Inside or out? Proving the escape radius for Julia sets

AA HL Ambitious Also in: Pure maths

Research question

For the iteration z → z² + c with complex numbers, why does any point that ever gets further than 2 from the origin escape to infinity, and how does the shape of the filled Julia set change as c moves along the real axis?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A proof about complex numbers with a practical pay-off: it is the stopping rule every fractal program uses. Exploring real values of c connects to fixed points you can calculate.

The mathematics you'll need

Complex number arithmetic and modulus
The triangle inequality
Proof of the escape criterion |z| > max(2, |c|)
Fixed points of z² + c and their stability
Iteration on the real line as a special case

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; iterate in a spreadsheet or GeoGebra.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain complex iteration with examples.
Prove the escape criterion.
Explore real c and find where the set becomes disconnected.
Find fixed points and decide stability.
Reflect on what a computer picture can and can't show.

Pitfalls that cost marks

Using the escape rule without proving it.
Overlap with existing Mandelbrot write-ups; keep the question about Julia sets and your own c values.
Images with no mathematics.

Showing personal engagement

Choose c values that matter to your question.
Predict the shape before plotting.
Check your proof with extreme examples.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Complex number arithmetic and modulus; The triangle inequality
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Generate the fractal or the iteration yourself, choose your own variation (a different rule, angle or starting value), and pursue a question you raised while exploring.

Reflection (D)

Reflect on the limits of the model: real coastlines and plants are only self-similar over a range of scales, and computer iterations carry rounding error. Say how that affects your numbers. For this idea, start with: using the escape rule without proving it — say how it affects your answer.

Use of mathematics (E)

SL: Geometric sequences and series for lengths and areas, logarithms for dimension, and iteration of functions, each calculated and checked numerically.

HL: Proof of a limit or dimension, complex-number iteration with a derived condition, or analysis of fixed points and their stability using derivatives.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Investigate period-2 points and when they are stable, or compare with the Mandelbrot set along the real axis.

Extending it for HL

Find fixed points and decide their stability with derivatives, or prove the limit you found numerically.

See a complete IA, marked

Our annotated exemplar How far can a stack of books lean over the edge of a table? (AA HL) asks a different question, but shows how a complete fractals & chaos exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).