What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented

TL;DR

Explore visual shrink proofs and the absence of regular polygons in square grids, cubic grids, and higher dimensions.

Transcript

welcome to another Mathologer video. today I'll animate some absolutely gorgeous visual shrink proofs for you. really nice super accessible and at the same time deep mathematics plus at the end I'll prove some really cool irrational trigonometry for you. something you may have always felt in your bones but never captured in words. but what's that? ... Read More

Key Insights

• ❎ Shrink proofs systematically demonstrate that regular polygons, except for squares, do not exist in square grids.
• 🔺 Equilateral triangles and regular hexagons can be found in 3D cubic grids, but other regular polygons are absent.
• ✋ Willy Scherrer's shrinking argument proves the absence of regular polygons higher than hexagons in cubic grids.

Questions & Answers

Q: How are shrink proofs used to show the absence of regular polygons in square grids?

Shrink proofs involve systematically shrinking regular polygons within a grid. By demonstrating that polygons become too small to fit within the grid's spacing, it is proven that regular polygons, except for squares, do not exist in square grids.

Q: Why do equilateral triangles and regular hexagons exist in cubic grids?

Equilateral triangles and regular hexagons exist in cubic grids because they can be constructed using the shifting property. However, other regular polygons, such as pentagons or heptagons, do not exist in cubic grids.

Q: What is the significance of Willy Scherrer's shrinking argument?

Willy Scherrer's shrinking argument proves that regular polygons higher than hexagons do not exist in cubic grids. This argument also applies to higher dimensional hypercube grids.

Q: How do shrinking proofs relate to trigonometric ratios?

Certain angles in the list of rational angles with simple trigonometric ratios (0, 30, 45, 60, 90, etc.) are proven to be the only rational angles with at least one rational trigonometric ratio. This is shown using shrinking arguments and the assumption of rational cosines.

Summary & Key Takeaways

• The video introduces the concept of shrink proofs and challenges viewers to count squares and equilateral triangles in a grid systematically.

• The speaker demonstrates various visual shrink proofs to show that equilateral triangles and regular polygons do not exist in square grids.

• The concept is extended to 3D cubic grids, where equilateral triangles and regular hexagons are present, but other regular polygons are not.

• The speaker presents Willy Scherrer's shrinking argument and explains how it proves the absence of regular polygons beyond hexagons in cubic grids.

• The proof is applied to angles and trigonometric ratios, showing that the cosines of certain rational angles are irrational.