The ancient Greeks felt that the line and the circle are the basic figures and the straightedge and compass are their physical analogues. Hence they were interested in what can be done with a straightedge and compass, i.e. what figures are Euclidean constructions. For example, given a line l and a point P on l, it is possible to construct the perpendicular to l though P using only a straightedge and compass as follows: with the compass draw a circle C0 with center P. Let A and B be the intersection points of l with C0. Next draw the circles with centers A and B which pass through B and A respectively. These two circles intersect in two points E and F say. The line segment EF is perpendicular to l.
If one is given segments AB and CD of lengths x and y respectively then one can construct segments of length x + y and x/y as follows. Let's assume for simplicity that x is less than or equal to y. Put the compass point on point A and the tip on point B. Now move the compass point to C and draw the circle of radius x. Draw the line through CD. This lines intersects the circle in two points say E and F. Then F is interior to CD then segment ED has length x + y and segment CF has length y - x. Now draw the circle of radius 1 with center at D. Also draw the perpendiculars to CD which pass through points F and D. The perpendicular through F intersects the circle of radius one in a point G. The segment GC intersects the perpendicular through F in a point H. Since triangle CFH is similar to triangle CDG, the length of FH is equal to y/x.
The Greeks knew that if one draws a square with side length x then the diagonal has side length equal to x times the square root of 2 which is the side length of a square with twice the area of the given square. So they wondered if it was possible to construct a segment of length equal to x times the positive cube root of 2 since this is the side length of a cube with twice the volume of a cube with side length x. That problem is called "doubling the cube". They also asked if it was possible to trisect any given angle (bisecting is easy) and if given any circle can one construct a square with the same area. The latter is called "squaring the circle". They could do all these things with other instruments (for example there is a linkage that can be used to trisect any angle).
It was not until 1837 that it was proven (by Wantzel and Gauss) that all three of the above problems are impossible. Today, students learn how to prove their impossibility using Galois theory. Most of the works of Evariste Galois (1811-32) were lost before anyone else read them. Before he died in a duel at age 21, he summarized his theories in a note to a friend, August Chevalier. Today his ideas are frequently applied in the fields of number theory and algebraic geometry.
Most theorems about constructions can be derived from a study of K the set of constructible numbers which is a subset of C the complex numbers. C can be identified with two dimensional real space by the map which assigns the number a + bi (where a and b are real and i is a square root of -1) to the point with coordinates (a,b). The definition of K is that it is the set of all points in C which can be obtained from straightedge and compass constructions from the initial point 0 and 1. It can be shown that if x and y are in K and y is nonzero then x + y, x-y, x*y, x/y and the square root of x (or y) is in K. So K contains the rational numbers and i. If points A, B, C and D have been constructed and X and Y are either circles or lines constructed directly from A, B, C and D then it is not hard to show that the points in the intersection of X with Y satisfy a quadratic or linear equation with coefficients that are derived from A, B, C and D by a combination of multiplication, division, addition, subtraction, and taking square roots. Hence K is the smallest subset of C containing 0 and 1 which has the property that if x and y are in K and y is nonzero then x + y, x-y, x*y, x/y and the square root of x (or y) is in K.
Doubling the cube of side length one requires constructing the cube root of 2. Squaring the circle of radius one requires constructing the square root of Pi. It can be shown using elementary trigonometry that if A is any angle then cos A = 4*cos(A/3) - 3*cos(A/3). Hence to trisect a sixty degree angle, one must construct the roots of the equation 8*x^3 - 6*x - 1. Using Galois theory (and the fact that Pi is a transcendental number) it is possible to show that the cube root of 2, the square root of Pi, and the roots of 8*x^3 - 6*x - 1 are not in K. Hence the famous three problems are impossible.
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