What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented
Mathologer・18 minutes read
The Mathologer video explores visual proofs and irrational trigonometry, including puzzles on counting squares and equilateral triangles in grids. Various proofs demonstrate the absence of equilateral triangles in square grids, using properties like shifting and shrinking polygons, with applications to rational trigonometric ratios.
Insights
- Edward Lucas and Joel Hamkins presented proofs against the existence of equilateral triangles in a square grid, showcasing the limitations of geometric shapes within this context.
- The shrinking argument, originating from Willy Scherrer in 1946, demonstrates that regular polygons beyond hexagons cannot exist in square grids, highlighting the historical significance of this mathematical concept and its implications for geometric patterns.
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Recent questions
How do mathematicians prove the absence of equilateral triangles in a grid?
By utilizing visual shrinking proofs and shifting properties.
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Summary
00:00
"Grid Geometry: Counting Shapes and Proofs"
- Mathologer video featuring visual shrink proofs and irrational trigonometry
- Puzzle: Counting squares in a grid systematically
- Challenge to find a formula for the number of squares in an n x n grid
- Puzzle: Counting equilateral triangles in a grid
- Explanation of finding triangles close to equilateral in a grid
- Edward Lucas' proof that there are no equilateral triangles in a square grid
- Introduction to Joel Hamkins' visual proof against equilateral triangles in a grid
- Properties of square grids: turning and shift properties
- Proof against regular polygons in a square grid using shrinking hexagons
- Introduction to a 3D cubical grid and challenges to find shapes in it
17:17
"Shrinking Polygons in Regular Grids"
- The star argument fails for any regular n-gon where n is a power of two.
- A third shrinking argument, using the shifting property, applies to all regular n-gons beyond hexagons.
- The shrinking argument dates back to 1946 in a paper by Swiss mathematician Willy Scherrer.
- The 2D square grid only contains squares as regular polygons.
- The 3D cubical grid contains equilateral triangles, squares, and regular hexagons as regular polygons.
- Higher dimensional hypercube grids also follow the shift property, containing equilateral triangles, squares, and regular hexagons.
- Hadwiger's application of Scherrer's shrinking polygons proves that specific rational angles have rational trigonometric ratios.
