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Re: Trisection
Posted:
Jul 22, 2002 9:17 PM


Steve Gray <stevebg@adelphia.net> wrote:
> There is an article in the Monthly comparing about 8 > methods for approximate trisection. It includes the number > of steps each one needs and the errors they give. See > The American Mathematical Monthly, May 1954, page 334, > by Jamison. I have not examined Mark's method relative to > any of those. Someone might consider writing an updated > version. But not me; I'm into constructions on Ngons > for now.
I looked up that article. It is short and insightful. It also inspired me to do a thorough error analysis of the the trisection algorithm proposed by Mark Stark in this thread; see:
http://mathforum.org/epigone/geometrypuzzles/quumeldplyr/74tzzy089efg@legacy
It turns out that Mark's construction is amenable to a pretty nice analysis and a precise expression can be obtained for the error. In his notation, if we let the measure of the angle AOB be T and the measure of the angle AOE be a, then the error e(T,a) is given by the expression:
e(T,a) = T/6  a/2  arcsin((1/3)sin(T/2  3a/2)).
I have placed the derivation on my web site. The statement of the problem is in
http://www.math.umbc.edu/~rouben/Geometry/trisectstark.html
and the proof in:
http://www.math.umbc.edu/~rouben/Geometry/trisectstarkproof.html
It turns out that the accuracy of Mark's construction sort of falls between the two methods described in the Monthly article mentioned by Steve. Definitive comparison is difficult because Mark's method involves an arbirary choice, while Jamison's doesn't. I am basing the comparison statement made above by setting a=T/4 in calculating the error e(T,a).
 Rouben Rostamian <rostamian@umbc.edu>



