How fractal is a fern? Box-counting dimension from photographs

AI SLAA SLAI HLAA HL Solid Also in: Modelling, Health & biology

Research question

What box-counting dimension do photographs of a fern, a tree's branches and a cauliflower give, how consistent is the estimate across box sizes, and does it differ between species?

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

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Why it makes a good exploration

Box counting turns a vague claim ('nature is fractal') into a measurement with an uncertainty, using logarithms and regression from the syllabus.

The mathematics you'll need

Counting boxes at several scales
Log-log transformation and linear regression
Gradient as the dimension
Range of scales over which a straight line fits; residuals
Uncertainty from repeated counts

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.

Modelling the data, step by step

This idea usually needs power or straight-line models. See it worked step by step, with a criterion tip at every step: Power y = a·xⁿ (log–log) · Straight line y = mx + c · Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Photograph plants against a plain background, overlay grids in GeoGebra (or print with grids) and count boxes at five or more scales.

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

Explain box-counting with a shape of known dimension (a line, a square).
Count boxes for each photo at several scales.
Fit log-log lines and compare gradients.
Investigate where the straight line breaks down.
Reflect on photo resolution and the limits of self-similarity.

Pitfalls that cost marks

Too few scales for a meaningful gradient.
Counting inconsistently between scales.
Ignoring the range where the plant isn't self-similar.

Showing personal engagement

Use plants from your garden or school grounds.
Validate the method on a shape with known dimension.
Compare plants that look 'more' and 'less' fractal.

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	Counting boxes at several scales; Log-log transformation and linear regression
AA HL	Good fit	Counting boxes at several scales; Log-log transformation and linear regression
AI SL	Good fit	Counting boxes at several scales; Log-log transformation and linear regression
AI HL	Good fit	Counting boxes at several scales; Log-log transformation and linear regression

Level: Solid. Needs some independent work beyond class examples. 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: too few scales for a meaningful gradient — 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

Compare box counting with a different method (for example the divider method used for coastlines) on the same image.

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 When can we launch? Modelling the tide at our sailing club's slipway (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).