In class, we talked about how dimensions can be non-integer values. We were given some examples of fractals, shapes with non-integer dimensions, and we were able to calculate the dimensions of some fairly simple fractals. But what about more complex shapes, that can’t be easily doubled, quadrupled, or so on? How does one measure, arbitrarily, the dimension of any shape?
Well, one of the ways to do so is to find the Hausdorff Dimension of a set. This concept of measuring dimensions was developed by (big surprise) Felix Hausdorff back in 1918.
The key idea is this: a “circle” with dimension 1 (a line) has its length vary proportionally to its “radius”. A circle with dimension 2 has its area vary proportionally to its radius squared, and so on with spheres and volume. To extrapolate, a “circle” with any dimension p would have its “p-volume” vary proportionally to its radius to the power of p. So, if p were 2.5, a 2.5 dimensional circle would, when doubled, increase in its 2.5 dimensional volume by 22.5 = about 5.66.
Now, hold onto your hats (if you don’t have a hat, go get one real quick, then hold onto it) because here’s where things get interesting. Let’s say you wanted to cover a 3-dimensional object with a bunch of smaller spheres, and measure the “4 dimensional volume” of the spheres. Well, we know that the 4-dimensional volume of an object is proportional to its radius4, so we can get an idea of how the 4-dimensional volume changes by simply adding together the radii4 of every sphere covering the object.
Now let’s take an arbitrary covering of our object by spheres and measure its 4-dimensional volume as above. We will get some number A. Now, how does that number change as our spheres shrink? Let’s say we, for example, replaced each sphere with 8 spheres, each having half the radius, and still managed to cover the object. Note that while the 3-dimensional volume of our spheres has remained the same (8 spheres of radius 1/2 have the same volume as 1 of radius 1), our 4-dimensional volume has been cut in half!
The main idea is that by shrinking the radii of our spheres, we can arbitrary decrease the 4-dimensional volume of the spheres covering our object. Since we can cover our object with spheres having arbitrary small 4-dimensional volume, it would make sense that the object would have 0 4-dimensional volume, which is consistent with how one would think about 4-dimensional area.
The key here is that this is true for any d-dimensional area if d is greater than 3, because our object is 3-dimensional. Thus, if we took the infimum of values for d such that this is true, we would get 3.
To reiterate, we had a 3-dimensional object. We were able to determine it was 3-dimensional because for any dimension d higher than 3, we could cover our object with spheres such that their d-dimensional volume was arbitrarily small. To be very precise:
The Hausdorff content of dimension d of an object is the infimum of numbers δ ≥ 0 such that there is some cover of the object by balls of radius r1, r2,… such that (r1d+r2d+r3d+…)< δ.
The Hausdorfff dimension of an object is the infimum of numbers d such that the hausdorff content of dimension d of an object is equal to 0.
Now, this approach matches quite nicely with our definition of integer dimensions, and provides a very nice way for us to expand that notion into other, non-integer dimensions. For example, this approach can actually be used in a surprising way: to find the dimension of coastlines.
Perhaps you’ve heard of the coastline paradox: that the more precise you try to measure the coastline, the longer it gets, seemingly without bound. This should signal to us that the coastline behaves as a fractal, and since the above method gives us a way to measure the dimension of arbitrary objects, we can use it to try and measure the dimension of the coastline.
Here’s how it’s done: first, cover the coastline in large circles and measure the sum of the radii all put to some power p. Then, shrink the circles and measure again the sum of the radii all put to the power p. If this number keeps getting smaller, you’ve overestimated the dimension of the coast. If the number keeps getting bigger, you’ve underestimated it. If you keep doing this, you can fine-tune your estimate of the dimension. In fact, Mandelbrot did this very thing, and got that the coast of Great Britain had a fractional dimension of about 1.25, while the coastline of South Africa had a fractional dimension of about 1.02. While these are just estimates, it’s still cool to see how abstract ideas such as this can be used to measure things in real life.