The YouTube channel 3Blue1Brown contains some excellent videos. The video linked to here on Fractal Dimensions is no exception. Beginning with self-similar shapes, like the Koch snowflake and Sierpinski Triangle, the commentator calculates their fractal dimension and then goes on to look at non-self-similar shapes such as the coastlines of England and Norway. The screenshot below displays the fractal dimensions for the coastlines of these two countries.
What emerges is that the fractal dimension is a measure of the roughness of a shape and that these shapes really exist, unlike mathematically perfect shapes like lines, circles, cubes and so on. These latter shapes have dimensions of 1, 2 and 3 corresponding to the one, two and three dimensions that they inhabit. A fractal shape like the one shown below however, can be considered to be halfway between a line and a two dimensional shape like a square. It has a fractal dimension of 1.5:
Here is the video:
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