Symmetry in Islamic geometric art: which wallpaper groups appear?

AA HLAA SL Ambitious Also in: Pure maths, Geometry & Voronoi

Research question

Which symmetry groups (rotations, reflections, glide reflections) are present in [several patterns from a named building or collection], and can they be described with transformation matrices?

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

Why it makes a good exploration

Islamic geometric art explores symmetry with extraordinary sophistication. Classifying real patterns — with rotations, reflections and matrices — connects AA HL transformations with art and history.

The mathematics you'll need

Isometries: rotations, reflections, translations, glide reflections
Transformation matrices and composition (HL)
Group ideas: closure, identity, inverses (explain)
Classifying by rotation orders

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

Use high-quality images from museum collections or your own photographs of buildings.

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

Introduce isometries and symmetry groups.
Analyse each pattern's symmetries.
Represent them with matrices and verify compositions.
Classify the patterns.
Reflect on the crystallographic restriction and the artists' methods.

Pitfalls that cost marks

Descriptive art history instead of mathematics.
Claiming all 17 groups appear without evidence.
Matrix calculations with no link to the pattern.

Showing personal engagement

Construct a pattern yourself with compass and straightedge.
Visit or research a specific building.
Explain why 5-fold rotation cannot tile the plane periodically.

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

Taking it further

Prove the crystallographic restriction (only 1, 2, 3, 4 and 6-fold rotations) using the trace of a rotation matrix.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).