How long is a coastline? Measuring fractal dimension

AA SLAA HLAI SLAI HL Solid Also in: Geometry & Voronoi, Modelling

Research question

How does the measured length of [a coastline] change with the ruler length used, and what fractal dimension does a log-log model give compared with a smoother coastline?

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

Why it makes a good exploration

Mandelbrot's famous question has a surprising answer: a coastline's length depends on your ruler. Measuring two coastlines at several scales gives data for a log-log model whose gradient is a fractal dimension.

The mathematics you'll need

Power laws and log-log regression
The divider (ruler) method or box counting
Interpreting the gradient as 1 − D
Measurement error and scale limits

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

Use an online map's measuring tool at several ruler lengths for two coastlines (one rugged, one smooth).

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

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 coastline paradox.
Measure each coastline with 5–6 ruler lengths.
Fit log-log models and compute D.
Compare coastlines and published values.
Reflect on map resolution and tides.

Pitfalls that cost marks

Too narrow a range of ruler lengths.
Inconsistent start and end points.
Formula for D without derivation.

Showing personal engagement

Choose a coastline you know.
Test the method on a computer-generated Koch curve with known D.
Discuss why the UK and Norway have such different values.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Validate your method on the Koch snowflake (D = log 4 / log 3) before applying it to real coasts.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).