What Are Triangular Squares and How Do They Prove Irrationality?

TL;DR

Triangular squares are equilateral triangles composed of smaller equilateral triangles, with the total count of mini triangles being a perfect square. These structures reveal that square roots like root 2, root 3, root 5, and root 6 are irrational by demonstrating that the equation X² + X² + X² = Y² lacks positive integer solutions. Additionally, nearest miss solutions approximate these irrational values effectively.

Transcript

Welcome to another Mathologer video. It's something that only a few experts among you will be aware of but pretty much every single Mathologer video features fresh maths, made up just for you. Every once in a while I even sneak in some of my own little theorems and proofs. Today's video is one of those videos. What I want to show you today are some... Read More

Key Insights

• 🔺 Triangular squares are equilateral triangles made up of mini equilateral triangles, with the number of mini triangles always being a square.
• ❎ The equation X^2 + X^2 + X^2 = Y^2 is used to prove the irrationality of square roots using triangular squares.
• 👋 Nearest miss solutions provide good approximations of square roots.
• 🍹 Gauss's notebook contains a famous theorem stating that every positive integer is the sum of at most three triangular numbers.

Questions & Answers

Q: What are triangular squares?

Triangular squares are equilateral triangles made up of mini equilateral triangles. The number of mini triangles in a triangular square is always a square, equal to the total width of the big triangle in terms of mini triangles squared.

Q: How does the video prove the irrationality of square roots using triangular squares?

The video proves the irrationality of square roots by showing that the equation X^2 + X^2 + X^2 = Y^2 has no positive integer solutions, using the concept of triangular squares. This is done for square roots such as root 2, root 3, root 5, and root 6.

Q: What are nearest miss solutions?

Nearest miss solutions are ratios that provide good approximations of square roots. In the video, these ratios are shown through propeller formations and choreographies using triangular squares.

Q: Why does the smaller equation not prove that three identical triangular numbers cannot add to another triangular number?

The smaller equation does follow from the larger equation, but it does not prove that three identical triangular numbers cannot add to another triangular number. This is a puzzle presented in the video, challenging viewers to figure out the reason behind this discrepancy.

Summary & Key Takeaways

• The video introduces triangular squares, which are equilateral triangles made up of mini equilateral triangles, and explains how the number of mini triangles is always a square.

• Using triangular squares, the video proves that the square root of 3, as well as other square roots, is irrational by demonstrating that the equation X^2 + X^2 + X^2 = Y^2 has no positive integer solutions.

• The video also explores nearest miss solutions, which are ratios that provide good approximations of square roots, and showcases choreographies that illustrate these mathematical concepts.