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Topic: Trisection
Replies: 19   Last Post: Feb 11, 2012 9:59 AM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Trisection
Posted: Jul 3, 2002 10:52 AM
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On 21 Jun 2002, Mark Stark wrote:

> Thanks for the kind words John. I'm afraid this is a case of "have CAD
> system -will play". It's the result of taking the word impossible as a
> challenge. The accuracy does vary with the size of the given angle and
> the quality of the first guess. The smaller the given angle the
> better. My error calculations were based on a medium sized angle (60
> degrees) and fair first guess (25 degrees). However, even a 90 deg.
> angle and a poor guess (45 degrees) yields a first iteration of 30.021
> deg. and a second iteration within 10E-7.

It's nice to hear from you. I didn't want to say so in my first
message, but it's a bit unfortunate that, although the construction
theoretically has this high accuracy, in practice it's going to be
weak because you must produce the short line ED to the decidedly
lone one EF. I expect this defect can be cured in some simple way,
and hope you'll work on this.

Here's a question: For a given angle CAB, let D (on the
straight line CB) and E (on the arc), vary in your manner (so
that CD = CE). Then what's the envelope of the line DE ?

The way my thoughts are running is this: if all such lines
that are tolerably near to the correct one were "understandable",
then one could use probably this to give an alternative finish to
the construction.

The neusis construction I was thinking of is this. Make your
original circle be a unit one, and draw also the unit circle
centered at C. Then adjust your ruler, which has two marks X
and Y one unit apart, so that X lies on this latter circle,
Y on the straight line CB, and so that it also passes through A.
Then the ruler will trisect the angle CAB.

This has the advantage over Archimedes' one (if you don't know
that, you can find it in "The Book of Numbers", which I wrote
jointly with Richard Guy), that it trisects the given angle "in
situ", as it were. It therefore combines very nicely with my
angle-trisector construction for the regular heptagon (which you
can also find in The BoN). However, I want to look at that
again, because for the version in The BoN, the angle to be
trisected is inconveniently small, and because there may also
be a nice way to economize by using something twice (once in
the angle-trisection, and once in the ensuing heptagon construction).

John Conway

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