Investigating Science 11–12 · Year 11
The length of the NSW coast: measuring a fractal pattern with dividers
Module 2: Cause and Effect – Inferences and Generalisations
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The idea
The measured length of a coastline grows as the measuring step shrinks, an irregular pattern Richardson found and Mandelbrot described with a fractal dimension, so a coastline has no single true length.
What you need
- a printed map of the NSW coast from Tweed Heads to Cape Howe at a known scale (1 : 1 000 000 or similar)
- drawing dividers, ruler, calculator or spreadsheet with log–log graphing
How to do it
- Set the dividers to a map distance equal to 100 km on the ground and walk them along the coast from one end to the other, counting steps; length L = number of steps × step.
- Repeat with steps of 50, 25 and 12.5 km.
- Plot log L against log(step) and draw the line of best fit.
- Calculate the fractal dimension D = 1 − gradient.
- Compare the class lengths with Geoscience Australia’s published NSW mainland coastline and with Richardson’s dimensions reported by Mandelbrot.
What you should see
The measured length rises every time the step is shortened and the log–log plot is close to a straight line with a negative gradient, so D lies between 1.0 and about 1.3. Mandelbrot (Science, 1967) reports from Richardson’s data D = 1.13 for the Australian coast, 1.25 for the west coast of Britain and 1.02 for South Africa’s; with D = 1.13 each halving of the step adds about 9 % to the length. Geoscience Australia gives 1973 km for the NSW mainland coastline and states that the figure depends on the scale of its 1 : 100 000 map data, which the class result illustrates.
What changes
- What you change
- divider step length
- What you measure
- measured coastline length
- What you keep the same
- same map
- same start and end points
- same walking rule at headlands
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A coastline has one correct length.
- Using a smaller step only makes the measurement more accurate, not longer.
- Irregular shapes cannot be described mathematically.
Safety card
Hazards
- sharp divider points
Controls
- carry and store dividers closed
Note
No hazardous chemicals or heat sources: record the activity in the school's RiskAssess risk assessment, following the NSW Department of Education Science safety and compliance page; the Chemical Safety in Schools package is not triggered.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Investigating Science Stage 6 Syllabus (2017), NESA; currentINS11-9INS11/12-1INS11/12-2INS11/12-4
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.