Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How long is a coastline? Measuring fractal dimension

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

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?

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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

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.

Real data for this idea

Primary source: OpenStreetMap. Read off coordinates (right-click, “Show address”) or use Export for a small area. Licence: Open Database Licence: credit “© OpenStreetMap contributors”; data you build from it must be shared alike.

OpenStreetMap is under the Open Database Licence: credit “© OpenStreetMap contributors”. Links checked October 2026; we link to the data and never copy it. More free data sets.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Power laws and log-log regression; The divider (ruler) method or box counting
AA HL	Good fit	Power laws and log-log regression; The divider (ruler) method or box counting
AI SL	Good fit	Power laws and log-log regression; The divider (ruler) method or box counting
AI HL	Good fit	Power laws and log-log regression; The divider (ruler) method or box counting

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

Reflection (D)

For this idea, start with the weak points: too narrow a range of ruler lengths; and inconsistent start and end points. Say how each one affects your answer.

Use of mathematics (E)

SL: power laws and log-log regression; the divider (ruler) method or box counting; interpreting the gradient as 1 − D; measurement error and scale limits, 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

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

Extending it for HL

Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.

See a complete IA, marked

What is the quickest way to deliver a leaflet to every house on my estate? (AI HL) 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 SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Power laws and log-log regression, the divider (ruler) method or box counting, interpreting the gradient as 1 − D and measurement error and scale limits.

Where can I get data for this IA?

Use an online map's measuring tool at several ruler lengths for two coastlines (one rugged, one smooth). Checked sources and tools: OpenStreetMap and GeoGebra.

Turn this idea into your IA

The planner opens with “How long is a coastline? Measuring fractal dimension” filled in: the question, the maths and the data sources.

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