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.
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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
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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!