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Date: Around 1993


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?



Replies:
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


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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