Constructions with a ruler and a compass are classical problems in plane geometry, which go back at least as far as the Ancient Greeks, and which have kept mathematicians around the world busy until the 19th Century. Loosely speaking, it involves finding ways to draw geometrical pictures using only an unmarked ruler, or straight edge, and a compass. A more precise definition will be given later; for the moment let us just keep this simple point of view.
In ancient times, pretty much all of mathematics was geometry. In around 300 BC, Euclid wrote Elements, a series of books containing probably all the mathematics known at that time, and he devoted the largest part of it to geometry.
The procedure followed by Euclid was as follows: he first introduced primitive terms - self-evident notions which are not defined by means of other terms, such as points and straight lines - and then a number of axioms, which describe the way the primitive terms relate to each other. The following two axioms are particularly relevant to construction problems:
Axiom 1: We can draw a straight line between any two points.
Axiom 3: We can draw a circle with a given radius around any point.
The straight edge and compass enable us to carry out in practice the operations described in Axioms 1 and 3.
Along with the capability of drawing lines and circles came the ability to construct points - these simply arise at the intersection of two lines, or two circles, or a line and a circle. Obviously, the intersecting figures shouldn't be drawn at random, and we will give a precise definition after describing an example.
An Example of Construction
Given two points A and B, we wish to construct a point obtained from B after a quarter turn around A. Here is the detailed procedure:
Draw the line (AB) passing through A and B.
Draw the circle Γ centred on A, with radius equal to the distance between A and B.
This circle intersects the line (AB) twice: once at B, once at a different point which we call B'.
Draw two circles, centred on B and B' respectively, with their radii equal to the distance between B and B'.
These two circles intersect at two separate points, say P and P'.
Draw the line (PP') joining these points.
The very first circle which was drawn (that is, Γ) meets the line (PP') at two points, say C and C'.
And so we have done it: either of the points C or C' is obtained from B after a quarter turn around A - one clockwise, the other anti-clockwise.
Now we are left to formulate in an abstract way the intuitive notion illustrated by the procedure above. You start with two given, separate points; by convention we agree that these points are constructed. Then, repeat the following steps as many times as you wish:
You can draw a line joining two constructed points.
You can draw a circle centred on a constructed point, with a radius given by the distance between two constructed points.
Any point which appears as the intersection between a line or circle constructed earlier is also called constructed.
Finally, any point on the plane which can be constructed by means of this procedure is called constructible.
There are two common variations of the construction problem. The first consists of giving a more elaborate figure to start with, rather than just two points. Typically, one can start with a triangle; this leads to questions such as the construction of its centre of gravity, or its circumcircle (the circle passing through all three of its vertices).
The second variation is the construction of more elaborate figures, for example straight lines or polygons. Technically, this just means constructing the points defining the figure. So for instance, constructing a square just means constructing its four vertices. Of course, if you are drawing a picture it is advisable to also draw the sides because it looks better.
In this section we give some known construction problems. We will not go into the detail of their solution, which is either similar to the example given above or too elaborate to describe. These constructions have been known since ancient times.
Equilateral triangles and squares.
Octagons, hexagons, and pentagons.
More difficulties arose when trying to deal with the following problems:
Given an integer n, construct a regular polygon having n equal sides3.
Trisection of an angle: given an angle, construct two straight lines dividing the angle into three equal parts.
Squaring the circle: given a circle, construct a square which has the same area as the surface encompassed by the circle.
Despite numerous attempts over the centuries to solve the last two problems, nobody ever came up with a correct construction. In the 19th century though, these questions were finally answered in a rather unexpected fashion, namely it was proven that trisecting the angle or squaring the circle are impossible using only a ruler and compass. For the polygon problem, the answer depends on the number of sides. For example, polygons with 7 ,9 or 42 sides are not constructible, but polygons with 17 or 257 sides are. The general answer involves Fermat numbers. In the next section, we will outline how mathematicians managed to prove that trisecting the angle and squaring the circle are impossible with a ruler and compass.
From Geometry to Algebra
What is Algebra?
While we certainly seem to have a good understanding of what 'geometry' is, it might be less clear what is meant by 'algebra', and why it should be of any help anyway. Let us briefly address both issues.
Roughly speaking, algebra is a field of mathematics dealing with quantities which you can add and multiply, pretty much in the same way that you do with ordinary numbers. In the context of this Entry, it will indeed be numbers which are involved, so we won't need to elaborate the notion any further.
Algebra can be used to translate geometric questions into equations, and this allows us to prove difficult results which seem intractable from a purely geometric point of view. Typically, to prove that a geometric construction is not possible you end up having to prove that some algebraic equations have no solution, which is usually much easier to do.
Now we explain how to obtain algebra (for us, this means numbers) from geometry. Since we are dealing with constructible points, we will want to define something like 'constructible numbers'.
First we need some considerations regarding distances in the plane. Remember that when speaking of constructible points, we start by giving two separate points A and B which we declare arbitrarily to be constructible. Now we also declare them to be at 'distance 1'. Then all other distances will be given by comparison to this one. For example, the middle point of the segment [AB] is at a distance 0.5 from both A and B alike. Or, if you draw a square the sides of which have length 1, then its diagonal has length √2.
The easiest way to attach numbers to constructible points is as follows. Define a number to be constructible if its absolute value4 is given by the distance between two constructible points. In other words, wherever you have two constructible points A and B, the distance between A and B is a constructible number. From what we said above, 1 and √2 are constructible numbers.
Properties of Constructible Numbers
The reason why constructible numbers are well suited for computations is that they enjoy the following remarkable properties5.
If the numbers x and y are constructible, then x + y and x - y are constructible.
If x and y are constructible, then the product xy is constructible. If y is non-zero, then also x/y is constructible.
If x is constructible, then √x is constructible.
While the first property is very easy to see, the second and third ones are more difficult.
With these properties, it is easy to see that any number which can be written out using only integers, the four standard operations and square roots is constructible. For example, this is a constructible number. A more important feature is that all constructible numbers can be written out in this fashion. This property can be described in more abstract but mathematically useful terms, which makes it possible, using standard algebraic techniques, to check whether a given number is constructible or not.
So What About Squaring the Circle Then, eh?
Now that we are able, in theory at least, to decide whether a number is constructible or not, how can this help us with trisecting the angle or squaring the circle? We use a proof by contradiction: assume that you can square the circle (or trisect the angle); then you compute the distance between two points which you can construct from that situation; finally you show that the distance you obtained is actually not a constructible number.
Let us illustrate the method in the squaring the circle problem. Assume you can construct a square the area of which is the same as the one encompassed by a circle of radius 1. Recall that the area of such a surface is equal to π. A square with the same surface would have a side of length √π. By definition this would imply that √π, and hence π itself is a constructible number. But this is known6 not to be the case.
Changing the Rules of the Game
The Napoleon Problem
It is said that French emperor Napoleon Bonaparte was a keen amateur mathematician, and that he would enjoy a little geometry on the eve of battles. The following result is sometimes referred to as Napoleon's construction, though it is more likely the brainchild of the Italian mathematician L Mascheroni7, whom Napoleon met while campaigning in Italy:
It is possible to construct the centre of a circle using only the compass.
In other words, assume that you are given a circle but no mark of its centre (say, you take a beer glass, put it upside down on a paper and draw a circle around the rim with a pencil). Then, using only a compass, ie drawing only circles, you can locate the centre of the circle8.
Actually, Mascheroni proves an even better theorem:
Any point which is constructible with a ruler and a compass can be also constructed using only the compass.
Of course, this doesn't mean that you can draw straight lines with a compass; nevertheless, in order to construct points, the ruler appears to be superfluous, albeit convenient.
The Ruler Alone
If you can forget so easily about the ruler and use a compass alone, what about doing it the other way round, forget the compass and use only an unmarked ruler? For one thing, you need to give more than 2 points to start with (usually 3 points); but even so you cannot construct as many points as you could using a compass. For example, you cannot construct the middle point of a segment with only a ruler.
Marked Rulers and Folding Papers
The last type of instrument we would like to mention is the marked ruler. Actually, unlike the rulers you can buy at your local stationery shop, it comes with no marking at all - but then you are allowed to report on it the distance between any two points which you have managed to construct. But no cheating: you have to construct points first if you want to mark their distance on the ruler.
What can you do with a marked ruler which you cannot do with an unmarked one? Assume that you have two straight lines L and L', and a point A outside of these. Also, assume that you have two markings on your ruler at some distance d. What you are allowed to do now9, but couldn't do before, is to draw a straight line passing through A and intersecting the lines L and L' at points B and B' in such a way that the distance between B and B' is precisely d.
With this instrument (and a compass), one can show that the problem of trisection of the angle becomes possible; the construction is referred to as Pappus's construction10, although it was already known to Hippocrates11. Thus, you can definitely construct more points with a marked ruler than with an unmarked one.
In 1980, H Abe provided a similar construction for trisecting the angle by folding a piece of paper.
Constructing Points: What is it Good for?
These constructions are very nice, but what are they good for really? Practically speaking, not much. Because of inaccuracies, due for instance to the thickness of the pencil one uses for the pictures, whatever construction one makes is approximative anyway. Thus, all one has to do to construct anything in practice is to construct it with sufficient precision, so that the error due to approximate construction is lower than the error due to use of pencil.
Geometrical constructions have been used for aesthetic reasons. In architecture, for instance, one can find geometrical features in the structure of cathedrals; also, the compass is featured on the symbol for Freemasonry. Of course, geometry plays a fundamental role in classical and modern painting alike, from da Vinci and Dürer to Picasso and Kandinsky. Then again, one might just be appealed by the abstract beauty of the underlying mathematics.
Finally, geometrical problems are more often than not a source of inspiration and motivation to develop further mathematical theories (such as Galois theory, field theory or calculus) in order to solve geometrical or practical problems. These in turn can be further developed, with practical, and sometimes unexpected, applications.