Classical problems of Greek geometry

Summary

The ancient Greeks were fascinated by geometry and faced three unsolvable construction problems: trisecting an angle, squaring the circle, and duplicating the cube, using only a straight edge and compass. Despite progress by mathematicians like Carl Gauss, it was ultimately proven impossible to solve these problems with the given tools. Analytic geometry, developed later, showed that the problems involved polynomials of degrees that couldn't be constructed with straight edge and compass, and the quest to solve them was finally laid to rest.

Takeaways

• 📐 The ancient Greeks were fascinated by geometry and had three major construction problems that they couldn't solve using only a straight edge and compass.
• 🔍 One of the problems was angle trisection, which asks if an arbitrary angle can be divided into three equal parts using a straight edge and compass.
• 🔳 The second problem was squaring the circle, which questions whether it's possible to construct a square with the same area as a given circle using only a straight edge and compass.
• 🔶 The third problem was duplicating the cube, which involves constructing a cube with twice the volume of a given cube using a straight edge and compass.
• 👨‍🎓 Carl Gauss, a German mathematician, made significant progress by constructing 17-sided polygons and determining which polygons could not be constructed using his method.
• 🚫 Pierre Wantzel, a French mathematician, proved in 1837 that angle trisection and duplicating the cube were impossible with a straight edge and compass.
• 📉 Analytic geometry, pioneered by René Descartes and Pierre de Fermat, was crucial in understanding the limitations of construction problems and involved representing points with Cartesian coordinates.
• 🔢 In analytic geometry, the constructibility of polygons was related to the roots of polynomials, with Gauss finding a way to construct distances whose polynomials had degrees equal to a power of two.
• ✅ Wantzel's proof showed that angle trisection and cube duplication led to cubic polynomials, which are of degree 3, and thus were impossible to construct with the allowed tools.
• ❌ Squaring the circle remained unresolved until Ferdinand Lindemann proved in 1882 that pi is transcendental, meaning it is not the root of any polynomial, making the construction impossible.

Q & A

What tools were the ancient Greeks limited to in their geometric constructions?

The ancient Greeks were limited to using only a straight edge (unmarked ruler) and a compass for their geometric constructions.

What types of polygons could the Greeks construct using just a straight edge and compass?

The Greeks could construct equilateral triangles, squares, regular pentagons, 15-sided polygons, and polygons with double the sides of basic constructible ones, such as 30-sided and 60-sided polygons.

What were the three major geometric construction problems that the ancient Greeks struggled with?

The three major problems were: 1) Trisecting an arbitrary angle, 2) Squaring the circle (constructing a square with the same area as a given circle), and 3) Duplicating the cube (constructing a cube with twice the volume of a given cube).

Why was trisecting an arbitrary angle difficult for the Greeks?

While some angles, such as a right angle, were easy to trisect, trisecting arbitrary angles using just a straight edge and compass proved difficult and resisted the Greeks' efforts. They could trisect an angle with a marked ruler but not with an unmarked straight edge.

What was the significance of Hippocrates of Chios in the attempt to square the circle?

Hippocrates of Chios made early progress toward squaring the circle by proving that a specific crescent shape, called a lune, has the same area as a triangle. However, his work did not lead to a full solution for squaring the circle.

Why was duplicating the cube considered an unsolved problem for the Greeks?

Duplicating the cube, or constructing a cube with twice the volume of a given cube, was impossible using only a straight edge and compass. Like angle trisection, the Greeks found it possible with a marked ruler but not otherwise.

What contribution did Carl Gauss make to geometric constructions?

Carl Gauss, in 1796, found a way to construct a 17-sided polygon and polygons with multiples of 17 sides (e.g., 34, 51, 68). He also proved which polygons could not be constructed using a straight edge and compass, such as heptagons and nonagons.

How did Pierre Wantzel contribute to solving the classic Greek geometric problems?

Pierre Wantzel, in 1837, proved that both trisecting the angle and duplicating the cube were impossible using just a straight edge and compass. He showed that these problems involved cubic polynomials, which could not be solved with the tools allowed by the Greeks.

What role did analytic geometry play in solving the ancient Greek construction problems?

Analytic geometry, developed by René Descartes and Pierre de Fermat, provided a way to represent geometric problems algebraically. This allowed mathematicians like Gauss and Wantzel to use polynomials to determine which constructions were possible with a straight edge and compass.

Why is squaring the circle impossible, according to modern mathematics?

Squaring the circle is impossible because it would require the number π (pi) to be the root of a polynomial with degree 2, which is not the case. In 1882, Ferdinand Lindemann proved that π is a transcendental number, meaning it is not the root of any polynomial.