Why Straightedge and Compass Only?

Date: 10/02/2002 at 13:39:17
From: Helen White
Subject: Geometry Constructions

My Geometry students want to know why constructions can only be done 
using a straightedge and a compass. They want to know why they can't 
just measure a line segment to copy it or use a protractor to 
construct an angle. What's the difference?  We have searched our book 
as well as some internet sites containing constructions, but to no 
avail.

Thanks,

Ms. White 
Wasatch Junior High

Date: 10/03/2002 at 09:14:43
From: Doctor Peterson
Subject: Re: Geometry Constructions

Hi, Ms. White.

Here is a brief answer I gave to a related question:

   The Importance of Geometry Constructions
   http://mathforum.org/library/drmath/view/52326.html 

There are two ways that I can see to explain the restrictive rules for 
constructions, which come to us from the ancient Greeks:

1. They are just the rules of a game mathematicians play. There are 
many other ways to do constructions, but the compass and straightedge 
were chosen as one set of tools that make a construction challenging, 
by limiting what you are allowed to do, just as sports restrict what 
you can do (e.g. touching but not tackling, or tackling but no 
nuclear weapons) in order to keep a game interesting. Other tools 
could have been chosen instead; for example, geometric constructions 
can be done using origami.

2. They are the basis of an axiomatic system, with the goal of 
ensuring that geometry is built on a solid foundation. Euclid wanted 
to start with as few assumptions as possible, so that all of his 
conclusions would be certain if you just accepted those few things. So 
he listed five postulates (in addition to some other assumptions even 
more basic); I've taken these from the reference given in my answer 
above:

    Postulate 1. 
    [It is possible] to draw a straight line from any point to any
    point. 

    Postulate 2. 
    [It is possible] to produce a finite straight line continuously
    in a straight line.

    Postulate 3. 
    [It is possible] to describe a circle with any center and radius. 

    Postulate 4. 
    That all right angles equal one another. 

    Postulate 5. 
    That, if a straight line falling on two straight lines makes the
    interior angles on the same side less than two right angles, the
    two straight lines, if produced indefinitely, meet on that side
    on which are the angles less than the two right angles. 

The first two postulates say that you can use a straightedge: line it 
up with two given points, and draw the line between them, or line it 
up with an existing segment, and draw the line beyond it. That's the 
first tool you are allowed to use, and those are the only ways you 
are allowed to use it.

The third postulate says you can use a compass to draw a circle, given 
the center and radius (or a point on the circle). That is the only way 
you are allowed to use the compass; you can't, for example, draw a 
circle tangent to a line by adjusting its radius until it _looks_ 
tangent, without knowing a specific point the circle has to pass 
through.

The last two postulates relate to angles, and are less associated with 
the construction process itself than with what you see when you are 
done.

So really the two tools Euclid required for a construction just 
represent the assumptions he was willing to make: if these two tools 
work, then you can construct everything he talks about. For example, 
you can use these tools, in the prescribed manner, to construct a 
tangent to a given circle through a given point; but it takes some 
thought to find how to do so (without just drawing a line that _looks_ 
tangent), and it takes several theorems to show that it really works.

Of course you CAN just measure a line or an angle, if your goal is 
just to make a drawing - and usually that will be more accurate than 
a complicated compass construction! But when you use only the tools 
allowed in this game, you are actually playing within an axiomatic 
system, getting a feel for how proofs work. You are simultaneously 
playing a challenging game, and doing one of the few things in life 
that can give you absolute certainty: if these lines and circles were 
exactly what they pretend to be (with no thickness, etc.), then the 
point I construct would be exactly what I claim it is. And it's that 
sense of certainty that the Greeks were looking for.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/