A Tisket, a Tasket, an Apollonian Gasket
Fractals made of circles do funny things to mathematicians
A Little Bit of Gasketry
To this point I have only written about the very beginning of the gasket- making process—how to inscribe one circle among three given circles. However, the most interesting phenomena show up when you look at the gasket as a whole.
The first thing to notice is the foamlike structure that remains after you cut out all of the discs in the gasket. Clearly the disks themselves take up an area that approaches 100 percent of the area within the outer disk, and so the area of the foam (known as the “residual set”) must be zero. On the other hand, the foam also has infinite length. Thus, in fact, it was one of the first known examples of a fractal—a curve of dimension between 1 and 2. Even today its dimension (denoted δ) is not known exactly; the best-proven estimate is 1.30568.
The concept of fractional dimension was popularized by Benoît Mandelbrot in his enormously influential book The Fractal Geometry of Nature. Although the meaning of dimension 1.30568 is somewhat opaque, this number is related to other properties of the foam that have direct physical meaning. For instance, if you pick any cutoff radius r, how many bubbles in the foam have radius larger than r? The answer, denoted N(r), is roughly proportional to rδ. Or if you pick the n largest bubbles, what is the remaining pore space between those bubbles? The answer is roughly proportional to n1–2/δ.
Physicists are very familiar with this sort of rule, which is called a power law. As I read the literature on Apollonian packings, an interesting cultural difference emerged between physicists and mathematicians. In the physics literature, a fractional dimension δ is de facto equivalent to a power law rδ. However, mathematicians look at things through a sharper lens, and they realize that there can be additional, slowly increasing or slowly decreasing terms. For instance, N(r) could be proportional to rδlog(r) or rδ/log(r). For physicists, who study foams empirically (or semi-empirically, via computer simulation), the logarithm terms are absolutely undetectable. The discrepancy they introduce will always be swamped by the noise in any simulation. But for mathematicians, who deal in logical rigor, the logarithm terms are where most of the action is. In 2008, mathematicians Alex Kontorovich and Hee Oh of Brown University showed that there are in fact no logarithm terms in N(r). The number of circles of radius greater than r obeys a strict power law, N(r)∼Crδ, where C is a constant that depends on the first three circles of the packing. For the “bugeye” packing illustrated in the second figure, C is about 0.201. (The tilde (∼) means that this is not an equation but an estimate that becomes more and more accurate as the radius r decreases to 0.) For mathematicians, this was a major advance. For physicists, the likely reaction would be, “Didn’t we know that already?”