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/ |
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