Why does a random game draw the Sierpiński triangle?

AA SLAA HLAI SL Solid Also in: Simulation, Probability

Research question

Why does repeatedly moving halfway towards a randomly chosen corner of a triangle produce the Sierpiński triangle, and what shapes appear if you change the ratio, the number of corners or the rules for choosing them?

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

The result looks like magic, which makes explaining it satisfying. Variations you invent make the exploration yours.

The mathematics you'll need

Transformations: enlargement by scale factor ½ about a vertex
Iteration and attractors
Why the empty middle triangle stays empty (proof)
Changing ratio and number of vertices
Fractal dimension of the results

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; simulate in a spreadsheet or GeoGebra.

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

Run the game and describe the result.
Explain it with enlargements about each vertex.
Prove the middle region stays empty.
Explore other polygons and ratios.
Find the ratio for a square that gives a fractal, and reflect.

Pitfalls that cost marks

Pictures without explanation.
Not explaining why the first few points don't matter.
Changing several rules at once.

Showing personal engagement

Play the game by hand with a die first.
Invent your own rule and predict the shape.
Find the rule that fills the square exactly.

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	Transformations: enlargement by scale factor ½ about a vertex; Iteration and attractors
AA HL	Good fit	Transformations: enlargement by scale factor ½ about a vertex; Iteration and attractors
AI SL	Good fit	Transformations: enlargement by scale factor ½ about a vertex; Iteration and attractors
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: pictures without explanation — 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

Write the game as a set of affine transformations with matrices and design a fern-like shape.

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 long does a game of Snakes and Ladders last on my grandmother's board? (AI 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).