Geometry and topology · Maths EE idea · Accessible
V − E + F = 2, and when it fails
A research question to start from
Why does Euler's formula V − E + F = 2 hold for convex polyhedra, and how does it change for polyhedra with holes?
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 classic proof with famous flawed proofs to evaluate, and an extension to the torus.
Mathematics you would need
- Planar graphs
- Proof by induction
- Polyhedra
- Euler characteristic
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
- Verify the formula on polyhedra you build.
- Prove it for connected planar graphs.
- Examine counterexamples (polyhedra with tunnels) and find the general formula.
Scope and difficulty
Accessible. Accessible; the counterexamples bring depth.
Pitfalls
- Only checking examples.
- Hidden assumptions in the proof.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Euler characteristic proof planar graph; polyhedra with holes Euler formula. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).