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One of the simplest examples of a fractal is the Koch Star.
We start with an equilateral triangle.
Then, each segment is divided into three equal parts,
An equilateral triangle is constructed over the middle part, and
Finally, the middle part is removed.
This process is repeated for all the resulting segments, ad infinitum. The result is the
Koch Star fractal
One strange observation is that the length of the boundary of the Koch Star is infinite, reasoning as follows - Suppose each segment of the initial triangle is of unit length so that the length of the boundary is 3. Then the subsequent lengths of the boundaries are 3(4/3), 3(4/3)
, ... Hence, the length of the boundary of the Koch Star is the limit of 3(4/3)
as n approaches infinity, which is infinite.
Prof. Ashay Dharwadker