Date: Jul 22, 2002 9:17 PM
Author: Rouben Rostamian
Subject: Re: Trisection
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 N-gons

> 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/geometry-puzzles/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/trisect-stark.html

and the proof in:

http://www.math.umbc.edu/~rouben/Geometry/trisect-stark-proof.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>