Compass and straightedge
|x + 3 = 7
x = ?
|This article/section deals with mathematical concepts appropriate for a student in early high school.|
Compass and straightedge constructions played an important role in the history of mathematics. Some constructions accomplished by the ancients led to developments in abstract mathematics. Significantly, problems that the ancients couldn't solve by these methods led to even more revolutionary developments in abstract mathematics.
To appreciate the significance of these constructions, it is important to realize that the ancients thought of mathematics in terms of geometric figures rather than the numbers, equations, and abstract things that we use today. These geometric constructions may be considered amusements today, but they were once the lifeblood of mathematics. The transition from thinking in terms of geometry to thinking in terms of numbers, formulas, and sets, happened very slowly, and wasn't really complete until the 19th century.
Rules of the game
You are given an abstract drawing surface (piece of parchment, clay tablet, patch of dirt, whatever), a writing implement (pencil, stylus, finger), a straightedge, and a compass.
- The straightedge is just a ruler with no markings. You can't measure; you can only draw straight lines.
- The compass can draw a circle through a given point, but, unlike modern compasses, it can't "remember" its radius from one circle to another. Once it is lifted off the paper, it "goes limp" and collapses.
The straightedge could be described as "infinitely long". A better description would be "as long as you need it to be", or "a construction will never fail because the straightedge wasn't long enough." The compass is similar—it can draw circles of any size.
The things you are allowed to do are
- Given any two points (you can recognize a point as the place where two lines—straight or circular—intersect), you can place the straightedge against them and draw a line passing through both.
- Given any two points, you can place the point of the compass on one of them, adjust it so that the pencil touches the other, and draw a circle. That circle will be centered on one and pass through the other.
The ancients (the Greeks were most famous for this) were able to do a remarkable amount of mathematics through reasoning about these geometric figures. Euclid's book The Elements is regarded as the best presentation of classical geometry. The methods of reasoning described by Euclid are widely used even today to teach logical reasoning and the principles of axioms, theorems, and proofs.
The mathematical "operations" that the ancient Greeks could do with these constructions include what we would now, in our number-centric world, would call addition,subtraction, multiplication, division, and square roots. They could also solve problems of proportion, bisect arbitrary angles, find areas of a many shapes, and construct some, but not all, regular polygons.
In modern times, compass-and-straightedge constructions were rediscovered by Euler, who resurrected them from the pages of Euclid's Elements and found that their simple geometric truths were a pleasant diversion from the calculus wars then raging between the Newtonian British and the Leibnizian German mathematical communities. In the twentieth century, compass-and-straightedge methods have become a popular form of recreational mathematics, and the technique is taught as a one-week unit in many middle school math classes.
The classical unsolved problems
There were three very famous geometrical problems that they couldn't solve. Listed by their popular names, they are:
- "Squaring the circle"—Given a circle, construct a straight line whose length is equal to the circle's circumference. Equivalently, construct a square with the same area. Equivalently, given a square (or rectangle), construct a circle with the same area or the same perimeter. This can be seen to be the problem of constructing the ratio π.
- "Duplicating the cube"—Given the length of the side of a cube, construct the length of the side of a cube with twice the volume. This can be seen to be the problem of constructing the cube root of 2.
- "Trisecting an angle"—Given an arbitrary angle, divide it into 3 equal angles. Some angles (e.g. 90 degrees) are easy to trisect, but the problem is to do it for any angle. The ancient Greeks could bisect arbitrary angles (hence divide them by 4, 8, 16, ...) but could not trisect them. Pythagoras himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. This problem is equivalent to solving cubic equations.
|x2 − 5x + 6 = 0
x = ?
|This article/section deals with mathematical concepts appropriate for a student in mid to late high school.|
It wasn't until nearly two thousand years later that mathematicians proved that the problems are in fact impossible. It can be shown that any quantity that can be constructed by compass and straightedge must be a number that is algebraic and has a minimal polynomial which is a power of 2. Pi is transcendental, so it is not the root of any polynomial at all. Duplicating the cube requires the solution to the equation x3 - 2 = 0, which is cubic. Trisecting angles also requires cubic equations.