Collapsing compass and infinite ruler?
Geometry is one of the oldest branches of mathematics: the roots go back to the ancient Greeks. Euclid and his coevals achieved brilliant results in this field, in particular, related to geometric constructions. “Geometric construction” means a process coming from a need to create certain objects in our proofs.
A construction is, in some sense, a physical substantiation of the abstract. The construction process describes how to “draw” figures from some given data if we can use only two special tools: a straightedge, which is an idealized ruler, and a compass.
To “draw” the figure means that, using the given data and the two tools we denote in the plane, or in the space sufficiently many points of the desired object such that a detailed plan, or blueprint can be created for the “builders” to “implement” the object. All compass and straightedge constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are:
- Creating the line through two existing points.
- Creating the circle through one point with center another point.
- Creating the point which is the intersection of two existing, non-parallel lines.
- Creating the one or two points in the intersection of a line and a circle (if they intersect).
- Creating the one or two points in the intersection of two circles (if they intersect).
The Greeks were very funny guys: they made their own life complicated since they set up some additional requirements about the tools and about the process. The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn starting from two given points: the center and a point on the circle.
Well, it’s fine – but the compass may collapse when it’s not drawing a circle! So, it is not allowed to “copy” distances in the obvious way: as soon as we lift up the compass from the paper it collapses. Although the straightedge is infinitely long, but, unfortunately it has no markings on it: no distances can be measured. In addition, it has only one straight edge, unlike ordinary rulers. It’s full capacity is to draw a line segment between two points or to extend an existing segment.
About the whole construction procedure we assume that it is exact: “eyeballing” is not allowed. Also, the process must terminate, that is, it must have a finite number of steps.
A typical and simple example is the construction of an equilateral triangle. Starting with just two distinct points, we can create a line through the points and two circles, using each point as center and passing through the other point. If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle.
Of course, Euclid and his friends, Hippocrates, Menaechmus and Hippia, among others were not satisfied with just constructing different triangles. They discovered how to construct sums, differences, products, ratios, and square roots of given lengths.
The most-used compass-and-straightedge constructions include:
- Constructing the perpendicular bisector from a segment.
- Finding the midpoint of a segment.
- Drawing a perpendicular line from a point to a line.
- Bisecting an angle.
- Mirroring a point in a line.
- Constructing a line through a point tangent to a circle.
- Constructing a circle through 3 noncollinear points.
Still they came a cropper with some jobs which were considered as the most famous construction problems:
- Squaring the circle: drawing a square the same area as a given circle.
- Doubling the cube: drawing a cube with twice the volume of a given cube.
- Trisecting the angle: dividing a given angle into three smaller angles all of the same size.
Of course, the question arises: weren’t the Greeks smart enough to complete these constructions, or it is simply impossible to complete them? For 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible. But how can we apply here any mathematical rules?
The key word is here: the constructible number. We call the number constructible, if there are two constructible points such that the length of the line segment is . Here we use the concept of constructible point: these are those points which can be constructed with the two basic tools applying a process subjected to the above mentioned rules. In any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry is equivalent to an axiomatic algebra, replacing its elements by symbols. Carl Friedrich Gauss, “Princeps mathematicorum” (in Latin), “the foremost of mathematicians” was the first who realized this, and used it to prove the impossibility of some constructions.
Although Gauss might have been the “Princeps” there was a “Prince”, too: the Prince “whom the Gods love” – a young man, lived less than 21 years only. Évariste Galois, the French kid who started to get bored in school at the age of 14, and began to take a serious interest in mathematics. That time he could not suspect that he wouldn’t have a long career: after 7 years he died from wounds suffered in a duel under questionable circumstances. Still his brilliant heritage is one of the most amazing stories in mathematics: he founded his theory which we call now Galois theory.
Apart from the beauty and depth of his theory it turned out that, as a side result, based on his results rigorous mathematical proofs can be given for the impossibility of different geometric constructions, like the three above. Also, his theory proved famous results about the impossibility to set up a solution formula for algebraic equations of degree greater than .
Most of his work is too complicated to explain or just mention here. “Don’t cry, Alfred! I need all my courage to die at twenty ” – these were his last words to his younger brother. He came and left like a comet. He was buried in a common grave of the Montparnasse cemetery whose exact location is unknown.
Several years after his death leading mathematicians started to recognize the importance of his monumental work in maths. Nowadays it is basic material in advanced algebra courses, not for the beginners. The freshmen just meet simple problems related geometric constructions, sometimes they are fighting with those, sometimes fail with them – if you want to test your skills on such problems, then look for the Geometry Problem Sheet on the page.