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It Took 2137 Years to Solve This

Another Roof・42 minutes read

The video explores the historical challenge of constructing polygons with only a compass and straight edge over 2,000 years, detailing the mathematicians involved, the tools used, and the advancements in math concepts to achieve this. From Euclid's Elements to Gauss's contributions, the development of constructible numbers and their application in creating regular polygons are discussed, culminating in Pierre Wul's classification of constructible polygons and the resolution of ancient geometric problems.

Insights

  • The video explores the historical journey of constructing polygons with a compass and straight edge, tracing back to ancient Greece and progressing through the Middle Ages, highlighting the pivotal role of mathematicians like Euclid, Al-Khwarizmi, and Gauss in advancing geometric constructions.
  • Constructable numbers, rooted in quadratic equations and involving rational coefficients, play a crucial role in the construction of regular polygons, with complex numbers and roots of unity enabling the embedding of polygons in the complex plane, ultimately leading to Pierre Wantzel's classification of constructible polygons, resolving age-old geometric problems and solidifying his legacy in mathematical history.

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Recent questions

  • What is the history behind constructing polygons?

    The video delves into the history of constructing polygons using a compass and straight edge, tracing the challenge back 2,000 years and exploring the progress made by mathematicians from ancient Greece to the post-enlightenment era.

  • What tools are used to construct polygons?

    The tools used for constructing polygons are a straight edge for drawing line segments and a compass for creating circles and duplicating lengths, essential for constructing regular polygons with equal side lengths and angles.

  • How did ancient Greek mathematicians face limitations in constructing polygons?

    Ancient Greek mathematicians faced limitations in constructing regular polygons due to their concept of numbers and the inability to work with certain irrational numbers, influencing their progress in geometry and number theory.

  • What role do complex numbers play in constructing regular polygons?

    Complex numbers, including imaginary numbers, play a crucial role in constructing regular polygons by allowing mathematicians to embed polygons in the complex plane, generating equidistant points for various polygons through specific angles.

  • Who classified which polygons are constructible?

    Pierre Wantzel classified exactly which polygons are constructible, solving ancient questions like doubling the cube and trisecting the angle problems. His work, known as the Gauss-Wantzel theorem, corrected historical misconceptions and recognized his contributions to mathematics.

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Summary

00:00

"Constructing Polygons with Compass and Straight Edge"

  • The video discusses the problem of constructing polygons using only a compass and straight edge, a challenge that took 2,000 years to solve.
  • It delves into the history of construction problems from ancient Greece to the post-enlightenment era, exploring the mathematicians involved and their progress.
  • The term "polygon" is defined as a series of line segments forming a complete chain, with regular polygons having equal side lengths and angles.
  • The tools allowed for construction are a straight edge for drawing line segments and a compass for circles and duplicating lengths.
  • The construction of an equilateral triangle is demonstrated as the simplest regular polygon using the compass and straight edge.
  • Euclid's Elements, a foundational mathematical treatise, sets the groundwork for compass and straight edge constructions with five postulates.
  • Euclid's constructions progress from an equilateral triangle to the construction of regular polygons like the square and pentagon.
  • The method of embedding polygons in a circle is crucial for constructing regular polygons, as shown in the construction of a regular decagon.
  • The limitations faced by ancient Greek mathematicians in constructing regular polygons were tied to their concept of numbers and the inability to work with certain irrational numbers.
  • The journey from plane geometry to numbers reveals that constructible numbers involve integers, basic operations, and square roots, influencing the construction of polygons.

15:41

Constructable Numbers and Regular Polygons in Mathematics

  • Using Compass and straight edge, a circle trick can be employed to duplicate lengths around a circle, aiding in construction.
  • The side length of a regular polygon being a constructable number is a key question.
  • The Middle Ages saw advancements in understanding equations and numbers, with mathematicians like Dianas of Alexandria and Al-Khwarizmi contributing.
  • Rational and irrational numbers were accepted, with the concept of rational and irrational numbers being defined.
  • Al-Khwarizmi developed a systematic approach to solving quadratic equations.
  • Constructable numbers are those that can be reached through a chain of quadratic equations.
  • Constructable numbers with one root follow a specific form: r + S root t, which is the solution to a quadratic equation with rational coefficients.
  • Applying the square root function multiple times leads to a chain of quadratics, allowing for the construction of more complex numbers.
  • Complex numbers, including imaginary numbers, play a crucial role in mathematics, allowing for the construction of regular polygons.
  • Regular polygons can be embedded in the complex plane using complex numbers, with specific angles generating equidistant points for various polygons.

30:58

Constructing Roots of Unity with Gauss's Method

  • Multiplying all terms by Z increases powers by one and subtracts the original sum.
  • Most terms cancel out, leaving only Z to the N and a minus one.
  • Z to the N equals 1, resulting in the final calculation equaling zero.
  • Gauss aimed to construct a root of unity using quadratic equations for constructible numbers.
  • Gauss split roots of unity into groupings with convenient properties.
  • Gauss's method works for nth roots of unity where n is prime.
  • Gauss applied his method to a heptadecagon with 17 sides.
  • Squaring operations repeat every eight steps, creating periodic groupings of roots.
  • Nice properties of sums and products of roots allow for the formation of a chain of quadratics.
  • Gauss proved the constructibility of the heptadecagon using quadratic equations and known coefficients.

44:41

Wul's Theorem Solves Ancient Geometric Problems

  • Pierre Wul classified exactly which polygons are constructible, solving ancient questions like doubling the cube and tricing the angle problems; his work, known as the Gaus-Wantel theorem, corrected history and gave him recognition for his contributions.
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