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Even more boring math (fractal)

  
 
odd jest on horn
19:33 / 05.12.02
Logos:
Now, if I may be allowed to go off on a slight tangent: Can someone explain to me (in general terms) the process for calculating fractal dimensionality: where something like a rocky beach has 2.68 dimensions or whatever. I ran across the concept in Gleick's Chaos book a number of years ago, and haven't been able to find someone who is both a) capable of understanding what's going on, and b) capable of explaining the thing in lay terms.

Imagine a circle in 2d. It's surface is pi*radius^2. So if you increase the radius of the circle, it's surface will increase proportionally to radius squared.

Ok, imagine a fractal. Draw a circle around it somewhere. Measure how much of the surface is filled. Double the radius of the circle. How much of the surface of the circle is now filled?

Let's the fractal has a dimensionality of 1.5, then it would fill 2^1.5/2^2
as much surface as before of the circle. (The surface of the circle quadripled but the surface of the fractal increased by 1.5^2 = 2.8.. )

same applies to spheres and fractals with dimensionality between 2 and 3

same applies to lines and fractals with dimensionality between 0 and 1

etc.
 
 
grant
13:44 / 06.12.02
OK, I'm stumbling at the "now imagine a fractal" point. How do I imagine a fractal? And what is "dimensionality"?
 
 
cusm
14:16 / 06.12.02
Does the shape contain the fractal entirely, or only a portion of it? In the later, if the shape grew, so would the fractal accordingly. So it looks like the goal is to calculate something like (shape area) - (fractal area) as the shape is changed in size, with the ratio of change being the dimentionality of the fractal?
 
 
odd jest on horn
22:37 / 06.12.02
grant: A fractal is a very squiggly, self-similiar shape.
Self-similiar means that it doesn't matter at which "zoom level" you're examining it, it will always look more or less the same. Something which isn't fractal but looks like one, and fractals have been used very successfully to simulate is a tree.

A (simplified) tree starts with a trunk. The trunk divides into, say, three main branches. Each main branch divides into three smaller opnes. etc. This is self similiarity. If you break a small branch off you'll get a miniature version of the tree. (minus leaves of course ;-)

Very squiggly: a fractal drawn on a piece of paper exists in 2D space. If you draw a line on a piece of paper, it's gonna have some area, the width of the pencil mark * the length of the line. In the idealized world of mathematics however, the line would have no area, but be one dimensional. Even if it were a curvy line, it would still have no area.

However, fractals are so squiggly, that even if you draw them with an idealized point pencil they're still gonna have a bit of area. Not quite as much area as say, a solid circle, but a bit of it. This requires fractals to be given dimensions that are not whole numbers but fractions (hence the name fractals).

One way of approximately measuring the dimension of (some) fractals is to draw a circle around a portion of a fractal, measure the area it occupies when drawn with a real pencil inside this circle, make the circles radius larger by a factor F, but not so large that it goes outside the fractal, measure the area again. Then you compare before and after:

fractaldimension = ln((Fa/Fb))/ln(F)
( Fb = area of fractal before, Fa = area of fractal after)

In our case with the 1.5 dimension fractal and the doubling of the circle, this would be:

dimension = ln(2.82/1.0) / ln(2.0) = 1.5

anywho this method only works for some fractals, those fractals that are "spread out" over the paper.

One fractal which is very vell known, and is not measurable in this way
is the Snowflake. It's very easy to construct. Draw a unilateral triangle. Draw 3 times smaller unilateral triangles, with their bases in the middle of each line of the first triangle. You know have the star of David. Now erase each of the bases. Repeat with 9 times smaller triangles for each of the 12 linesegments you have now. You now have something which looks a bit like a snowflake. Repeat ad infinitum, the triangles shrinking b
 
 
odd jest on horn
22:58 / 06.12.02
(continued cuz I pressed Enter at an unfortunate time, posting my previous message)
One fractal which is very vell known, and is not measurable in this way
is the Snowflake. It's very easy to construct. Draw a unilateral triangle. Draw 3 times smaller unilateral triangles, with their bases in the middle of each line of the first triangle. You know have the star of David. Now erase each of the bases. Repeat with 9 times smaller triangles for each of the 12 linesegments you have now. You now have something which looks a bit like a snowflake. Repeat ad infinitum, the triangles shrinking to a third of their size each time and the number of triangles added each time multiplying by four.

You get a very squiggly snowflake.

In this case however there are no circles to draw :-( So how do we proceed to find the dimension? Well each time we iterate the drawing process, we are adding 4 times as many triangles as before, and each of them is 3 times as small as before. It turns out that the dimension of this fractal is ln(4)/ln(3) = 1.2619..

Actually for all fractals, you can take the number of self-similiar components next level below that make up this level, take the numbers logarithm, then take the logarithm of the size difference. Divide as above and voila, you have your fractal dimension. But in this case you have to know the iterative function that creates the fractal.

There is prolly an "experimental" method to find this fractals size too, but I don't know it. That is when you have fractal, but not the iterative function that created it, and want to measure it's dimension.

Also just do google search on "fractals". Gazillions of pictures, and prolly a dozen or so pages that explain this fractal dimension concept better than I do :-P
 
 
odd jest on horn
22:59 / 06.12.02
and cusm: it's log and division instead of subtraction, but you've got the right idea.
 
 
Lurid Archive
18:48 / 07.12.02
Topic abstract?

I believe that you are describing capacity dimension, odd jest. There are lots of other versions of dimension which try to extend the notion from the "obvious" cases - a point is 0D, a line is 1D, a plane is 2d etc. Notionally, lots of "objects" have dimension, though to claim that a real object has a fractional dimension is to say something about it in infinite detail - fine for idealisations, more shaky for real objects. Especially in a quantum world.
 
 
—| x |—
07:37 / 09.12.02
Notionally, lots of "objects" have dimension, though to claim that a real object has a fractional dimension is to say something about it in infinite detail - fine for idealisations, more shaky for real objects. Especially in a quantum world.

How so Lurid? That is, how come a fractal dimension is shaky in a quantum world? I know that you've expressed this thought before: that fractals and the quantum world don't mix, but I would appreciate your reasoning (I have an intuition, but I also think there is an intuition that says the contrary as well). Clue me in please!
 
 
Lurid Archive
14:35 / 09.12.02
Imagine a fractal tree. Suppose each branch has ten smaller branches, getting smaller and smaller at every stage. Now imagine another tree next to the fractal tree which has one million branches. The second tree is much like the first tree in that each branch has ten branches which get smaller and smaller. The only difference is that in the second tree the branching stops at a certain point.

Now imagine that you are short sighted. You take off your glasses and someone spins you round. Which tree is which? In the right situation, you can't tell the difference. I could be wrong, but it seems to me that the uncertainty principle (or any limit of accuracy in measuring equipment) presents you with short sight that you are unable to correct.

BTW - the fractal tree has a fractional dimension, whereas the second tree does not. Also, you can freely replace a million with any (finite) number.
 
 
—| x |—
08:35 / 10.12.02
Yah, I think I get the gist of what you are saying Lurid--it is kinda' how I thought you were going to go.

However, simply because we can't get inside Plank's Constant (I think this would be a correct correspondence to your use of the uncertainty principle?) doesn't rule out that an infinitely divisible structure occurs within that minute range: it is our perception and ability to measure that is limited, but this does not entail that the universe is limited in this manner. Of course, on the flip side, we can't assert that it isn't either.

Sometimes I "flip" the picture, so to speak. What if we start at the level of quanta as the base level, and then delve into the fractal as we get (relative to us) larger in scale? I mean, it's like when you look at the Mandelbrot Set, right, it's one big structure, but as you get into it, there is more and more detail. So what if the whole subatomic realm is like this one big structure, and then as we move further from the microscopic, we get more and more variety of detail?

I think what is contrary to the limit case that you suggest is that, as far as I know, which admittedly might not be very far, there aren't really discrete particles, per se but only particles when measured as such; that is, I seem to recall that the quantum realm is composed of continuously interacting elements that do not hold a stable form: a particle's existence, as such, is dependent upon its interactions with other particles.

Shrug, I dunno'. Seems to me that there isn't such a well defined distinction between discrete and continuous anymore. I would say that manifestation is somewhere in between or interdependent upon such a dichotomy; that is, from one view there is discrete objects, from another there is an interwoven continuum--perhaps like those "figure-ground" pictures with the old and young lady, or the cup and the faces.
 
 
Lurid Archive
19:00 / 11.12.02
> it is our perception and ability to measure that is limited, but this does not entail that the universe is limited in this manner - mod

OK. Two points. First, as far as I understand this is wrong. It is not "our" inability to measure, it is a universal inability to measure. It is the nature of the universe that imposes this limitation. (You seem to admit this later on.)

Second, even if it were simply a human inability to measure, this sits badly with the kind of certainty required. It is like using a ruler, accurate to the nearest millimetre, to give a measurement to the nearest trillionth of a millimetre (in fact, its much worse than that). I'd be bemused in that situation, if someone told me my piece of string was 10 millimetres and one trillionth (but not two trillionths, heavens no).

Your points about continuity then serve to undermine your point, rather than reinforce it. If things are continuous - I think that QM implies that they are, in a sense - then how does it make sense to describe an object with ultra precision? It doesn't.
 
 
—| x |—
08:03 / 13.12.02
First, as far as I understand this is wrong. It is not "our" inability to measure, it is a universal inability to measure. It is the nature of the universe that imposes this limitation.

Hmm...so, wrong to you? Or wrong to some collection of people, but not patently wrong?

You say that it is not our inability to measure; yet, in the same breath, you say it is a "universal inability to measure"--what is the difference? Both imply that there are observers doing the measuring--a measurement isn't something that spontaneously occurs on its own!

You see, what I wonder about is how we move from the analysis of data that is obtained through observation to asserting that what is uncovered by the analysis are facts about the universe in itself--outside of observation--as opposed to facts about how the universe looks to us, or perhaps as features of our relation to the universe. Maybe it is the nature of the universe which imposes this limitation; however, I see no way to validly infer this from our observations.

...even if it were simply a human inability to measure, this sits badly with the kind of certainty required.

What kind of certainty is required?
Who is it that needs to be certain and why?
What is this certainty required for?

Your points about continuity then serve to undermine your point, rather than reinforce it.

But, I would agree with the idea that it might be impossible to describe objects with ultra precision. How does this undermine my point?
 
 
Lurid Archive
12:13 / 13.12.02
You say that it is not our inability to measure; yet, in the same breath, you say it is a "universal inability to measure"--what is the difference?

Yeah, you are right in that this is a philosophical stance. I think the statement "earth doesn't have two moons" is the same as "it is impossible for anyone to observe two earth moons". You seem to be making a distinction. That is, you allow the possibility that earth has two moons, while accepting that it isn't possible to observe them. However, any assertion you might make about an object's existence (or perhaps more accurately, the properties of an object) is ultimately rooted in observation.

As such, we can agree to disagree. We agree that no observer, hypothetical or real, can overcome the uncertainty principle while staying within the logical confines of our universe (however, you could easily postulate a god with magical powers without such limitations, who could also see two moons). I think this means that the universe inherently has this limitation, you do not.

What kind of certainty is required?
Who is it that needs to be certain and why?
What is this certainty required for?

Sorry, I thought we were talking about fractals whose existence requires ultra precise geometry. Perhaps you mean that there exist fractals in the universe, which cannot be verified to be fractals by any inhabitant. We are back to the previous problem. I think that to assert that there are fractals in nature, you would have to be able to point to the existence of one. To do so reqiures that you know the geometry of an object beyond what the uncertainty principle allows. Hence you have to be certain to know a fractal, in a way that is impossible. No doubt you disagree.

But, I would agree with the idea that it might be impossible to describe objects with ultra precision. How does this undermine my point?

Again, I thought you were asserting the existence of natural fractals whose very definition requires discreteness to an infinite degree. But you might take the view that this discreteness is simply "hidden" by continuity. I think the total separation of existence and even universal observability don't make sense - this is my point.
 
 
—| x |—
08:22 / 17.12.02
I think the statement "earth doesn't have two moons" is the same as "it is impossible for anyone to observe two earth moons".

I don't know know Lurid. They seem like two different statements to me; moreover, I think that most forms of analysis would assert that they're not the same. But let's not argue semantics, OK?

I think the total separation of existence and even universal observability don't make sense - this is my point.

I might agree with this, at least in some ways (and perhaps on some days), depending on how you filled it out (we could debate the fine points and details if only for snicks). But, are you saying that your point is that existence is observation?
 
 
Lurid Archive
09:01 / 17.12.02
I think that most forms of analysis would assert that they're not the same.

I don't think so. Though perhaps I should stress the point that you are glossing over. If we have an object which cannot even hypothetically be observed - this is the sense I which I am using "everyone" and "universal" - then I think it doesn't make sense to assert its existence. Its not that existence is observation, just that you can only get at it through observation. I tend to think that statements like "there is an unobservable pink fairy on my shoulder" are meaningless.

Tell me, do you think the earth has a single moon? Or is it just that no one has yet observed the second? If the former, how do you justify it?
 
 
—| x |—
12:06 / 17.12.02
Lurid, I think that we can argue back and forth until we are, as is sometimes said, blue in the face, and it isn’t going to accomplish anything: you are not going to convince me that I am wrong, and I am not going to convince you that I’m right. I think this might be, from my end anyway, because, like I’ve said, in some sense, I do agree with what you are saying, but I also do not think that you are absolutely right—there are several ways to skin a cat.

If I may be so bold: your stance seems largely reflective of a pragmatic view. If you can’t observe something, then you do not think it exists. Fair enough. On the other hand, on the view I’ve been putting forth, I am not willing to make such a strong claim: if I can’t observe something, then I don’t know whether it exists or not—I’m not willing to commit one way or the other.

Perhaps what you’d now want to say, like you’ve already said, is that it at least has to be hypothetically possible to observe something for it to exist. Again, in some ways I’d agree with you. However, what I’d wonder about is this very notion of ‘hypothetical’: a hypothesis is not neutral, nor is it a view from nowhere—it occurs from within some way of understanding the world. However, because we are human our understanding is necessarily limited: we do not appear to be omnipotent! In other words, it seems to me that what we think is hypothetically possible does not rule out what might be possible beyond what we can hypothesize.

On the other hand, your position appears to require that it has established what is absolutely possible and what is impossible: it seems to go beyond hypothesis. After all, you do talk about the logical constraints of the universe and require that there is some universal view which we as limited humans somehow have access to. Again, I don’t see how these might not merely be the logical constraints of our way of being in the world, or perhaps the constraints of our current understanding of the world in which we are embedded. Anyway, I do not think that we can say with any sense of absolute certainty what is possible and what is not.

In other words, I agree with you when you say that we might simply have to agree to disagree.

Oh, I think that the earth appears to have one moon.
 
 
Lurid Archive
14:36 / 17.12.02
I agree with a lot of that. Absolute certainty isn't possible and any hypothesis is fallible. But I think that this is implicit in language. One tends not to say things like, I appear to have two arms, London appears to be the capital of england, planes appear to fly, etc, etc. You are doing it for emphasis, sure, but you are ultimately using lack of absolute certainty as an argument for the plausibility of existence.

Also, my original comment was that the uncertainty principle means you can't have fractals. Of course, if you accept the fallibility of the hypothesis, then the conclusion fails. I'll definitely concede, on those terms, that you can have fractals in a quantum universe. Of course, in that putative quantum universe, the laws of quantum mechanics won't be obeyed... (Appear not to be obeyed?)
 
  
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