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?
Adapt it: change the place, the data or the comparison until the question is yours.
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
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
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.
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
- 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.
Taking it further
Show that joining points k and 2k on a circle produces a cardioid envelope.