Welcome back to Numberholic! I’m your host, Sejari, and it’s always a pleasure to have you here. Today, we’re diving into the fourteenth episode of Proven Yet Hard to Believe.
Today’s topic is related to the problem of constructing figures.
The ancient Greeks believed that “the most perfect shapes are the straight line and the circle, and thus, the gods hold these in the highest regard.” Based on this belief, they restricted their geometric constructions to the use of only a straightedge and a compass, which could draw only straight lines and circles. One of their remarkable achievements was the construction of a regular pentagon.
However, certain problems proved difficult to solve using only a straightedge and compass.
There were three particularly challenging problems:
Doubling the volume of a cube.
Trisecting an arbitrary angle.
Constructing a square with the same area as a given circle.
These difficult construction problems contributed greatly to the development of geometry. In this video, we will focus on the problem of doubling the cube. The first significant breakthrough came around 440 BCE, when Hippocrates of Chios reformulated the problem into one of constructing two mean proportionals between two given line segments of lengths s and 2s:
1 : x = x : y = y : 2,
x^2 = y, y^2 = 2x,
x^3 = 2
After Hippocrates’ reformulation, the problem of doubling the cube was reduced to finding a method for constructing the two mean proportionals. But … that doesn’t mean the problem has been solved easily. Since this was not feasible using only a straightedge and compass, various alternative solutions were devised by Greek mathematicians.
Among those who tackled these problems was the great philosopher Plato. He devised a mechanical instrument to determine the side length of a cube with double the volume.
His device consisted of a fixed right-angled structure with two perpendicular moving arms, one attached to each leg of the structure. Additionally, there was a cross-shaped ruler with markings in a 1:2 ratio. By fitting the cross-shaped ruler into the structure’s arms and adjusting its position so that the moving arms met precisely on the ruler, one could determine the required side length.
However, Plato ultimately rejected this approach, stating, “Mathematics should be pursued through reasoning alone, without reliance on mechanical tools—only with a straightedge and compass can a problem be meaningfully solved.”
A completely new approach emerged over two millennia later in Europe, where construction problems were translated into algebraic problems. This transformation began with René Descartes, who introduced the coordinate plane. Finally, in 1830, a young French mathematician, Évariste Galois, made a groundbreaking discovery, proving that some constructions—including doubling the cube—were impossible.
European mathematicians converted geometric construction problems into algebraic ones, utilizing the properties of numbers and polynomials.
Let’s explore how geometry transitioned into algebra.
Constructions using a straightedge and compass ultimately involve finding intersections: intersections of lines, of a line and a circle, and of two circles. The process begins by setting a unit length (1) and defining a reference point. From there, new points are determined by drawing lines and circles and finding their intersections.
The intersection of two lines corresponds to solving a linear equation. The intersection of a line and a circle, or two circles, corresponds to solving a quadratic equation.
Arithmetic operations can be represented geometrically through construction: addition, subtraction, multiplication, and division. The quadratic formula, which involves square roots, is fundamental in this process. Geometric constructions can therefore be viewed as a sequence of operations that expand a set of numbers.
Starting from the natural numbers , performing arithmetic operations repeatedly leads to the set of rational numbers (Q). Introducing square roots, like square root of 2, forms an extended set, denoted Q(√2), and further applications of arithmetic and square root operations create an ever-expanding number set.
By continuing this step-by-step process, increasingly complex numbers can be constructed. But can all numbers be constructed in this way?
To analyze constructible numbers, we introduce a labeling system: the degree of the polynomial for which the number is a root.
Rational numbers are roots of first-degree polynomials.
Numbers involving one square root have a degree of 2.
Numbers involving nested square roots have a degree of 4.
More generally, the degree follows powers of 2.
Starting from rational numbers, adding numbers of degree 2 forms a new layer, followed by another layer of numbers with degree 4, and so on. Thus, a constructible number must have a degree that is a power of 2.
Thus, a number’s constructibility is determined by the degree of its minimal polynomial, which must be a power of 2. However, another problem arises: what if two polynomials are multiplied, increasing their degree? For example:(…)
Here, an eighth-degree polynomial is formed, rendering our labeling system ineffective.
To address this, we ensure that the polynomials used are irreducible, meaning they cannot be factored further. This criterion confirms that the degree of a constructible number remains a power of 2.
Galois provided the key insight: all roots of an irreducible polynomial are symmetrically equivalent. For example, in x^2 – 2 = 0, both roots (√2 and -√2) behave identically in arithmetic operations. This symmetry principle underlies Galois theory, which ultimately proved that certain polynomials, such as x^3 – 2, cannot be solved using only arithmetic and square roots.
Returning to the cube duplication problem, solving x^3 – 2 = 0 requires extracting the cube root of 2. Since this polynomial is irreducible over the rational numbers, the cube root of 2 is not constructible with a straightedge and compass.
Of course, these conclusions were only reached more than two thousand years after the Greeks first posed the problem.
That’s all for today’s episode of Proven Yet Hard to Believe! Until next time, keep exploring, keep questioning, and stay Numberholic!