The Science of Comprehension

How We Understand Abstract Concepts

Part IV—B

Impossible Geometric Constructions

The concept of there being an equation which is unreachable within a system reminds me of the impossibility of certain geometric constructions using only a straightedge and compass, so I'll digress here briefly to give a conceptual overview of how that is proved. The following excerpt is from Charles C. Pinter A Book of Abstract Algebra:

Greek geometers devised ways to cleverly use the ruler and compass, unaided by any other instrument, to perform all kinds of intricate and beautiful constructions. They were so successful that it was hard to believe they were unable to perform three little tasks which, at first sight, appear to be very simple:
  1. Doubling the area of a cube
  2. Trisecting an arbitrary angle
  3. Squaring the circle
The first task demands that a cube be constructed having twice the volume of a given cube. The second asks that any angle be divided into three equal parts. The third requires the construction of a square whose area is equal to that of a given circle. Remember, only a ruler and compass are to be used!

Mathematicians, in Greek antiquity and throughout the Renaissance, devoted a great deal of attention to these problems, and came up with many brilliant ideas. But they never found ways of performing the above three constructions. This is not surprising, for these constructions are impossible! Of course, the Greeks had no way of knowing that fact, for the mathematical machinery needed to prove that these constructions are impossible—in fact, the very notion that one could prove a construction to be impossible—was still two millennia away.

The final resolution of these problems, by proving that the requested constructions are impossible, came from a most unlikely source: it was a by-product of the arcane study of field extensions, in the upper reaches of Modern Algebra.1

The basic idea to proving that these geometric constructions can not be done is to first convert the geometric problem to an algebraic problem:

Doubling the area of a cube  –  Given a cube with a side of length 1, construct another cube with a side of length 3root2
Squaring the circle  –  Given a circle of radius 1, construct a square with side length of rootpi
Trisecting an arbitrary angle  –  It suffices to come up with one counterexample. Given an angle of 60°, construct an angle of 20°. This corresponds to constructing a line of length cos 20°.

The geometric constructions have been transformed into algebraic equations:

Doubling the area of a cube  –  Construct a line of length 3root2
Squaring the circle  –  Construct a line of length rootpi
Trisecting an arbitrary angle  –  Construct a line of length cos 20°.

Constructible Numbers
Consider what points on a grid we can reach by using just a straightedge and compass, given a line of length 1 to start with. If we can construct lines of length a and b, then we can also construct lines of length
a + b, a - b, ab,
a/b (if b ≠ 0)

Since we can construct all lines with length a/b, we can construct all the rational numbers. All rational numbers are reachable using straightedge and compass.

The only other thing we can do is obtain the length of a diagonal line between two points on the grid.

Since for a triangle a2 + b2 = c2 the length of our new line c will be a2b2

Thus any constructible real number is a number obtainable by using the square root a finite number of times.

The proof that the numbers 3root2 , cos 20°, and rootpi , can not be reached by using a finite number of square roots requires a lot of Abstract Algebra (which I think is a fun and interesting subject, but I won't cover it here).

The conceptual point I want to make though is that in order to prove these geometric constructions are not possible using only a straightedge and compass, we first transform the geometric problem into an algebraic problem. This is conceptually similar to converting equations into unique numbers, which is what Gödel did. We are taking the problem out of one field and into another where it is easier for us to handle.

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1. Charles C. Pinter A Book of Abstract Algebra (second edition, 1990)