Maths IA · Algebra and functions · topic 13 of 97
The Mandelbrot set: which numbers stay bounded? · IB Maths IA idea: guide, sample graph and step-by-step plan
prove it, then draw itSquare a number, add a constant, repeat. For some constants the numbers blow up, for others they stay put, and the border between them is infinitely detailed. You're going to find part of that border exactly.
- ✓ you like complex numbers and want a use for them
- ✓ you want to prove something about a picture
- ✓ you can write or adapt a few lines of code to count iterations
The big idea
Pick a complex number c and iterate z → z² + c starting from 0. The Mandelbrot set is every c for which the sequence stays bounded. A clean proof shows that once |z| passes 2 the sequence escapes for ever, so software only has to test against 2. Then you can do what software cannot: find parts of the boundary by algebra. Where a fixed point stops being stable you get a cardioid, and where a two-step cycle stops being stable you get a circle. Polar and Euler forms do the work. Test your exact boundaries against pixel counts, then estimate the area by random sampling and judge how many iterations are enough.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Show that if |c| ≤ 2 and |z| > 2 then |z²+c| > |z|, and that the sequence grows without bound.
→ A proof by inequality, then a test: the escape rule you give your program is something you proved.Solve z² + c = z for the fixed point, say when it is stable, and find the boundary in polar and Euler form.
→ A curve you derive and plot over your own pixel image. Compare the two and say where they differ.Find where a two-step cycle appears and when it is stable.
→ A second exact region and a second algebra problem. Does it touch the cardioid where your algebra says it should?Count random points that stay bounded after N steps, and see how the estimate moves as N grows.
→ A probability and error question: what is the margin of error, and which way is the bias?Fix c, then colour starting values z by whether they escape. Compare c inside and outside the Mandelbrot set.
→ A link between one picture and many: connected and dust-like shapes. Keep it to one clear comparison.Measure how the iteration count needed grows as you zoom towards the boundary.
→ A fitted model for the cost of accuracy, with a sensible limit stated.Your Maths IA research question: run the check
exact or estimated?Write your own question first. Then make sure it ticks every box:
- Defines the iteration, the starting value and what 'bounded' means with a clear test.
- Says which boundary or quantity you will find exactly and which you will estimate.
- States how you will compare algebra with a picture: pixels counted, points sampled, with the sample size.
- Says how many iterations you use and why, and what you do about points that are not decided.
- Is narrow enough for 12 to 20 pages, with every proof on the page and not hidden in code.
The set-up
One rule, one starting point, one question. Draw your own orbit for a value of c you choose and mark where it ends up.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
The iteration, what 'bounded' means and what you predict before running any code.
Cartesian, polar and Euler forms, with modulus and argument defined and used.
The proof that |z| > 2 escapes, with every inequality justified.
Fixed points, two-step cycles and the stability condition leading to the cardioid and the disc.
Your pixel image with the exact curves drawn over it, and a measured gap.
The sampling method, the estimate for several iteration limits and a margin of error.
Back to your aim: what you proved, what you only estimated, with numbers.
Finite iterations, floating-point rounding, a boundary that is never fully pinned down. Say which limit matters most.
Where every definition, picture and published value came from.
What your Maths IA graph could look like
plot against NA made-up data set to show the form of a strong graph, not the answer: an area estimate against the iteration limit, with the exact area of two regions as a floor. Yours will look different. example only
- The estimate depends on N, so plot it against N rather than quoting one number.
- Axis labels say what is counted and in what unit, even if the unit is only square units of the c-plane.
- Mark the exact area of the regions you derived. It is a floor, not a result.
- Plot the error bars from your sample size. A dot with no error bar is half an answer.
- Another graph to consider: the number of steps to escape along a line through the plane.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what the Mandelbrot set is. Before coding, guess whether c = 1, c = −1 and c = i stay bounded, then iterate by hand to check.
- Set up the complex numbersReview modulus, argument and Euler form. Say which form you will use for the fixed-point work and why.
- Prove the escape testProve that once |z| > 2 the sequence escapes. Write down the inequality in each line.
- Trial run (1 lesson)Code or build the iteration for a grid of 20 by 20 values of c. Check three points against your hand work before going on.
- Find the fixed-point boundarySolve for the fixed point, state when it is stable and get the cardioid in Euler form.
- Find the two-step boundaryFind the two-step cycle, state when it is stable and get the circle.
- Draw them over the picturePlot your exact curves over a fine image and measure how well they agree.
- Go further, with a reasonEstimate the area by sampling, or compare Julia sets for two values of c. HL: justify the sample size and say which way the estimate is biased.
- Reflect as you goAfter every result, write one line on what is proved, what is estimated and what is assumed. These lines become your evaluation.
- Cut, check, then sendCheck every proof is complete, every graph has labelled axes and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Complex numbers in polar and Euler form – Write for the boundary of stability and expand into .
- Inequalities and proof – Prove with the triangle inequality that if and then , so the orbit escapes.
- Quadratic equations – Solve for the fixed points, and for the points of a two-step cycle.
- Moduli of complex numbers – Find when the multiplier of a cycle has modulus less than 1: for a fixed point and for a two-step cycle.
- Probability and sampling – Estimate area as the fraction of random points that stay bounded times the box area, with a margin of error from the binomial model.
Key equations
Syllabus links
- 1.12Complex numbers in Cartesian formHL
- 1.13Polar and Euler forms of complex numbersHL
- 1.14Conjugate roots, De Moivre's theorem, roots of complex numbersHL
- 1.15Proof by induction, contradiction and counterexampleHL
- 2.7Quadratic equations, inequalities and the discriminant
- 4.5Basic probability and expected frequency
- 1.12Complex numbers in Cartesian formHL
- 1.13Polar and Euler forms of complex numbersHL
- 4.5Basic probability and expected frequency
Read up first
Data and sources to start from
Iterate the map in Python or a spreadsheet over a grid or random sample of c values, and compare counts with the exact regions.
References to cite
- Peitgen, H.-O., & Richter, P. H. (1986). The beauty of fractals: Images of complex dynamical systems. Springer.
- Devaney, R. L. (2003). An introduction to chaotic dynamical systems (2nd ed.). Westview Press.
- Mandelbrot, B. B. (1980). Fractal aspects of the iteration of z → λz(1 − z) for complex λ and z. Annals of the New York Academy of Sciences, 357, 249–259.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
survived N steps is not proof- Treating a pretty picture as the maths. The proofs and the exact curves are the marks.
- Testing against 2 without proving why 2 is enough.
- Reporting an area from one iteration limit, when the estimate moves as the limit changes.
- Calling a point 'inside' because it survived N steps. It might escape later.
- Mixing up the c-plane (the Mandelbrot set) with the z-plane (a Julia set).
- Using a graphing tool's colours as evidence, with no count of what is actually in the set.
More Maths IA ideas like this
Roots of unity: a pentagon by ruler and compass?●●●
Chaos: does a butterfly change the weather?●●●
Newton's method: how fast, and when does it fail?●●○
Fonts and logos: how do Bézier curves work?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.