Geometry and topology · Maths EE idea · Accessible
Why are there only five Platonic solids?
A research question to start from
Why are there exactly five regular convex polyhedra, and what happens to the argument if convexity is dropped?
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 short complete classification you can prove two ways and then test at its edge.
Mathematics you would need
- Angle sums at a vertex
- Euler's formula
- Diophantine inequalities
- Duality
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 classification using angles at a vertex.
- Prove it again using Euler's formula and compare.
- Discuss the Kepler–Poinsot star polyhedra.
Scope and difficulty
Accessible. Accessible; the comparison of proofs and the non-convex case carry the essay.
Pitfalls
- History of Plato.
- No second approach.
Where to start reading
Search a library catalogue or a university's open lecture notes for: five Platonic solids proof Euler formula; Kepler-Poinsot polyhedra. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).