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From: andrews@yale.ARPA (Thomas O. Andrews)
Newsgroups: net.math
Subject: Re: A geometrical construction poser
Message-ID: <627@yale.ARPA>
Date: Mon, 10-Feb-86 14:45:18 EST
Article-I.D.: yale.627
Posted: Mon Feb 10 14:45:18 1986
Date-Received: Wed, 12-Feb-86 21:12:38 EST
References: <964@decwrl.DEC.COM>
Reply-To: andrews@yale-cheops.UUCP (Thomas O. Andrews)
Organization: Yale University CS Dept., New Haven CT
Lines: 65
Summary: 

In article <964@decwrl.DEC.COM> binder@dosadi.DEC (You are what you do when it counts.) writes:
>I've got a real poser for the geometry wizards out there.  Back in Engineering 
>Drawing class, oh, some 25 years ago, I learned a technique for the construc-
>tion of a regular inscribed pentagon.  The instructor, who was not a geometer,
>stated that its validity was not proven and was probably unproveable.  In the 
>following figure, I've faked it as best I can without a graphics tube, and
>I've described the procedure.  Is there anyone who can prove that this little 
>gem is valid, or that its validity is unproveable?
>    
>    (long detailed description of constructrion procedure which I have not
>       read.)
Pentagon constructions are relatively easy, if you know the length of the side
of a triangle.  It is downright foolish, though, to claim that a construction
of a pentagon is correct but not provably correct.  People on this net seem to
have Goedel on the brain.

Specifically, in the coordinate plane, given the parameters for two lines, 
we can *compute* the coordinates of the point of intersection.	Similarly, we
can compute the the coordinates of the intersection point(s) of two circles or
a line and a circle.  Thus, given any construction with compass and straight 
edge, we can pick one point on the plane as an origin, choose arbitrary axis,
conveniently label some length "1" and compute the coordinates of the point(s)
constructed.  This is difficult and unpleasant, but it can be done.  

  The sort of "non-existance proofs" in geometric constructions are:
	1)  There is no construction of a regular heptagon with straight edge 
           and compass.
	2)  Given only two points which are 1 unit apart, the distance 
			(cube root of)(2)   can't be constructed.
	3)  Given only two points which are 1 unit apart, the distance 
			pi   can't be constructed.
	4)  There is no way to construct a 10 degree angle (and hence no way
		to trisect an arbitrary angle.)
	
   In none of these statements is it suggested that constructions exist
which are correect but not provably correct.  They instead say that these
constructions cannot be made at all, no matter how long you diddle with your
compass and straight egde, as long as you follow two simple rules.

	1)  Lines can only be constructed through pairs of points already 
	  constructed.  (A constructed point is either a point given in the
	  premise of the problem, or the intersection of two figures.)

	2)  Circles can only be constructed if the center point has already
	  been constructed, and at least one point on the circle has been
	  constructed.  (in other words, if a radius is constructed.)

   These two rules make it impossible for ramdomess to enter into constructions.
   
   Interesting point about constructions:

	1)  All constructions possible with a straight edge and compass are
	also possible with only a compass, and, indeed, this compass can be
	"rusty"; that is, stuck in one position.

	2)  If you draw a single circle, and then throw away your compass,
	the straight edge alone can be used to make all possible constructions.

	Courant and Robbins, *What is Mathematics* has a particularly good
	elemtary dicussion of these two statements.  


-- 
					      Thomas Andrews
					      andrews-thomas@yale