Geometry and topology · Maths EE idea · Solid
The best shape for a fence
A research question to start from
Among polygons with n sides and a fixed perimeter, why does the regular polygon enclose the greatest area?
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 precise optimisation you can prove for triangles and quadrilaterals, then generalise.
Mathematics you would need
- Heron's formula
- Calculus optimisation
- Inequalities (AM–GM)
- Symmetry arguments
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
- Prove the result for triangles with Heron's formula.
- Extend to quadrilaterals.
- Discuss the general n-gon and the circle as a limit.
Scope and difficulty
Solid. Solid.
Pitfalls
- Assuming the maximum exists.
- Numerical search instead of proof.
Where to start reading
Search a library catalogue or a university's open lecture notes for: isoperimetric inequality polygons; Heron formula maximum area AM-GM. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).