MATH325: Lecture 10


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Constructions using a Compass and Straightedge

Axioms, Reprise

Where did Euclid's postulates come from?

The first postulate says that given two points, we can connect them with a straight line segment. In practice, we do this by lining a ruler up next to the two points and drawing the segment with a pencil or marker.

The second postulate says that you can continue that segment indefinitely. Obviously you can extend it a short distance just by drawing to the end of the ruler rather than stopping at the endpoints. To extend it further, you must move the ruler. But you can move the ruler as many times as you need to to keep extending it. In practice, errors creep in and the line stops being straight, but in theory one could extend the segment forever.

The third postulate says that you can draw a circle with any center and radius. Your compass will let you draw small circles on any planar surface, and to draw large circles you can use a stake in the ground and a rope or string.

The fourth and fifth postulates guarantee that the surface you're drawing on (and your ways of measuring that surface) are flat and smooth like a white board or sheet of paper.

Straightedge and Compass

Euclidean geometry is built on these five postulates and, roughly speaking, describes the types of figures and diagrams we can draw precisely using only a ruler and compass. Over the course of the next few weeks we will learn more of what we can do with these techniques.

I hope by this point in the semester you're not satisfied by "roughly speaking" about important concepts like this one. What are we leaving out? What can or can't we do with a ruler and compass that we can (or can't) do with Euclid's postulates?

Think about this for at least three minutes before you look at this answer.

Because you can do more with a ruler and compass than with a straightedge and compass, we will try to remember to say "straightedge and compass" when we describe what we're doing rather than "ruler and compass". The phrase "ruler and compass construction" is common and almost always describes what we will call "straightedge and compass construction". (It is interesting to study what more can be done if you allow the use of a ruler -- or any marks on your straightedge -- and some mathematicians have done so.)

Constructible

Before we get started, we need to add one more assumption to our list of postulates -- we assume that lines and circles intersect in the way that we're accustomed to and that we can identify those points of intersection. You'll recall from our proof of the Side-Side-Side theorem that our textbook proves this in Chapter 9, and that it apparently never occurred to Euclid that this needed proving.

Definition of Constructible: A figure is constructible if it can be recreated in a finite number of steps by connecting (given or constructed) points by lines and drawing circles with (given or constructed) centers through (given or constructed) points.

Constructing a point
How does one construct a point or a radius? Points are constructed by the intersection of two lines, two circles or a line and a circle. For example:

Problem 1: Given two points A and B, construct a point B' on line AB so that B' is not equal to B and |AB| = |AB'|.

Discussion of Problem 1: This is a scary looking request! If this point B' even exists, where will we find it?

Once you've figured out that B' is some sort of copy of B on "the other side of" A, the problem gets much easier.

Construction: Use your straightedge to construct line AB (or as much of it as you'll need). Then set your compass so that it will draw circles with radius equal to |AB| (adjust your compass so that one of its pointy bits is on A and the other is on B). Then use the compass to draw a circle centered at A with radius |AB|.

As expected, the circle about A with radius AB intersects the line AB in two points. One of those points is B, and the other is point B'. By locating this point of intersection we have constructed a point B' whose distance from A is |AB|, which lies on line AB, and which is not equal to B.//

Note: it's cumbersome to write "the circle centered at A with radius |AB|", so the textbook has introduced notation to make this easier. We'll use CP(r) to denote the circle centered at point P with radius r.

Constructing a distance
We've seen an example of the construction of a point. How do we "construct" a radius or distance?

Because there are no marks on our straightedge, our only tool for recording or remembering distances is our compass. If we can construct segment AB, we have also constructed length |AB|.

Problem 2: Given points A and B, construct a distance that is twice distance |AB|.

Discussion of Problem 2: You should be able to solve this by carefully studying the solution to problem 1.

non-Constructible

The game of dreaming up new lengths, points and figures and attempting to construct them is a fun one, and is the content of chapter 3. (Given points A and B, can you construct a distance that's half of |AB|? How about |AB| times the square root of two? One third of |AB|? Three times |AB|? Pi times |AB|?) The game becomes even more interesting when you start to realize that there are some simple things you can dream up but cannot construct.

The Greeks realized that it probably was not possible to use a ruler and straightedge to construct a regular 7- or 9-sided polygon, a ray trisecting a given angle (unless it was a very special angle), or a distance that was a multiple of the cube root of two. However, there was no proof that it couldn't be done until the early 1800's! It was the work of Carl Friedrich Gauss that made this proof possible; the proofs presented in the book relies on the notion of a field from abstract algebra and so won't be covered in this class.

Some Constructions

Using a compass, it is easy to copy distances. Perhaps this is why constructing regular polygons is so popular. We'll start by constructing an equilateral triangle:

Theorem 3.2.2: Given two points O and P, we can construct an equilateral triangle. In particular, we can construct an equilateral triangle with side length |OP|.

Discussion of Proof: All we start with are two points and the distance between them -- that doesn't give us much to work with! Initially we can construct line OP and the two circles CP(OP) and CO(OP). As it turns out, we don't even need the line.

Proof: Construct circles CP(OP) and CO(OP). These circles intersect in two points -- call one of those points Q. Triangle OPQ is an equilateral triangle.//

Problem 3: Given two points O and Q, construct the midpoint of segment OQ.

Hint for Problem 3: Look to Theorem 3.2.2 and Figure 3.1(b) for inspiration. If you find the problem too easy, write down the details of your construction step by step.