IA idea · Matrices, transformations & Markov chains

Animating a logo with transformation matrices

AI HLAA HL Solid Also in: Art & music, Trig & navigation

Research question

How can a sequence of rotations, reflections, enlargements and shears be combined into single matrices to animate a logo smoothly, and how do determinants and the order of operations explain what you see?

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

Transformation matrices are in AI HL, and a design you choose gives the exploration a creative purpose. Showing why the order matters and what the determinant measures goes beyond the textbook.

The mathematics you'll need

2 × 2 transformation matrices
Composition and non-commutativity
Determinant as area scale factor; sign and orientation
Interpolating between transformations for smooth animation
Translations with 3 × 3 (homogeneous) matrices (new)

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; take coordinates from your own logo or design.

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

Set up the logo's coordinates.
Build each transformation and check it.
Compose them and show order matters.
Use determinants to predict area changes.
Design a smooth animation and reflect on interpolation choices.

Pitfalls that cost marks

Matrices applied without explaining what they do.
Not checking results on a simple shape.
Interpolating matrix entries directly, which can squash the shape; explain the problem.

Showing personal engagement

Animate a logo you designed.
Find two transformations that do commute and explain why.
Fix the squashing problem yourself.

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	2 × 2 transformation matrices; Composition and non-commutativity
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	Good fit	2 × 2 transformation matrices; Composition and non-commutativity

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)

Collect the data for your matrix yourself (counting transitions, measuring a shape), and choose the states or the transformation from a situation you care about.

Reflection (D)

Question the model's assumptions: is the process memoryless, are the probabilities constant, does the transformation preserve what it should? Say how each affects your conclusion. For this idea, start with: matrices applied without explaining what they do — say how it affects your answer.

Use of mathematics (E)

SL: Matrices are in AI HL. At SL, keep to small matrices you explain carefully as new mathematics, with every multiplication shown once and the result interpreted.

HL: Transition matrices, powers, steady states and eigenvalues used correctly, with diagonalisation or a general result derived, and the long-run behaviour interpreted in context.

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

Interpolate rotations through angles rather than matrix entries, or extend to 3D.

Extending it for HL

Diagonalise the matrix to find a formula for the nth state, or compare the steady state with what your data actually shows.

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 matrices & markov 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).