Number theory · Maths EE idea · Solid
Farey sequences and Ford circles
A research question to start from
Why do neighbouring fractions in a Farey sequence satisfy bc − ad = 1, and how do Ford circles turn this into geometry?
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 number-theory result with a striking geometric picture: two representations you can connect in one argument.
Mathematics you would need
- Fractions in lowest terms
- Mediants
- Tangent circles and coordinate geometry
- Proof by induction
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
- Build Farey sequences by hand and observe the mediant property.
- Prove the neighbour property by induction on the order.
- Define Ford circles and prove that circles of neighbouring fractions are tangent.
- Use the picture to explain a result about rational approximation.
Scope and difficulty
Solid. Solid and self-contained.
Pitfalls
- Pretty diagrams without proofs.
- Losing the thread between the two parts.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Farey sequence mediant property; Ford circles tangent. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).