Visualising Pythagoras: ultimate proofs and crazy contortions

TL;DR

Pythagoras's theorem is explored, debunking the misconception that it was discovered by Pythagoras himself. The video presents various beautiful and simple proofs of the theorem, as well as its generalizations in different shapes and dimensions.

Transcript

Welcome to another Mathologer video A squared plus B squared equals C squared. Forget about Euler's formula and company, Pythagoras's theorem beats them all in just about every conceivable way, at least in my books. Ok so finally a Mathologer video about THE theorem of theorems. My main mission today is to chase down the all-time greatest, simplest... Read More

Key Insights

• ⌛ Pythagoras's theorem was not discovered by Pythagoras but was known to the Babylonians before his time.
• 🎁 There are numerous beautiful and simple proofs of Pythagoras's theorem, with the video presenting a few examples, including those involving triangles within a square and parallelograms.
• 📔 Euclid's proof of Pythagoras's theorem, from his book "Elements," is more comprehensive and detailed than other proofs.
• 😒 Pythagoras's theorem can be generalized to other shapes and dimensions, such as through the use of areas instead of distances.
• 🇬🇵 Several interesting Pythagorean facts were mentioned, including de Gua's theorem and the existence of Pythagorean triples.
• 🔆 Pythagoras's theorem has applications beyond right-angled triangles, including in trigonometry and higher-dimensional geometry.
• 😤 A new book by the Mathologer team, featuring mathematical articles with an Australian theme, has been published by the American Mathematical Society.

Questions & Answers

Q: Was Pythagoras the first to discover Pythagoras's theorem?

No, Pythagoras did not discover the theorem. It was known to the Babylonians before his time, and there is no evidence to support the claim that he produced a rigorous proof for it.

Q: How is Euclid's proof of Pythagoras's theorem different from other proofs?

Euclid's proof, found in his book "Elements," is more rigorous and detailed than other proofs presented in the video. It uses geometric constructions and logical deductions to demonstrate the theorem's validity.

Q: Can Pythagoras's theorem be generalized to other shapes and dimensions?

Yes, Pythagoras's theorem can be generalized. For example, it can apply to any shape as long as the areas are considered instead of distances. It also holds true in higher dimensions, where it relates to volumes and hyper volumes.

Q: What are some interesting Pythagorean facts mentioned in the video?

Some fascinating Pythagorean facts include de Gua's theorem, which relates the areas of right-angled triangles and a triangular pyramid's base, and the existence of Pythagorean triples, which are sets of three positive integers satisfying the theorem.

Summary & Key Takeaways

• Pythagoras's theorem, commonly attributed to Pythagoras, was actually known to the ancient Babylonians before he was born.

• The video presents several elegant proofs of Pythagoras's theorem, including the arrangement of triangles within a square and using parallelograms, as well as Euclid's proof from his book "Elements."

• The cosine rule, which generalizes Pythagoras's theorem for all triangles, is introduced, along with the concept of Pythagorean triples and higher-dimensional versions of the theorem.