Fold a strip, unfold it: the dragon curve

AA SLAA HL Solid Also in: Pure maths, Art & music

Research question

If you fold a strip of paper in half n times and open every fold to a right angle, what sequence of left and right turns do you get, and can you predict the nth turn and prove the curve never overlaps itself?

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

You can make every stage with paper, so the investigation begins with your own hands. The turn sequence has a pattern you can find, describe with a rule and prove.

The mathematics you'll need

Sequences generated by a recursive rule
Predicting the nth term from its binary representation
Proof by induction
Transformations (rotation by 90°)
Counting segments and the curve's bounding box

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; fold paper strips and record turns; check the sequence on OEIS.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

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

Fold strips and record turn sequences.
Find the recursive rule.
Find a rule for the nth turn using binary.
Prove the rule by induction.
Reflect on why the paper and the ideal curve differ after a few folds.

Pitfalls that cost marks

Recording turns inconsistently.
Conjectures with no proof.
Ignoring the physical limit on folds.

Showing personal engagement

Make the curve yourself.
Fold with different angles and draw the result.
Tile the plane with copies and explain why it works.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Sequences generated by a recursive rule; Predicting the nth term from its binary representation
AA HL	Good fit	Sequences generated by a recursive rule; Predicting the nth term from its binary representation
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: Solid. Needs some independent work beyond class examples. 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: recording turns inconsistently — 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 folding at angles other than 90°, or prove four dragon curves tile around a point.

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 Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) 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).