A really Satisfying Proof about Triangles on a Lattice.

Опубликовано: 03 Июль 2022
на канале: Dr. Will Wood

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In this video we'll discuss an important result in lattice geometry: All primitive lattice triangles have and area of 1/2. Incidentally, the method of proof shows that all primitive lattice parallelograms have an area of 1.

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References and Notes:

Ref. 1 This proof is entirely based on Garbett (2010) - https://documents.kenyon.edu/math/Gar...

Ref 2. This video shows how the determinant of a matrix gives the scaling of an area under that matrix transform - • The determinant | Chapter 6, Essence ...

Note 1. If (m+i,n+j) was not a visible point, then this would imply that there was a lattice point within one of the primitive triangles: a contradiction.

Note 2. Assume the image of the parallelogram was not primitive. There would be a lattice point in it's interior or on the boundary. Since the determinant of the matrix is 1, we can use the inverse of the matrix to give the original parallelogram. However, since points on or within the boundary remain so under the transform, this would imply the original parallelogram was not primitive: a contradiction.

Note 3. An interval [x,x+1] on the real line always contains an integer.

FAQ : How do you make these animations?
Animations are mostly made in Apple Keynote which has lots of functionality for animating shapes, lines, curves and text (as well as really good LaTeX). For some of the more complex animations, I use the Manim library. Editing and voiceover work in DaVinci Resolve.

Supporting the Channel.
If you would like to support me in making free mathematics tutorials then you can make a small donation over at
https://www.buymeacoffee.com/DrWillWood
Thank you so much, I hope you find the content useful.

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