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?
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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
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.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Not a natural fit | The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics. |
| AA HL | Good fit | Complex numbers in Cartesian and polar/Euler form; Fixed points of iteration and the derivative condition |f′(z)| < 1 |
| 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: pretty pictures without mathematics; and using the derivative condition without explaining it. Say how each one affects your answer.
Use of mathematics (E)
HL: 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, each explained in your own words and carried out correctly.
How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Derive the circle of period-2 attracting cycles: |c + 1| < 1/4.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.
See a complete IA, marked
Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete pure maths exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.
What mathematics does this IA use?
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 and verifying the cardioid shape.
Where can I get data for this IA?
Generate images or iteration tables with a program or online tool to support the analysis. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “Why is the main body of the Mandelbrot set a cardioid?” filled in: the question, the maths and the data sources.
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