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Wednesday, January 15, 2014

Polygons, Circles and Limits, Oh My… an interesting errata


I was re-reading parts of the classic, and widely acclaimed, "Mathematics and the Imagination" by Edward Kasner and James Newman when I came across an old math chestnut that struck me as just a bit odd -- I'll explain why in a moment, and perhaps someone can further clarify the situation, but first, here's the chestnut, in case you're not familiar with it, regarding polygons, circles, and limits:

1)  Consider a circle of radius "1". Draw an equilateral triangle inside it. Then draw the largest possible circle inside that triangle, and then a square inside that circle, followed by another circle, followed by a pentagon… (keep going with circles and regular n-gons). As one approaches an infinite n-gon, what will be the limit of the radius of the consequent circle that fits inside?

2) Now, reverse the situation: start with a circle of radius 1. Circumscribe it with an equilateral triangle. Circumscribe that triangle with a new circle followed by a circumscribed square… then a circle… followed by a pentagon, a circle, etc. As you approach an infinite n-gon, what is the limit of the radius of the circumscribing circle?
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If you answer too quickly without thinking it through, the tendency for many folks is to respond that in case #1 the circle approaches a radius of zero, and in case #2 the radius approaches infinity. ...But that would be wrong (not even very close).

As the number of sides of the polygon increase, the size (perimeter) of the polygon (going in either direction, larger or smaller) fairly quickly stops increasing or decreasing appreciably, and a limit is rapidly reached. But this is where the errata enter.  Kasner and Newman have computed the limit of the radius in the first case to be approximately 1/12 (of the initial radius), and in the second case, the inverse, or about 12 (times the radius you begin with).

BUT, when I read those values somehow they didn't sound quite right to my memory, so I checked further, and sure enough all other citations of this problem I found listed approximately .115 (instead of Kasner/Newman's .083 or 1/12) as the so-called "polygon-inscribing constant" (limit) and 8.70, instead of 12, as the approximate "polygon-circumscribing constant" value -- quite a difference! See John Derbyshire's treatment of the problem for one example of the calculation involved (that all others seem to agree with):

http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2004-08.html

In examining further, I noticed Kasner/Newman used "n - 2" as the final (limiting) denominator of the multiplying sequence they employed, while everyone else simply uses "n". I presume this is the source of the difference/error, but WHY they used n - 2, I don't know (or if there has ever been any dispute over the correct approach?). The K/N book was originally published in 1940 and the edition I have is from 1989, so I'm wondering if this has ever been corrected in later editions???
At any rate, if anyone is familiar with any further interesting history to this problem or errata please let us know. It's occasionally comforting to see errors in mathematics that get passed on along the way, demonstrating that math is indeed, a very human endeavor! ;-)


1 comment:

Paul Coombes said...

I'm afraid this does not really add anything of much interest but Wolfram do list this under Foundations of Mathematics > Theorem Proving > Flawed Proofs >
see http://mathworld.wolfram.com/PolygonInscribing.html