Are powerlaws fractals?

[Chillheimer]

I'm trying to visualize the fractal aspect of powerlaws and came up with this draft:


It's obviously correct for a simple x² power law
((side-question: is it actually 1/x² in my example? and how would I have to name/mark the x&y axis so it is correct?))

but is it also true for other power laws like these?

I'm aware that for the real world examples you can't zoom in infinitely. It's more about the general idea, to show the self similarity in one easy to understand image.

What do you think? Any idea how to improve this and make it actually correct? I've probably made some obvious mathematical mistakes.. never been good at math..

[SamTiba]

I would say that this is not really a 'fractal aspect' of the power law.

In my opinion it is more the impact of infinity on the formula. The visualization of the power law is a transition of a behavior with x^-k (mostly with negative exponent) to a straight line, so we have a part that goes to infinity which is transferred to the part near the x-axis of this line.
It should be normal to have a denser area near the x-axis then because you have less space for more numbers there because of the divergence.

You can also clearly see that this is not the case for the 'real' examples, they don't get stuffed at the end, because the probability distribution is different in comparison to the natural numbers (which are evenly distributed before the logarithmic scaling).

(I hope you understood my point even though my vocabulary to explain it sucks)

[Khashishi]

It looks like you are plotting a set of points somewhat similar to:
and you are labeling the points with the number .
Obviously, it is self similar, but a line segment is also self similar to pieces of itself, and we don't call it a fractal.

So, a more precise question is, is the set of points a fractal? That is, does it have fractal dimension? If you look at the box counting dimension, it is 0 (see below). There's a lot of points clustered around (1,0), but only a countable infinity of points. AFAIK, you need an uncountable number of points to make something with dimension greater than 0.
So, no, it's not fractal.
Edit: Err, I spoke too soon. The Mandelbrot set has integer fractal dimension 2, but it's silly to call it not a fractal. But the boundary of the Mandelbrot set has higher Hausdorff dimension (2) than the topological dimension (which is 1). I don't think you have such a property for this above set of points.

box counting dimension:
If you take boxes of size 2^(-k), for counting numbers k, then you see that it takes k+1 boxes to cover the shape. Therefore, the box counting dimension is
limit_k->infinity  log(k)/(log(1/2^(-k)) = limit_k->infinity log(k)/k = 0

[Chillheimer]

First of all:

I'm aware that for the real world examples you can't zoom in infinitely. It's more about the general idea, to show the self similarity in one easy to understand image.

Romanesco Broccoli is not a mathematical fractal, because it's not infinite. yet it is the most commonly used example - with good reason.
I know the purists don't agree (cue Sockratease joining the discussion) but I have a different definition:
If it is self-similar over at least 3 orders of magnitude it can be considered a natural fractal


What is left out in your perspectives is the aspect of time.
To make a power law a pure mathematical fractalyou need to give it infinite time.

Take an image of the mandelbrot-set and with just 10 iterations:

there is no obvious fractal aspect to it. No self-similarity, no scale invariance.
Yet we all know it is a fractal.
all you have to do is iterate more (use more "calculation time", to gain more samples)

Now take one of the classic examples for a powerlaw distribution, earthquake-size - which is also often used as example for fractals in nature.
In this example, time takes the place of iteration depth:


you can watch that chart with a "low iteration count" or time window of one year. And you will only see one magnitude 8+ earthquake.
But when you watch it with a "high iteration count" or time window of 10 million years you get amuch larger sample size with 10 million 8+ earthquakes per year and probably a few thousand 9+ or even 10+ events.
And if you were able to take an infinitely large time window (sample size) you would have the distribution heading into infinity at any point.

I really don't see why this should not be fractal. You zoom in, change the scale, but find the same pattern.

If you have a pattern like this:
.                                                .                        .            .      .  . ......
that keeps getting smaller and smaller on the right side (beyond my physical, visible .resolution) you do have a pattern that looks the same no matter how close you zoom in.
which basically the (non-mathematical) definition of fractals.

saying it is not fractal, because we are only watching it at a .limited resolution is like saying a low iteration image of the mandelbrot-set is not a fractal.
same for earthquakes, for which we only have a limited time resolution(iteration depth) of collected data.
and as long as time is still running, we are still iterating.. so the big one will come - it's just a question of when..

real infinity is always a purely mathematical concept. we know nothing in reality that is actually infinite. except the actual size of the universe, which we are not sure about.
what's the point of mandelbrots "fractal geometry of nature" if we discard everything just because it is "not infinite". then nothing is fractal.
of course earthquakes are limited by natural causes. so are city sizes. there won't be a 10billion city, because the amount of people is limited.

[Sockratease]

First of all:Romanesco Broccoli is not a mathematical fractal, because it's not infinite. yet it is the most commonly used example - with good reason.
I know the purists don't agree (cue Sockratease joining the discussion) but I have a different definition:
If it is self-similar over at least 3 orders of magnitude it can be considered a natural fractal...

FINALLY!!

That's the first time I have seen you actually quantify a definition for this stuff!!

Congratulations  

I can accept things like this a lot better with a more Mathematical definition.  That was the one thing I kept pushing you to produce before I could accept "Real World" fractals as existing.

Looks like you did it at last  

I still feel they only exhibit fractal properties and are not actually Fractals, but if that were to become a Universal definition - I have no problems with it.

If you ever stated that definition before - I missed it.

[Chillheimer]

wohoo!
Damn, no, I never stated that definition before in our ongoing 2-3 year "battle"..
great, now we're talking!

I still feel they only exhibit fractal properties and are not actually Fractals..

Right. But I see it as complimentary addition to the mathematical definition.
It's the definition for natural fractals.


I just stumbled upon this quote by Einstein, which fits perfectly:

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”

[Sockratease]

wohoo!
Damn, no, I never stated that definition before in our ongoing 2-3 year "battle"..
great, now we're talking!

I agree with Uncle Albert that Math has an uncertain relationship with reality, but the mathematical definitions work for Empirical Tests!

Without them any such discussions are just Philosophical Discourses with no Science behind them  (as I see it).

Up until that, and I held off saying this until we had some common ground, the discussion reminded me of The Witch Trial in Monty Python And The Holy Grail!

"We have found a witch, may we burn her?"

"How do you know she is a witch?"

"Well, she looks like one!"

Needless to say, we couldn't weigh your examples to see if they weighed the same as a duck, so the whole thing was a joke to me and I couldn't take it seriously.

And I don't think Benoit Mandelbrot ever said "The Coastline Of Brittan Is A Fractal" - I think the point of his book was that it exhibited "Fractal Geometric Properties" - and to me that distinction is HUGE!

Perhaps that is a smaller distinction in your native language, but in my language that is an ENORMOUS difference.

But let's not start going backward now...

So if we find things that do not meet this definition, we are back to "The Universe" as a Set and whether a Set can contain things that are not members of that set  

...

What was that about Power Laws again?