What Really Happens When You Measure A Coastline?

Say that you are tasked with figuring out how to measure a coastline (i.e., the length). To do this, you are given a very detailed map of an arbitrary island. It sounds like a relatively simple task, right? Well, the catch is that you will have to get as accurate a measurement as possible. Whenever we talk about accurate measurements, science is involved.

In this article, I will guide you through the process of how we could try to solve this problem. Along the way, you will realise that hidden behind this problem is a strange, yet profound phenomenon. Understanding this phenomenon would give you a broader perspective of areas in science that we hardly understand as human beings. Let’s get started.

This essay is supported by Generatebg

Measuring Using Scales

Let’s say that we have a scale that is 1 Kilometre (Km) long. We can now use this scale to measure the length of the island’s coastline. We can be certain that it is not the most accurate way. But this will enable us to understand how to proceed.

It appears that using our 1 Km scale, we managed to measure that the island’s coastline is 9 Km long. Of course, with the 1 Km scale, we did not manage to capture the bends, twists, nooks, and crannies. So, how do we capture these details using a scale? How about if we use a smaller scale? Right, let’s use a 0.5 Km scale (half the size of our original scale) and see what results we get.

This is an interesting result. Using a scale that is 0.5 Km long, the coastline length has increased from 9 Km to 14 Km. The basic idea we get out of this is the following:

While measuring irregular/jagged coastlines, whenever we increase our measurement’s resolution (i.e., reduce the scale’s size), the coastline length only increases.

This happens because, as we refine our resolution, the number of bends and twists that we capture only increases.

What is the Right Scale?

So far, we have established that reducing the size of the scale is the right way to proceed. This way, we are guaranteed to get the most accurate measurement at some point. The next question is: what is the right size for the scale?

We first need to zoom into the details of the island’s coastline. Then, we need to figure out what the size of the smallest bend or twist in the coastline is. Once we figure this out, we could use a scale that is sufficiently small enough to measure this bend or twist. If this scale can measure the smallest bend or twist, it can be successfully employed to measure the entire coastline accurately.

We have a plan! So far, so good. Let’s go ahead and zoom into the island’s coastline.

The first time we zoom into the little blip in the southwestern section of the island, we come across something strange. The zoomed-in section looks largely like the bigger section that it belongs to. Okay, this is strange, but strange things happen, right?

It just happens to be the case that this section also has a smaller micro blip. The second time around, we zoom into this smaller micro blip. Surprise, Surprise! This zoomed image reveals a section that also looks like the other two parent sections it is a part of. Surely, I must be pranking you, right? Just what is going on here?

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Fractality

Unfortunately, I’m not pranking you here. It turns out that coastlines have a special property. No matter how much we zoom in, they seem to largely feature structures that resemble the parent structures.

We typically think that if we keep zooming in, the features of the coastline would straighten out at some point. But contrary to this intuition, coastlines never straighten out regardless of the zoom level. They seem to feature smaller and smaller bends and twists that resemble the bigger bends and twists (down to the microscopic level, and beyond).

To help you understand this, imagine a tree whose branch resembles the whole tree. Now, a sub-branch within this branch resembles the branch that resembles the whole tree, and so on. Another example: Imagine a picture in which you are smiling. Someone zooms into your eyes in the picture and surprisingly finds a copy of the entire original picture (with you smiling, of course!). They now zoom into the eyes of your picture that they found in the eyes in the original picture, only to experience the same loop again. This loop keeps going on without any sign of stopping.

In a rough sense, this phenomenon is known in mathematics as fractality. You might be sceptical about my arbitrarily drawn island. But here is an animation of researchers trying to measure the coastline of Great Britain.

You can see that the coastline length keeps on increasing as the measurement’s resolution is increased.

The Paradox

In the end, it turns out to be the case that it is impossible to measure coastlines accurately. Because of their fractal nature, coastlines seem to have an infinite length as we keep zooming in. This phenomenon was originally recorded by Lewis Fry Richardson. Later on, self-appointed “fractalist” Benoit Mandelbrot (one of my all-time heroes) contributed significantly to the research on this phenomenon and several related phenomena.

As for the coastline length measurement, as long as we agree to arbitrarily fix the scale to a particular size, we can measure a meaningful number. But keep in mind that all other coastlines have to be measured using the same scale. Otherwise, the numbers we get out of different scales would be meaningless and misleading when compared. Due to their fractal nature, coastlines produce different numbers at different scales.

To conclude, it is a bit of an anti-climax, but it really is impossible to measure coastline lengths accurately. And this phenomenon is not just limited to coastlines.

Fractality seems to occur all over nature: from plant leaves and fingerprints to planets and galaxies. It is nature’s expression of randomness that still stumps the best brains we have as human beings. Perhaps, certain phenomena are simply not meant to be fully understood by human beings after all!

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References: Lewis Fry Richardson and Benoit Mandelbrot.

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Further reading that might interest you: What Is The Shortest Road Connecting 4 Cities? and The Thrilling Story Of Calculus.