Date: Jul 3, 2002 10:52 AM
Author: John Conway
Subject: Re: Trisection


When I read this, I was immediately suspicious, partly because the
claimed accuracy sounded incredible, and partly because the construction
involves an arbitrary choice (which, so to speak, "contradicts" the
particularity of the error-estimate given, which is probably just for
one value that that choice).

So I started to try to disprove it, but since I've now verified
quite a bit of it, I'm beginning to change my mind...

On 20 Jun 2002, Mark wrote:

Come on Mark, tell us your full name, and how you came on this
remarkable idea!

> Just wanted to share this, I worked it out a couple of years ago and
> was surprised by the accuracy:
>
> Near Exact Trisection:
> 1. Start with an unknown angle <90 deg., label the vertex A.
> 2. Draw an arc with origin at A crossing both lines of the angle at
> points B and C.
> 3. Draw line BC making an isosceles triangle.
> 4. Using point C as the origin, draw an arc crossing line BC and the
> earlier arc somewhere between ¼ and ½ way between points C and B.
> Label where this new arc crosses line BC point D.
> Label where this new arc crosses the first arc point E.
> 5. Draw line DE and extend it well past A . If line DE passes
> exactly through point A (it won’t) stop, your first guess was an exact
> trisection.


My first "hope" was that this would be wrong (I say "hope", because
if it were wrong, I'd be absolved from checking the rest). But it's
entirely correct, and already intriguing, in that it leads to a nice
"neusis" construction whose simplicity rivals Archimedes'.

Congratulations, Mark!

> 6. Extend line AC well past point A, step off 3 times length AC from
> point A and label the new point F.
> 7. Swing an arc of length AF with A as the origin that crosses the
> extended line DE near point F.
> Label the intersection G.
> 8. Draw line GA and extend it to intersect the original arc from step
> 2.
> Label the intersection E’.


I haven't yet checked all this, but plan to. I'm already quite
impressed...

> Line AE’ is a good (within less than 1/1000 degree) trisection.
> However this is only the start. Repeating the process from step 4
> using CE’ as the arc radius results in a trisection to within 10E-11
> degrees. Each subsequent iteration improves the trisection by several
> orders of magnitude.


...and plan to study this construction very carefully.

Regards, John Conway