Math Forum Discussions

Rouben Rostamian

Posts: 193
Registered: 12/6/04

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

Plain Text

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>