Geometry and topology · Maths EE idea · Solid
Which pentagons tile the plane?
A research question to start from
What conditions on angles and sides allow a convex pentagon to tile the plane, and how can a given family be proved to tile?
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 modern story with concrete conditions you can verify with geometry.
Mathematics you would need
- Angle conditions at vertices
- Isometries
- Periodic tilings
- Geometric constraints
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
- Show why regular pentagons cannot tile.
- Prove that one or two known families tile, with your own diagrams.
- Discuss how the classification was completed and what was proved by computer.
Scope and difficulty
Solid. Solid; do not attempt the whole classification.
Pitfalls
- Listing the 15 types.
- Diagrams without proof.
Where to start reading
Search a library catalogue or a university's open lecture notes for: pentagon tilings types conditions; convex pentagon tiling proof. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).