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Looking for information on fractals through google I have read several time that one characteristic of fractals is :

  • finite area
  • infinite perimeter

Although I can feel the area is finite (at least on the picture of fractal I used to see, but maybe it is not necessarly true ?), I am wondering if the perimeter of a fractal is always infinite ?

If you think about series with positive terms, one can find :

  • divergent series : harmonic series for example $\sum_0^\infty{\frac{1}{n}}$
  • convergent series : $\sum_0^\infty{\frac{1}{2^n}}$

So why couldn't we imagine a fractal built the same way we build the Koch Snowflake but ensuring that at each iteration the new perimeter has grown less than $\frac{1}{2^n}$ or any term that make the whole series convergent ?

What in the definition of fractals allows or prevent to have an infinite perimeter ?

4 Answers 4