Enter the Hausdorff Dimension to measure coastlines

Illustration for article titled Enter the Hausdorff Dimension to measure coastlines

Alternate dimensions aren't outside our everyday experience. In fact, we can use exotic dimensions, like fractional dimensions, to measure things as common as coastlines. Find out about the Hausdorff Dimension before your next trip to the sea.


A point has zero dimensions. A line has only one dimension to its name. A plane has two, and a voluminous shape has three. But what about a line that shoots back and forth over a page in such a complicated pattern that can't really be defined as a one-dimensional object? We could say that it only takes three points (provided they're not all on one line) to define a plane. So the scribbly line lies within a plane, but does that really define the line? Mathematician Felix Hausdorff said no. The line still technically has no width. It only carves out width. According to Hausdorff, to describe the line, one needs fractional dimensions. These dimensions define many fractal shapes, such as the Menger Sponge, a fractal that is all surface area and no volume and has a dimensional value of 2.73.

The most famous example of the fractional dimensions are coastlines. Geology, weather, tides, and plate tectonics all come together to give coastlines a natural squiggliness (that's the technical term). The length of coastlines vary depending on how they are measured. Grab a ruler and a map and measure the distance from Maine to Florida by making one line from the tip of one to the tip of the other — you'll get a relatively short measurement. Measure it in increments of an inch (on the map) and you'll be able to follow the curve of some of the coast in between. Because you're measuring more curves, you'll get a longer measurement. Go out and measure the coast with a yard-long ruler by hand and you'll get an even longer measurement. Follow the curves of each individual atom and you'll be dead long before you're done. The point is, the "length" of the coastline varies depending on the size of the object the coast is measured with.

This expansion of the size of a measured object — which was dependent on the relative size of the measuring device — was noticed in both coastlines and fractal structures. Mathematicians graphed how the change in scale of the measuring device affected the change in the size of the object, and took the slope of the line made on the graph to assign the fractional dimension of the object. In coastlines, the fractional dimension of the coast of South Africa is a mere 1.02, while the curvy coast of Britain is an impressive 1.25. How do you think your nearest coastline measures up?

Image: Nasa Images. Via Wolfram Math World, University of Wisconsin and Vanderbilt.


Corpore Metal

This is going to sound like needless reiteration of what Esther just said above but—

As I recall Mandelbrot actually published an article once titled "How Long is the Coast of Britain?" In it, among other things, he pointed out that we could go to the insane level of running a tape measure along the individual atoms within grains of sand, rocks or outcroppings on the beaches. This taken to the limiting case with self-similarity in fractals, Britain's coast is infinitely long.

But here's an interesting and often overlooked point, especially when we fit fractals in with chaos dynamics and emergence: The lesson to be learned here is that in some cases, as detail goes up, relevance goes down.

By the way, I just wanna thank Esther again for these great math fact articles. She's like Dave and physics. We can always refer back to these just to quickly summarize points.