IA idea · Pure maths, number & proof
Why is the main body of the Mandelbrot set a cardioid?
Research question
For which complex numbers c does the iteration z → z² + c have an attracting fixed point, and why does this region form exactly a cardioid?
Adapt it: change the place, the data or the comparison until the question is yours.
Why it makes a good exploration
The Mandelbrot set's famous heart shape can be derived with AA HL complex numbers: a fixed point is attracting when |2z| < 1. Proving the boundary is a cardioid turns a computer image into a theorem.
The mathematics you'll need
- Complex numbers in Cartesian and polar/Euler form
- Fixed points of iteration and the derivative condition |f′(z)| < 1
- Parametric curves: c = e^(iθ)/2 − e^(2iθ)/4
- Verifying the cardioid shape
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
Generate images or iteration tables with a program or online tool to support the analysis.
- 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
- Explain the iteration and the Mandelbrot set.
- Find fixed points and the attracting condition.
- Derive the boundary parametrically.
- Show it is a cardioid.
- Extend to period-2 points (the circle to the left).
Pitfalls that cost marks
- Pretty pictures without mathematics.
- Using the derivative condition without explaining it.
- Losing track of the parameter θ.
Showing personal engagement
- Generate your own images and describe what you tested.
- Derive the period-2 disc yourself.
- Reflect on how a simple rule produces complexity.
See Criterion C: personal engagement for what examiners look for.
Taking it further
Derive the circle of period-2 attracting cycles: |c + 1| < 1/4.