IA idea · Art, music & design
What curve do straight lines make in string art?
Research question
When straight threads join equally spaced points on two perpendicular lines, what curve do they envelop, and how can calculus prove it is a parabola?
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Why it makes a good exploration
String art makes smooth curves from straight lines. Proving what the curve is — using families of lines and derivatives — is an elegant exploration you can also build.
The mathematics you'll need
- Equations of families of lines
- Envelopes (explain the idea) using differentiation
- Parabolas and their properties
- Extending to circles and other point spacings
Where the data comes from
Build a string-art piece and photograph it; use GeoGebra to model it.
- GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
A possible outline
- Describe the construction precisely.
- Write the family of lines.
- Find the envelope and show it is a parabola.
- Verify with your built model.
- Extend to lines between points on a circle.
Pitfalls that cost marks
- Asserting the curve without proof.
- Lines defined inconsistently.
- No link between the physical model and the algebra.
Showing personal engagement
- Make your own string art.
- Predict the curve for a new arrangement before building.
- Explore cardioids from circle-based string art.
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 | Equations of families of lines; Envelopes (explain the idea) using differentiation |
| AA HL | Good fit | Equations of families of lines; Envelopes (explain the idea) using differentiation |
| 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
Reflection (D)
For this idea, start with the weak points: asserting the curve without proof; and lines defined inconsistently. Say how each one affects your answer.
Use of mathematics (E)
SL: equations of families of lines; envelopes (explain the idea) using differentiation; parabolas and their properties; extending to circles and other point spacings, 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
Show that joining points k and 2k on a circle produces a cardioid envelope.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL art & music exploration adds.
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 SL and AA HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Equations of families of lines, envelopes (explain the idea) using differentiation, parabolas and their properties and extending to circles and other point spacings.
Where can I get data for this IA?
Build a string-art piece and photograph it; use GeoGebra to model it. Checked sources and tools: GeoGebra.
Turn this idea into your IA
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