Compass & Straightedge Construction and the Impossible Constructions
From Math Images
|Creating a regular hexagon with a ruler and compass|
|Creating a regular hexagon with a ruler and compass|
Let's assume we only have a compass and a unmarked straightedge. How could we construct certain geometric shapes and prove certain theorems? That was the problems that Euclid pondered not only because those were probably the only instrument that he had at his time but also he wanted to construct his theorems with as few assumptions, or axioms, as possible. In this picture, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but it should prompt you to ask a question: is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what is not? The issue will be addressed later.
A Compass is a tool which can be used for drawing circles. It has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. A normal form of compass is shown on the right but there are many other variants, crude or precise. For example, a crude form will be pinning a thread of string on the plane and fixing a pencil/pen at a certain distance away from the pin. We usually see this kind of crude compass in military field operations. Look out for such a compass in HBO's TV series Band of Brothers second episode when the paratroopers were trying to figure out where they would be dropped on D-Day based on the distances of their preparatory flights.
A straightedge is a tool which can be used for drawing straight lines, or segments thereof. It should be noted that the difference between a straightedge and a ruler is that the former has no graduations on it (and does not allow any markings as well) while the latter has divisions on them according to certain unit system, be it the metric or the British customary system.
Compass & Straightedge Construction is the construction of points, lengths, angles, and circles using only idealized straightedge and compass. The straightedge is infinite in length, has no markings on it and only one edge. The compass collapses when lifted from the page, so may not be directly used to transfer distances. However, it turns out that this restriction makes no difference due to the Compass Equivalence Theorem which was stated as Proposition II of Book I of Euclid's Elements. It stated that from a given point, it was possible to construct a straight line equal to a given straight line using collapsible compass. Euclid's proof for the Compass Equivalence Theorem will be presented after the section of Basic Construction.
We start by familiarizing ourselves with Euclid's three Postulates in his books Elements.
"Let it be grantedIt should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleases. Then from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. He could not say "I wanted to spread the legs of the compass centimeters apart (or any specified denominations)" because it is later shown that is not constructible. As a result, he could not (so couldn't we) however, choose any two points as he please. Compass and straightedge constructions only allow us to start with something we have and create "stuff" we don't. The three postulates were stated in ancient Greek and both its old and modern translations have not been consistent with each other. What is stated above is an assortment of different translations that give the least amount of confusion about what could and could not be done with a compass and a straightedge. Throughout his book, Euclid used only these three operations to do all his plane geometry: the drawing of straight line through two given points, and circle with a given center to pass through a given point. The terms Euclidean construction, construction, construct and constructible all refer to Euclid's three operations repeated any finite number of times.
- that a straight line may be drawn from any one point to any other point;
- that a line segment may be extended into a straight line;
- that given any straight line segment, a circle may be described having the segment as radius and one endpoint as center. "
The constructions below are some basic ones from where many more constructions and operations are possible and they are by no means exhaustive. The resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Congruency and theorems on congruent triangles are directly derived from Euclid's Postulates. Try proving the theorems yourself!
Given points and and the straight line passing through it. Construct a line that bisects line segment .
It should be noted that as well. Line is the perpendicular bisector of line segment .
Given angle , construct a line that bisects the angle.
Given a point on a line, construct a line that is perpendicular to the given line through the given point.
Given two points and the straight line passing through, construct a line that is parallel to the given line through another given point.
Given a circle and another given point, construct a line that is tangent to the circle.
This part reconnects with the previous section about the issue of compass being collapsible. Euclid's proof is presented in its originality. Additional comments are contained in the parenthesis.
From a given point to draw a line segment equal to a given line segment.
Let be the given point, and the given straight line : it is required to draw from the point a straight line equal to .
Because the point is the center of the circle , is equal to . And because the point is the center of the circle , is equal to and , parts of them are equal therefore the remainder is equal to the remainder .
But it has been shewn that is equal to ; therefore and are each of them equal to . But things which are equal to the same thing are equal to one another. Therefore is equal to .
Wherefore from the given point a straight line has been drawn equal to the given straight line .∎
It should be noted that from , we could "duplicate" in all directions by construct a circle centered at with radius .
From the few basic constructions, you would have probably realized that the different possibilities seems infinite. However, by intuition, we know that the possibility could not be infinite. Hence, mathematician are curious to find out what are constructible and what aren't and for this purpose, the language of pure geometry seems to have "limited vocabulary". Back in ancient times, mathematicians had limited algebraic knowledge and were more familiar with geometry. But in modern times, the reverse is true. Hence, today's mathematicians go back to their familiar realm of Algebra and try to find the link between geometry and algebra.
Algebraicization? This is certainly a strange word to you. You probably cannot even find it in a dictionary or encyclopedia entry. However show this to any mathematician or math majors, they will immediately tell you what that is all about. Algebraicization is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture Ruler & Compasses,
"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. For this reason, problems that can be solved with ruler only are called linear problems, and those that can be solved with ruler and compasses are called quadratic problems. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only."
Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The phrase "a certain analytical process" refers to the the parallel operations of a geometric operation in the realm of algebra. The word "corresponds" means "could be translated into". In addition, he did not mean "exactly" as we take it to mean "if and only if" today. He meant "strictly conforms to". To water down his statement, we could roughly take him to mean that every step of a compass and straightedge construction can be translated into algebraic processes, which is solving linear and quadratic equations, and it turns out that the solutions to those solutions are the points, or lengths, that we can construct. Any number that is not the solution of a linear solution, a quadratic equation, a combination of both or a repeated combination of both is henceforth impossible to construct.
In order to algebraicize straightedge and compass construction, we begin by choosing a specific length to be considered one unit. Then, every time we construct a figure, we think of it instead as constructing a set of numbers representing the lengths of the constructed line segments.Compass & straightedge construction only allow us to construct points that are at intersections of lines and/or circles. This means that every time we do a construction, we can use equations of circles and lines from coordinate geometry to figure out what numbers are being constructed. In this way, a geometric process is translated into an algebraic process.
Firstly, we define "1" on a straight line by specifying the length between any two point. But some will say, hey, in Euclid's postulates, he did not say you can choose any points; a point is the intersection of two lines, two circles or one line and one circle. That is a good observation. To have a line, we have to have given points. Then we draw the straight line. Choose one of the two given points, and use that as the center for our compass. Now, spread the legs of the compass by a distance between the two given points. The intersection is a new point. With the new point, we now have one arbitrarily chosen points.
Note, the "1" does not necessarily have to measure meter nor foot since we have chosen two random points. It could measure a new length of your choosing. Then, once you have chosen that length to be "1", you have to stick to this specification throughout your construction.
Next, it is very obvious that we could construct all the integers, that is (or = ). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, and , and for the sake of this discussion and clarity, we are talking about positive integers here. Next, I will show that from and , we could construct , and .
To construct , we will use as the center and use as radius. The two points of intersection with the line will be and .
To construct , we have , , and on the straight line.
The distance between and that point is .
Similarly, we could construct .
The distance between and that point is .
I will leave the proofs to you since they are very simple using similar triangles.
Therefore, it has been proven that we could construction all the rational numbers since and are any arbitrary integers.
The natural question to ask right now is that what else is possible to construct? It is not hard to think of numbers that are not rational. For example, is constructible. Construct a unit square and the diagonal is of length . So is it possible to construct given any constructible number ? It turns out that we could. See below for method.
Again, I will leave the proof to you as well using similar triangles.
Next, we moved to the general solution of the problem.
Assume we have two points and with coordinates and . Take an arbitrary point on the line.
By similar triangle, .
Rearranging the above we have
Since , , and are constant we can express this as which is the general expression of a straight line.
Now, if we have two lines specified by four given points, ,...., with coordinates . The intersection of the two lines, will satisfy two equations
You may say that the there might not be a solution. True the two lines do not have to intersect. But if they do, we only need the operations of addition, subtraction, multiplication and division to find the point.
Now, we move onto circle. Say we have circle centered at some point with coordinates and radius . We know that the explicit expression for a circle is . Hence, if that circle intersects with one of the straight lines, then the points of intersection will satisfy
To solve for the points of intersection, we only need the operations of addition, subtraction, multiplication and division along with the extraction of square roots. Therefore, from this analysis, we have turned geometric problem into algebraic problem and come to the conclusion that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.
What I have presented above is a simplified version of the derivation towards the theorem. To see a rigorous proof of this theorem at a college level, refer to the text below which is mainly taken from I. N. Herstein's Topics in Algebra, Second Edition. You need some knowledge in Linear Algebra and/or Abstract Algebra. Also see Constructible Numbers. You should not be discouraged should you find it hard to understand. Instead, you should be marveled by the simplicity and elegance of the algebraic proof.
Now, having derived what is possible to construct, I will briefly introduce a few of the impossible constructions.
Number 2, 3 and 4 are the so-called Geometric Problems of Antiquity. Though they have been proven impossible to construct with straightedge and compass, it does not deter amateur mathematicians to come up with false proofs even today.
Proof that is impossible to trisect.
If we could trisect by compass and straightedge, then the length would be constructible. Since . Substituting and , we obtain . Thus is a root of a cubic polynomial over the rational field. Since this polynomial is irreducible over the rational field and its degree is 3, is not constructible. Thus cannot be trisected.
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