Hello & welcome to the forum Bin Jiang!

What a nice coincidence - I am working on an episode for a new youtubechannel about fractals. The topic is power laws and we are investigating the strong connection with fractals.

I'm not sure if your definition is actually "defining enough" but I share your viewpoint.

The curve shown in slide 8 (of the above presentation) contains 7 recursively defined bends. It is not fractal, using traditional definition, since the number of bends is too small (=7) to meet a power law. Instead, there are far more small bends than large ones, or the scaling of far more small bends than large ones recurs twice.

@Sockratease: Nothing stops a single line to be fractal. It depends what you do with it. A simple line is just that. No iteration, nothing is happening.

You need to put points on it to give it "more meaning". You DO something with it. Be it a boring Add point A and point B. Or adding scale invariance through a powerlaw. Then the distribution of points undeniably has fractal characteristics - no matter if they are on a straight line or not.

now you will come again with "but it's not infinite".

that depends on iteration. if you iterate infinite times, it will be infinite. if you iterate just x times, as Bin Jiang suggests in slide 8, of course it will not be infinite, because your RESOLUTION is limited by ITERATION.

Take the second iteration of the koch curve here:

you say: this single image is not a fractal.

but the fractalness it is not about that still frame. the fractalness is emergent from the recursive process.

I ask you: when

does this koch-curve become fractal?

at what iteration? the 3rd? the last one? infinite?

we go out in nature and have a look at still frames like a tree, growing very slowly.

you say it is not a fractal. but thats just the same thing as watching the 2nd iteration of the rkoch curve and not recognizing the fractal principle that created it.

you have to watch the RECURSIVE PROCESS that is responsible for fractals.

In natural fractals, TIME is the eqiuvalent to mathematiocal Recursion and Iteration.

That's one of the core principles of my personal fractal theory.

What your big mistake is Sockratease , you leave that core principle out of the whole picture. Time/Iteration!

Without it, NOTHING is fractal. Just as you say.

so to bring this together:

I am getting more and more confident that we should include powerlaws into the definition.

I think that this is what Bin Jiangs attempt for a relaxed definition is basically saying.

The question that arises is:

1. Do all fractals follow power laws?

I'd say yes

2. Are actually all power-laws fractals?

I'd say yes again.

3. Are these two concepts basically the same "thing"? what is the difference?

I'm not sure yet.

Opinions?

In this thread I came up with a similar definition of fractals in nature (self similar over at least 3 orders of magnitude:

http://www.fractalforums.com/fractals-in-nature/are-powerlaws-fractals/Nice discussion Bin Jiang!