In the last few days, social media users have been sharing and staring at an extremely well-made gif that has been described as both a genius idea and the most annoying thing on the Internet. The animated image shows the aerial view of a coastline as the camera slowly moves in. The clip seems to go on for ages, continuing to zoom in, but really it is just two seconds long. We all got fooled by the power of fractals and the coastline paradox.
The paradox states that the coastline of an island doesn't have a well-defined length. This might seem counterintuitive but it is correct. The measurement of the length of a coastline depend specifically on the units used to measure it.
A standard example of this paradox is the measurement of the British coastline. In units of 100 kilometers (62 miles), the main British isle is 2,800 kilometers (or about 1,700 miles). But if we half our “ruler” to just 50 kilometers (31 miles), then Great Britain gains about 600 kilometers (373 miles) more of coastline, reaching a total of 3,400 kilometers (or about 2,100 miles).
So what gives? Well, the issue is that we are trying to measure the coastline like it was a square or an octagon, but that cannot work. A coastline is best approximated by a fractal, a special geometrical curve whose complexity changes with the measurement scale. Depending on your unit, the total length might vary wildly.
The term fractal was coined by Benoit B. Mandelbrot (and no, the B in his name doesn’t stand for Benoit B. Mandelbrot), and he was instrumental in the discussion of the coastline paradox.
There are many types of fractals with slightly different characteristics, but the one that matters in this gif the most is self-similarity. The patterns in such fractals are identical at all scales, which means if you zoom in on any part of the figure, you will see the same structure repeated over and over again.
Although fractals are designed to mimic natural objects, such as the coastline or even Romanesco broccoli, humans don’t easily spot the self-similar patterns. This means you can exploit this to create a truly maddening gif.