The Pi ParadoxSince ancient times it has been known that the ratio between the circumference of a circle, and its diameter remains constant for all circles. Determining this ratio to great accuracy was one of the motivating problems for calculus. We call this ratio pi. We know it to be a transcendental, irrational number, approximately equal to 3.14159. So why does this approximation make pi appear to be exactly four? 
Written 2000 Formatted 2010 

The Kuhn Polygon for approximating pi:

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With each step of this method the area of the Kuhn Polygon becomes closer to pi*r^{2}. However, the perimeter remains four, and thus, fails to approximate the circumference. Compare this to the Sierpinski triangle referenced in books on chaos and fractals.  
