Geometry and topology · Maths EE idea · Accessible
Pick's theorem and lattice polygons
A research question to start from
Why does the area of a lattice polygon equal I + B/2 − 1, and why does the formula fail in three dimensions?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A surprising formula with an inductive proof and a famous counterexample in 3D.
Mathematics you would need
- Lattice points
- Triangulation
- Proof by induction
- Reeve tetrahedra
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Test the formula on polygons you draw.
- Prove it for rectangles, triangles, then all polygons.
- Show with Reeve tetrahedra that no such formula holds in 3D.
Scope and difficulty
Accessible. Accessible and self-contained.
Pitfalls
- Testing without proof.
- Skipping the triangle case.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Pick's theorem proof triangulation; Reeve tetrahedron. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).