Infinite perimeter, finite area: designing your own snowflake curve

AA SLAI SLAA HL Accessible Also in: Pure maths, Art & music

Research question

If you change the Koch snowflake's rule (a different angle, a square bump, a different number of segments), how do the limiting area and the growth of the perimeter change, and which rules make the curve cross itself?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

The classic snowflake is in many textbooks, so the original part is your own variation. Geometric series give exact answers you can check against drawings.

The mathematics you'll need

Geometric sequences for the number and length of segments
Sum to infinity for the area
Trigonometry for the area of each new bump
Comparing limits across rules
Fractal dimension log N / log(1/r) (new)

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

No data needed; draw the first stages in GeoGebra to check your formulas.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
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

Derive perimeter and area for the classic curve.
Generalise to a rule with angle θ.
Find the limiting area as a function of θ.
Find when the curve overlaps itself.
Compare dimensions and reflect on what 'infinite perimeter' means physically.

Pitfalls that cost marks

Reproducing the textbook snowflake with nothing new.
Errors in counting segments at each stage.
No check against a drawing.

Showing personal engagement

Invent your own rule.
Draw the stages yourself.
Find the angle that maximises the limiting area.

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	Geometric sequences for the number and length of segments; Sum to infinity for the area
AA HL	Fits, but add an HL technique	Geometric sequences for the number and length of segments; Sum to infinity for the area
AI SL	Good fit	Geometric sequences for the number and length of segments; Sum to infinity for the area
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: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Generate the fractal or the iteration yourself, choose your own variation (a different rule, angle or starting value), and pursue a question you raised while exploring.

Reflection (D)

Reflect on the limits of the model: real coastlines and plants are only self-similar over a range of scales, and computer iterations carry rounding error. Say how that affects your numbers. For this idea, start with: reproducing the textbook snowflake with nothing new — say how it affects your answer.

Use of mathematics (E)

SL: Geometric sequences and series for lengths and areas, logarithms for dimension, and iteration of functions, each calculated and checked numerically.

HL: Proof of a limit or dimension, complex-number iteration with a derived condition, or analysis of fixed points and their stability using derivatives.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Find the dimension as a function of the rule's parameters and the rule that gives the largest dimension without overlap.

Extending it for HL

Find fixed points and decide their stability with derivatives, or prove the limit you found numerically.

See a complete IA, marked

Our annotated exemplar Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) asks a different question, but shows how a complete fractals & chaos exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).