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Old 06-24-2003, 12:45 PM   #1 (permalink)
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The Koch curve

This isn't really a purely philiosophical question, but I didn't know where else to post it.

Its a mathematical question (but a rather philiosphical mathematical question )

Anyway, I have just finsihed a book on chaos theory, and one thing that really interested me is fractals. I have a question regarding the Koch curve.

Looking at a koch curve, it is easy to see that there is a definite region which is enclosed by the curve. There is also a definate region which is outside of the curve. My question involves the area of "fuzzyness" where the two regions meet. If we were to choose a point inside this region of less then obvious certainty, is there a way to prove without doubt that a point is or is not enclosed by the curve?

It would be possible to prove if a point is inside the curve, it is simply a matter of going through the various iteration until one of the new triangles overlaps the point. However, assuming that after, say, 100 iterations, the point has not been covered, how can we tell that after further iteratons the point will not be covered?

Is there a method to prove that a point lies outside the curve? If so what is it?

BTW, the book I have just finished is Chaos: Making a New Science by James Gleick. If anyone has suggestions for further reading I would be grateful. Preferably nothing too heavy, as you can see I am new to the subject, and I am not a mathematician.
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Old 06-24-2003, 01:37 PM   #2 (permalink)
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I don't think it is possible, or at least I think there are points where it's not possible, but I'm not sure.

For more fun chaos, read the Chaos Cookbook. I read it when I was a teenager so it's not too tricky. There's lots of practical chaos too. I believe it comes with a CD these days.
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Old 06-24-2003, 04:27 PM   #3 (permalink)
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I'm not up on my chaos theory, but mathematically speaking, I believe that one must essentially prove that a point is (or is not) exactly ON the curve before examining which side of the curve it is on.

The only way I can think of doing this is to see if the coordinates of the point satisfy the equation of the curve (which, of course, may not be known = you're screwed).

In other words, it may be infinitely close to the curve without actually being on it, which means that no matter how many iterations you do, it will never fall on the triangle.
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Old 06-24-2003, 05:18 PM   #4 (permalink)
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Great book, read it years ago.
If you really want to throw your noodle for a loop
try "Infinity & the Mind" by Rudy Rucker or his "The Fourth Dimension"

Within math, you can somehow figure out the definite,
because you have set up the environment.

Within reality, you will never have an absolute definite,
because I all depends on the scale observed.
Only if you precisely define the scale, then you MIGHT be able to get a definite...
but then you are going back to mathemathical definitions, not reality.
This is one of the reasons Calculus was invented, to be able to handle the infinite variables and the scales & ranges.

The math & science are tools to help you categorize, model or "encompass" the object you are observing,
but this is not reality.

But if you are being real, then there is not real way to absolutely do it,
because of the scale is always reduced for measurement purposes,
thus is the whole idea behind "Chaos Theory"
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Old 06-25-2003, 06:48 AM   #5 (permalink)
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Actually I think it may be possible with calculus.

Graph the first iteration of Koch's Curve, then find the function that makes up that curve. Then integrate to find the area underneath that curve. Repeat this with the second iteration of the curve, and, after finding say 10 or so of these areas (I know, too much work, I wouldn't want to do it either), you can estimate the average area underneath a Koch Curve, then, you can predict a best fit graph line and find a probability that, in some iteration n number of iterations in the future, that point will be beneath the best fit graph line. No to well versed in calc, but it seems it would work in theory.
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Old 06-25-2003, 02:47 PM   #6 (permalink)
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Quote:
Originally posted by RatherThanWords
Actually I think it may be possible with calculus.

Graph the first iteration of Koch's Curve, then find the function that makes up that curve. Then integrate to find the area underneath that curve. Repeat this with the second iteration of the curve, and, after finding say 10 or so of these areas (I know, too much work, I wouldn't want to do it either), you can estimate the average area underneath a Koch Curve, then, you can predict a best fit graph line and find a probability that, in some iteration n number of iterations in the future, that point will be beneath the best fit graph line. No to well versed in calc, but it seems it would work in theory.
As far as I know, the area is no problem, it has a definite area, which can be calculated, but is irational, like pi

My take on it would be that the area would be made up of the sum to infinity of a converging sequence.
Assume the area of the first triangle = 1

Code:
Then toal area = 1   +   3*1/9   +   12 * 1/9 * 1/9   +   48 * 1/9 * 1/9 * 1/9 ....

=1  +  1/9^n * (3*4^n)
so you could use this to calculate the area of the koch curve, using a simple summation. The above equation may not be exactly right, but I think it is close.

The point is that you have a shape, with a known finite area, but witout a way to determine if a point is within it! Also, the length of the perimeter of the curve is infinite!
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