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Hi there!
I'm still working hard to get my head around some parts of the whole fractals in the universe thing.

One question popped up that I don't find an answer to:

Is there a name or a variable for the difference of relative detail that can be seen in different areas of the m-set?
I know, it's infinite, but if we agree upon this fixed resolution, there can be very different amounts of information. Attached is a picture that should explain what I mean, left side has relatively little detail, the right side as lots of detail.
If you safe each as a jpeg, the different amount of information becomes clear..

so, any scientific name for that kind of stuff?
and if not, what would you call it, how would you describe it?

regards,
chilli

I believe the variable in that example, if they are both the same location, would be "Iterations" - or maybe "Bailout"

I often misunderstand questions, but in this case it looks like it would be simply the number of iterations (again, if they are the same location!).

If different locations, then the word would simply be Detail, I suppose  :gum:

hmm, there might be a word for it, but since i do not recall it ... what about: (logarithmic) density ?

if it had been iterations i now ould be like  :banginghead: :crazyeyes:
but I indeed mean different locations, but at the same zoom-level.

mh.. but iterations and bailout are clearly closely related. with too little iterations you won't see the details. and there are far more "divided" regions with bailout.

logarithmic density describes it not bad, I think.
(I had just hope there was an existing, official term for this - maybe someone else remembers?)

I knew I was misunderstanding something!

I thought the images were of the same location and if they were, iterations would be the only variable that could do anything like that.

Sorry.

But, yeah.  I have no idea what sort of term would be called for here either   :blindfold:

Isn't it something like 'amount of information' you are aiming for?
If we make a scale from 0 to 100, one color only=0, indistinguishable noise=100.
The left of your image=10 and right=50 or something like that.
:)

arrh, its a shame i have the word at hand

... but cant the density described as "average iterations" for the whole image? the left one would have a lower average iteration compared to the right one, problem is just that the infinite areas of the minibrot outnumber anything ;)

yes, that's exactly what I'm talking about - i just wonder if this has been described somewhere already.

If it pops into your mind at anytime - please bump this thread with it! :)
(german word would be perfectly fine!)

I don't see that as a problem, this is actually the reason that brought me to the question. And I actually like "average iterations". It comes pretty close..

entropy?

Another related buzzword is "Minimum Description Length":

http://en.wikipedia.org/wiki/Minimum_description_length

There's a few relevant concepts from information theory. One that relates to how compressible data is is Kolmogorov complexity. Only problem is, for Mandelbrot images you can represent the image with a Mandelbrot program and the resolution, center, and magnification. That leads to all Mandelbrot images having a Kolmogorov complexity of a ln m + b, for some constants a and b, with m the image magnification (which, as it increases, demands the center coordinates be specified with increasing precision). Well, technically that's an upper bound; if you get a blank image there will be a more compact description still.

For a heuristic that will strongly contrast your left image and your right image, you could do worse than average distance estimate of pixel divided by image width in Mandelbrot units. If at the same magnification the left image would have a higher average distance estimate than the right, as the left has giant open areas but almost every pixel in the right hits or lands adjacent to a filament. Dividing by image width in Mandelbrot units is the same as multiplying by magnification, and has the effect that similarly sparse images at different magnifications would give similar numbers. The left image should have an average distance estimate of a bit under half the image width, since the only filaments are close to the image borders. The normalizing division makes that into simply 0.5, independently of the actual magnification of the left image.

Aside from the minibrot, I think the standard deviation of the iterations would be better than the average (mean).  Consider two variations of the left side, one with a bailout of 4 and another with a bailout of 4e(somebignumber).  Clearly, the larger the bailout, the larger the average number of iterations, but both of these versions would have the same information.  The standard deviation gets to the amount of variation over the image, which is one aspect of the information content.

now that you mention it, i dont think iterations are suitable, just because that bailout is an artificial number and if choosen high enough, there would just be a distinguisation between inside/outside

wouldnt the average distance estimation be a better candidate to describe the complexity better ? any point "inside" would not count, all the others distance estimations averaged would come out to "how many points are close to a mset"

If you zoom in to a sparse area like the left, then close to a minibrot, and then center the zoom again, you will pass a new sparse area but with much higher iterations also on the lowest pixels.
So I think the difference of pixels in the same view is more important, except for pixels inside any minibrots.

I would say the max iteration number minus the iterations that are skipped thanks to series approximation.

Maybe I found it:
http://en.wikipedia.org/wiki/Lacunarity