Imagine that you’re taking a GIS class and your instructor tasks everyone with coming up with the answer to, “what is the length of the coastline of Maine?” Everyone downloads a different GIS data set to calculate the length and everyone comes back with a completely different answer to that question.

The phenomenon is known as the ** coastline paradox**. Depending on the scale of the data used, the measurements of coastlines will vary because differing resolutions of data covering the same geographic area can yield remarkably different measurements of length. Coastal paradox exposes the difficulty in defining the true length of coastlines, which are naturally irregular and fractal-like in shape.

Questions such as: “How long is the coastline of Australia?” or “did you know that the coast of the U.S. state of Maine is longer than the coast of California?” become harder to answer consistently and depend greatly on the resolution of the GIS data used to measure the coastlines in question.

## What is the Coastline Paradox?

So what is the coastline paradox? It is a paradox that occurs when measuring a coastline that causes the total length of the coastline to increase each time you measure it with a smaller unit of measurement, due to the extra features that can be measured.

What does that mean? Just imagine you were told to manually measure the length of a jagged feature maybe a map and you have got no ‘thread’ to measure with. What do you do? You get a ruler, right?

However, the result you get will now be based on the length and size of the rule you use. The smaller the ruler the more precise your measurement will be and vice versa. Using a few straight lines to approximate the length of a curve will produce a low estimate. That is why you use a thread to get the most precise measurement possible.

The coastline is the most obvious example of this situation due to its fractal-like (jagged recurring pattern) properties. This phenomenon was first observed by Lewis Fry Richardson in 1961 when calculating the lengths of national boundaries and is sometimes referred to as the Richardson effect. In 1967, Mandelbrot’s expanded on the concept of a coastline paradox with a paper posing the question “How long is the coastline of Britain?” Mandelbrot made the statement, *‘‘Geographical curves are so involved in their detail that their lengths are often infinite or, rather, undefinable.’’*

## An example of the coastline paradox

At the center of the coastline paradox is the observation that as the scale of measurement becomes smaller and more detailed, the measured length of the coastline increases. This is because smaller scales can capture more of the coastline’s intricate contours and jagged edges, which are overlooked at larger scales.

This series of three simple maps showing Marina Del Rey near Los Angeles California illustrate how different scales of GIS data will calculate coastlines differently.

The first map below is taken from a highly generalized, small scale layer of the counties for the entire United States. This dataset has an extremely generalized coast line for this area. The is almost no detail in the coastline and the harbor is not represented at all. A calculation of the length of this coastline would simply be the three straight edges of the coast.

The map below shows the same coastline with a layer of all counties for the state of California. While still a small scale layer, the coastline shows more detail. The Marina del Rey harbor is represented by a small inlet on the map. Calculating the length of the coastline for the same area as the map above will now also factor in the small detail that can be seen along the coastline.

In a large scale layer created to show just the County of Los Angeles boundary, the coastline for this area contains the highest level of detail, and a recognizable harbor is represented. Now calculating the length of the coastline will include more detail include measuring the interior of the harbor.

## Is the coastline paradox real?

Some scholars argue that the coastline paradox, while acknowledging the complexities of measuring a coastline, does not mean that coastlines ultimately “have indefinite lengths.” Researchers with the Griffith University Gold Coast in Australia published a 2023 study in the *Journal of Coastal Research* countering that premise of the coastline paradox.

In the paper McNamara and Da Silva argue that the length of a coastline is real and can be measured. To create definite measurements, they approached the measuring of coastlines based on two concepts. The first is that the authors noted that there is no universally accepted definition of a coastline, so in the paper, the authors defined “coastline” as being where the sea touches the shore at the average water level. The second is addressing the measurement of all the concavities of a coastline as a measurement of their spans. The authors argued that the *“concept of ‘‘span’’ is well defined in other branches of engineering.”*

With a more discrete definition of a coastline, the authors contend that the finite length of a coastline can then be calculated using available geospatial technologies and data such as satellite altimetry, photogrammetry, LIDAR, or ground surveying. The authors of the paper stress that the ability to measure coastlines important for coastal communities and planners to better understand and react to changes in coastlines and rising sea levels.

## Consider map scale when calculating lengths of a coastline

The coastline paradox highlights how tricky and limited it is to measure natural, uneven shapes such as coastlines, especially when using different levels of detail in Geographic Information System (GIS) data. So the next time you are tasked with measuring the length of a coastline, river, or other lengthy feature, consider the resolution of the GIS data you need to use. As discussed in an earlier article, larger scale GIS data sets tend to show more detail than smaller scale data.

*This article was originally written on October 30, 2014 and has since been updated.*

## References

Coastline paradox: http://mathworld.wolfram.com/CoastlineParadox.html

Mandelbrot, B. B. “How Long Is the Coast of Britain.” Ch. 5 in *The Fractal Geometry of Nature. *New York: W. H. Freeman, pp. 25-33, 1983.

Mapping Monday: The Coastline Paradox: http://blog.education.nationalgeographic.com/2013/01/28/mapping-monday-the-coastline-paradox

McNamara, G., & Da Silva, G. V. (2023). The Coastline Paradox: A New Perspective. *Journal of Coastal Research*, *39*(1), 45-54. https://doi.org/10.2112/JCOASTRES-D-22-00034.1

Richardson, L. F. (1961). The problem of contiguity: an appendix to statistics of deadly quarrels. *General systems yearbook*, *6*, 139-187. https://doi.org/10.2307/2981156

The Coastline Paradox: How can one coastline be two different lengths?: http://www.richannel.org/the-coastline-paradox