The Pi Paradox

Since 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:

Start by approximating a circle with a square. Let the radius be 1/2 so that the diameter is 1. This creates a square with a perimeter of 4.

circle and square

Create a better approximation by cutting out the corners of the square to meet the circle. We now have a better approximation of the circle, but using various methods, such as projection,we can show that the perimeter is still four.

cut square

We can continue to cut out the corners to make the irregular polygon a better approximation for the circle, but the perimeter remains four, regardless of the number of steps in our approximation.

cut square again

 

 

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  With each step of this method the area of the Kuhn Polygon becomes closer to pi*r2. 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.  
 

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