IA idea · Art, music & design
Symmetry in Islamic geometric art: which wallpaper groups appear?
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?
Make it yours, then agree it with your teacher.
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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
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.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Fits — ambitious at SL | Isometries: rotations, reflections, translations, glide reflections; Group ideas: closure, identity, inverses (explain) |
| AA HL | Good fit | Isometries: rotations, reflections, translations, glide reflections; Transformation matrices and composition |
| 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
Reflection (D)
For this idea, start with the weak points: descriptive art history instead of mathematics; and claiming all 17 groups appear without evidence. Say how each one affects your answer.
Use of mathematics (E)
SL: isometries: rotations, reflections, translations, glide reflections; group ideas: closure, identity, inverses (explain); classifying by rotation orders, each explained in your own words and carried out correctly.
HL adds: transformation matrices and composition, each explained in your own words and carried out correctly.
How art & music explorations reach the top bands in general: Art, music & design IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Prove the crystallographic restriction (only 1, 2, 3, 4 and 6-fold rotations) using the trace of a rotation matrix.
Extending it for HL
This idea already has HL mathematics in it: Transformation matrices and composition. Make that the centre of the exploration and use it with full rigour.
See a complete IA, marked
Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete art & music exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL and AA SL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.
What mathematics does this IA use?
Isometries: rotations, reflections, translations, glide reflections, transformation matrices and composition (HL), group ideas: closure, identity, inverses (explain) and classifying by rotation orders.
Where can I get data for this IA?
Use high-quality images from museum collections or your own photographs of buildings. Checked sources and tools: GeoGebra.
Turn this idea into your IA
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