Date: Tue, 28 Jul 1998 09:44:55 -0700
From: John_Mitchell@intuit.com (John Mitchell)
To: Dave Rusin <rusin@math.niu.edu>
Subject: Re[2]: Angle trisection - in H^2

     Thanks for the reply to my sci.math post about angle trisection in 
     H^2.
     
     Your description of the unit disk model of the hyperbolic plane is 
     correct, but I don't think the reduction to the Euclidean case works. 
     One problem arises from the fact that, as you mention, the hyperbolic 
     and Euclidean centers of a given circle are generally different. 
     Because of this, drawing a hyperbolic circle through a given point 
     with a given center is different from drawing a Euclidean circle 
     through a given point with a given center, and it is not clear (to me) 
     that the former construction can be reduced to the latter (or to any 
     other Euclidean construction). I haven't thought much about it, but I 
     doubt that this reduction is possible, since hyperbolic constructions 
     are represented in coordinates by transcendental functions, whereas 
     Euclidean constructions are represented by (linear and quadratic) 
     algebraic functions.
     
     One might also consider scaling down a putative hyperbolic trisection 
     to the infinitesimal scale to obtain a Euclidean trisection (which we 
     know to be impossible), but the hyperbolic plane admits no scaling 
     transformations (unlike the Euclidean plane), and anyway the limit of 
     a valid construction may not be a valid construction (degeneracies may 
     arise).
     
     I received the following email reply from George Martin:
     ----------
     John,
     
     No, angle trisection in H^2 is generally impossible
     with ruler and compass. (I do not understand your second tool [draw a 
     circle through two given points], since this can be done with the 
     first (ruler) and the third (compass).)
     
     BUT, you can square the circle!
     
     See the last chapter of my book:
     Foundations of Geometry and the Non-Euclidean Plane Springer 1975 
     (recently reprinted)
     
     George
----------
 I haven't yet had a chance to look at his book (and by the way, he's 
right about my second construction - it doesn't make much sense; I was a 
little hasty in writing up my question). I'm not sure what he means by 
"squaring the circle" in the hyperbolic case, since the definition of a 
square is problematic.

A question which I didn't ask in my sci.math post, but which underlies 
the question I did ask, is whether or not there is a purely geometric 
proof of the impossibility of angle trisection in the Euclidean plane. 
The usual proof depends upon an algebraic interpretation of the problem. 
Since this interpretation fails for other geometries such as the 
hyperbolic plane or the sphere, it would be nice to understand the 
problem in a way that would extend naturally to other geometries. Maybe 
George Martin's book discusses this.

Thanks again,
John Mitchell
San Diego, California

______________________________ Reply Separator 

Subject: Re: Angle trisection - in H^2
Author:  Dave Rusin <rusin@math.niu.edu> at Internet
Date:    7/27/98 11:25 PM

In article <35B686C5.7F10B60B@intuit.com> you write:
>Is it possible to trisect an arbitrary angle using only "ruler and 
>compass" constructions - in the hyperbolic plane?
>
>So, you're given two rays (geodesics) forming an angle at some point, 
>and you're allowed to:
>- draw a (geodesic) line between any two known points
>- draw a circle (in the hyperbolic metric) through two given points 
>- draw a circle with a given center and through a given point
> - etc.
     
Forgive my rusty memory, but if we view the hyperbolic plane as the 
unit disc with a novel metric, aren't the hyperbolic lines just
ordinary circles (meeting the unit circle perpendicularly), and hyperbolic 
circles also ordinary circles (whose center is not necessarily the
center in the hyperbolic sense)? If so, then you are given no more 
construct than in Euclidean geometry, and so you cannot trisect 
any angle not trisectible in the Euclidean sense.
     
dave