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Welcome to Fractal Forums > Fractal Math, Chaos Theory & Research > Mandelbrot & Julia Set > The 3D Mandelbulb > Theory > Fractal geometry?
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Author Topic: Fractal geometry?  (Read 5592 times)
Description: I need some help understanding more fractal geometry
0 Members and 1 Guest are viewing this topic.
Zojirushi
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« on: December 21, 2010, 12:42:34 PM »
So, I understand the gist of what constitutes a fractal. What I'm having trouble with is the concept of fractal dimension, and how far it extends to measurements. I get why the (box-counting) dimension of the koch curve is log(4)/log(3). So... If I have a koch curve, which at the start of the construction is 1 meter long (0 iterations, a straight line), how many meters to the log(4)/log(3) "is it"?
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cKleinhuis
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« Reply #1 on: December 21, 2010, 01:10:44 PM »
it has infinite length wink when using infinite iteration
and the curve grows in length at each iteration step  by log(4)/log(3) ... i think please correct me if i am wrong...

the koch kurve at iteration 0 consists of 1 line of length 1
the koch kurve at iteration 1 consists of 4 line segments of length 1/3 of the original length
the koch kurve at iteration 2 consists of 16 line segments of length 1/6 of the original length
... and so on

the fractal dimension is a measure for the growth of the line when using smaller measurements.
« Last Edit: December 21, 2010, 01:15:30 PM by Trifox » Logged
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Zojirushi
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« Reply #2 on: December 21, 2010, 10:38:55 PM »
(The curve grows in length by 4/3 every iteration)
but ya, I get the whole idea of "if you try to measure in the wrong dimension, you get degenerate results". So, how could you measure a fractal in the right dimension? One that isn't a whole number?
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David Makin
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« Reply #3 on: December 21, 2010, 11:06:32 PM »
I think I understand what you're getting at but you can't use the term "length" here - a 1 dimensional object has "length" and a 2 dimensional object has "area" but fractal objects of dimension between 1 and 2 have simply a "size" defined by the fractal dimension itself.
So in the case of a koch curve created from an initial line segment of unit length I would say that the size of the koch curve produced is 1 units^(dimension of koch curve) - whereas if the initial line segment were 2 units then the koch would be 2 i.e. 2 units^(dimension of koch curve).
However this is just an educated guess, I don't know if the idea is strictly correct.
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« Reply #4 on: December 22, 2010, 01:04:29 AM »
"If I have a koch curve, which at the start of the construction is 1 meter long (0 iterations, a straight line), how many meters to the log(4)/log(3) "is it"?"

Its actually 1 mlog(4)/log(3).
However, if the initial line is 2m long then the koch curve is 2.398 mlog(4)/log(3)
More generally, if a unit length fractal is 1md, then one with side length s is sd md.
For example a unit length square has area 1m2, and a square with length 2 has area 4 m2.


I recently gave this some thought in http://www.fractalforums.com/mathematics/how-long-is-the-coastline-of-great-britain/.
[More formally, the measurement (e.g. the 2.398) is the 'hausdorff measure', fractals don't have a 'lebesgue measure' as that only works for integer dimensions]
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Zojirushi
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« Reply #5 on: December 22, 2010, 08:26:09 AM »
ahhh this is making my head hurt, but thanks for the reading material in hat post! it's a lot to think about.

One thing that seems important, that I really have no idea where to begin with, is comparisons of measures in fractal dimensions. If you go looking at fractal measures (I'll call 'em that for now, unless that term means something else) with the same mindset as in euclidian geometry... 1m^(2.000001) really isn't comparable to 1m^2. (Like comparing how big a point is compared to a line)

It seems like anything I've dealt with in math that is "close" or "kinda", is resolved with statistical something or other. :\

Also... I've been having a really interesting thought... Can the notion of simplexes be extended to fractal dimensions? IE, given a real, positive dimension, create the analogous simplex.

._. crazy stuff. I have so many questions but too little skill to math 'em out as of yet...
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