold]

1.
There is an algorithm for the construction of synthetic fractals which does not need any explicit iteration or feedback. It is called "spectral synthesis".

2.
Iteratively adding more detail does not necessarily result in fractals. For example spline curves are smooth, but they can be constructed iteratively by subdivision. One particularly illustrative variant is called "corner cutting".


In other words, fractals are probably not being clearly distinguished by these two properties.

[tumnus123]

Thanks @hobold, I appreciate the specific examples!  After 20min of online research, I wasn't able to understand how spectral synthesis can be used to construct synthetic fractals... can you point me to a page for the specific algorithm?

The corner cutting algorithm is interesting:
http://www.idav.ucdavis.edu/education/CAGDNotes/Chaikins-Algorithm/Chaikins-Algorithm.html

The limit shape is not self-similar, but it seems like the increasingly small cuttings from a given corner (i.e., the pieces being trimmed off) might be...?


 
 

[hobold]

I wasn't able to understand how spectral synthesis can be used to construct synthetic fractals... can you point me to a page for the specific algorithm?

The idea is to approximate "fractal Brownian motion" with an inverse Fourier transform, starting from a suitable spectrum. I don't know a good presentation on the web, but these slides do have a chapter on this and other algorithms:

http://www.cg.tuwien.ac.at/courses/Fraktale/PDF/fractals7.pdf

The limit shape is not self-similar, but it seems like the increasingly small cuttings from a given corner (i.e., the pieces being trimmed off) might be...?

Well, any smooth curve is locally approximated by its tangents. That is, if you keep zooming further and further into a specific point on a curve, eventually what you see will be indistinguishable from a straight line. And a straight line is self similar in the sense that any small part of it looks like a straight line, too. :-)

The limit curve of a (suitable) corner cutting process will be smooth. Another example where this idea is being applied is for modeling of smooth objects with subdivision surfaces.

[taurus]

I'm really no expert, but this might be of interrest
http://www.keithlantz.net/2011/11/using-fourier-synthesis-to-generate-a-fractional-brownian-motion-surface/

[eiffie]

This IS interesting. Thanks.
I will offer my two cents although it is probably worth much less than that.
In practice an FFT is iterative - sorting amplitudes into different frequency buckets - but I get the point that theoretically you can view it as non-iterative as the buckets get infinitesimally small. Similarly you could contrive a Mandelbrot formula that could do the rotation, scale and offset in tiny increments and say it becomes non-iterative at some point.

[tumnus123]

You're welcome, @effie... thanks for the contributions to this thread!   

I'm wrestling with the "corner cutting" example (as I feel I understand it far better than the "spectral analysis" example)... I find myself wanting to identify where it "goes wrong" and produces a spline curve instead of a fractal.  What if I try to compare it to the Koch curve? For comparison purposes, the initiator I'll use for the spline curve is two legs of an equilateral triangle, and rather than cut off the corners at the 1/4 mark, let's cut at the 1/3 mark. This way, the construction of the two kinds of curves will be as similar as possible. :

Koch curve	Spline curve
Starts with a straight line segment	Starts with a bent line segment
Remove from the segment the middle third	Remove from each segment the third adjacent to the joint
Add two segments to form an equilateral triangle (without a base) in the gap where the middle third was	Add one segment that spans the gap where the thirds were
With each iteration, the number of line segments increases 4x	With each iteration the number of line segments increases 1.5x
With each iteration, the length of each new segment is one-third the length of the previous segment	With each iteration, the length of each new segment is two-thirds the length of the previous segment
dimH=log(4)/log(3)=~1.26186	dimH=log(1.5)/log(3/2)=1

So we might say that the reason that the corner cutting algorithm doesn't produce a fractal is because the rate at which new line segments are being produced is equal to the  rate at which the length of the segments is being shortened...? It'd be like trying to construct the Koch curve by replacing the middle third with two segments each exactly 1/2 the length of the third -- you get a straight line. 

That's a bit silly, but is there an interesting idea here re: what's going on with the corner cutting algorithm?   

[hobold]

So we might say that the reason that the corner cutting algorithm doesn't produce a fractal is because the rate at which new line segments are being produced is equal to the  rate at which the length of the segments is being shortened...?

You are on the right track. Spline curves (with a finite number of control points) have finite arc length. The Koch snowflake has infinite (actually undefined) arc length.