Pi — “Diameter vs. Radius” Paragraph (Formerly On Wikipedia)
©2006 Darel Rex Finley.  This complete article, unmodified, may be freely distributed for educational purposes.

In late summer 2006, I noticed that Wikipedia’s article on the number pi contained no mention of why pi goes only halfway around the unit circle, so I added a paragraph about it.  I was very careful to keep my addition factual and unopinionated, but no matter — boom — it was almost immediately deleted by someone who seemingly watches the article closely to keep it constantly in line with what he thinks it should contain.  His only explanation for deleting my content was that he “can’t find anything in the immediate suggestion that I would use.”  Nice.

Rather than engage in a post-delete-post-delete battle for which I have no time or patience, I have decided to repost the entry here, out of the Wiki-censors’ reach, for all to read freely.  Enjoy:

Diameter vs. radius

Today, the radius of a circle is widely accepted as its most important measure.  For example, the “unit circle,” around which radian angles are measured, has a radius of 1.  But at the time when the greek letter pi was first being associated with the circle’s circumference, it was thought that the diameter was the important measure of a circle.  Since the diameter is exactly twice the radius, this introduces an arbitrary factor of 2 into pi-related phenomena, such as:

One time around the unit circle is 2pi.
1/4 of the way around the unit circle (a right angle) is 2pi/4.
1/n of the way around the unit circle is 2pi/n.
One full cycle of a sine or cosine wave spans a width of 2pi.
A half cycle of a sine or cosine wave spans a width of pi.
Radian angles of 0 and 2pi are equal; 0 and pi are not.
If early mathematicians had measured their circles by radius, they perhaps would have assigned pi the value of 6.283.., and this factor of 2 would not be necessary.

Update 2007.10.07 — A person named Eptavian just notified me of this interesting article in the same vein, by Bob Palais.

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