Impossible Geometrical Constructions

"Impossible" Geometric Constructions

Three geometric construction problems from antiquity puzzled mathematicians for centuries: the trisection of an angle (dividing a given angle into three equal angles), squaring the circle (constructing a square with the same area as a given circle), and duplicating the cube (constructing a cube with twice the volume of a given cube). Are these constructions impossible? People tried for centuries to find such constructions. It was not until the development of “Abstract algebra" in the nineteenth century that it was proven these constructions were impossible. Whether these problems are possible or impossible depends on the construction "rules" you follow. In the time of Euclid, the rules for constructing these and other geometric figures allowed the use of only an unmarked straightedge and a collapsible compass. No markings for measuring were permitted on the straightedge (ruler), and the compass could not hold a setting, so it had to be thought of as collapsing when it was not in the process of actually drawing a part of a circle. Following these rules, the first two problems were proved impossible by Wantzel in 1837, although their impossibility was already known to Gauss around 1800. The third problem was proved to be impossible by Lindemann in 1882. The impossibility proofs depend on the fact that the only quantities you can obtain by doing straightedge-and-compass constructions are those you can get from the given quantities by using addition, subtraction, multiplication, division, and by taking square roots. These numbers are called Euclidean numbers, and you can think of them as the numbers that can be obtained by repeatedly solving the quadratic equation. These three problems require either taking a cube root or constructing pi. A cube root is not a Euclidean number, and Lindemann showed that pi is a transcendental number, which means that it is not the root of an algebraic equation with integer coefficients, making it too non-Euclidean.

"Impossible" Geometric Constructions Trisection of an angle Given an angle, construct an angle one third as large. The problem must be solved for an arbitrary angle. (Some angles, such as 90 degrees, can be trisected easily.) Angle trisection using straightedge and compass is equivalent to solving a cubic equation. Even when it is restricted to integer coefficients, the construction can only calculate the solution of a limited set of such equations. Since it can be shown that the equation for trisecting a 60-degree angle cannot be solved using only an unmarked ruler and compass, a general method of trisecting an angle is not possible.

Squaring (Quadrature of) the circle Given the radius of a circle, construct the side of a square of the same area. (Or, construct a square equal in area to the area of any given circle.) A circle and square have an equal area only if the ratio between a side of the square and a radius of the circle equals the square root of pi. Lindemann proved that two line segments cannot be constructed to have lengths in this ratio and therefore this method cannot square the circle.

Doubling (or Duplicating) the cube Given the side of a cube, construct the side of a cube that has twice the volume. (Also called the Delian problem.) Doubling a cube whose edge equals 1 yields the equation x3 = 2, whose solution (the length of a side of the larger cube) is the cube root of 2. The problem cannot be solved because the so-called Delian Constant (the cube root of 2) is not a Euclidean number.