Ask A Scientist Mathematics Archive

Question: What is the length of the boundary of a fractal?  If it is 
infinite, then how can it enclose a region of finite area?
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There are many mathematical situations like this that are related 
(as  are fractals) in a process continuing to infinity.  Another example  of a 
finite area formed by a line that goes to infinity is simple. The area between 
the x-axis between 0 and 10 and the function 1/square root(x) (the reciprocal
of the square root of x) is finite, while the function itself goes to infinity 
when x gets close to zero.  These interesting occurrences will be clearer when 
you study calculus.  Math and science have a lot of non-intuitive situations 
like this, making it fun.  
Samuel B. Bowen
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Some fractals are "curves". The Koch Snowflake curve is an 
example. The ordinary curves such as circles, parabolas, etc. which are 
normally studied in school mathematics are "locally straight".  Draw such a 
curve with your graphing calculator and then zoom in on a point of the curve 
several times (i.e. put it under a high-powered microscope, so to speak) the 
portion that you are viewing will eventually come to look like a straight 
line.  A fractal curve, on the other hand, will not look straight, no matter 
how much you zoom in.  A great book on the subject is Fractals for the 
Classroom by Peitgen et. al.  It is published by Springer-Verlag with the 
cooperation of the National Council of Teachers of Mathematics.
Robert Allan Chaffer
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