A Tisket, a Tasket, an Apollonian Gasket
Fractals made of circles do funny things to mathematicians
For many physical problems, the classical definition of the Apollonian gasket is too restrictive, and a random model may be more appropriate. A bubble may start growing in a randomly chosen location and expand until it hits an existing bubble, and then stop. Or a tree in a forest may grow until its canopy touches another tree, and then stop. In this case, the new circles do not touch three circles at a time, but only one. Computer simulations show that these “random Apollonian packings” still behave like a fractal, but with a different dimension. The empirically observed dimension is 1.56. (This means the residual set is larger, and the packing is less efficient, than in a deterministic Apollonian gasket.) More recently, Stefan Hutzler of Trinity College Dublin, along with Gary Delaney and Tomaso Aste of the University of Canberra, studied the effect of bubbles with different shapes in a random Apollonian packing. They found, for example, that squares become much more efficient packers than circles if they are allowed to rotate as they grow, but surprisingly, triangles become only slightly more efficient. As far as I know, all of these results are begging for a theoretical explanation.
For mathematicians, however, the classical, deterministic Apollonian gasket still offers more than enough challenging problems. Perhaps the most astounding fact about the Apollonian gasket is that if the first four circles have integer bends, then every other circle in the packing does too. If you are given the first three circles of an Apollonian gasket, the bend of the fourth is found (as explained above) by solving a quadratic equation. However, every subsequent bend can be found by solving a linear equation:
For instance, in the “bugeye” gasket, the three circles with bends a=2, b=3, and c=15 are mutually tangent to two other circles. One of them, with bend d=2, is already given in the first generation. The other has bend d'=38, as predicted by the formula, 2+38=2(2+3+15). More importantly, even if we did not know d', we would still be guaranteed that it was an integer, because a, b, c and d are.
Hidden behind this “baby Descartes equation” is an important fact about Apollonian gaskets: They have a very high degree of symmetry. Circles a, b and c actually form a sort of curved mirror that reflects circle d to circle d' and vice versa. Thus the whole gasket is like a kaleidoscopic image of the first four circles, reflected again and again through an infinite collection of curved mirrors.
Kontorovich and Oh exploited this symmetry in an extraordinary and amusing way to prove their estimate of the function N(r). Remember that N(r) simply counts how many circles in the gasket have radius larger than r. Kontorovich and Oh modified the function N(r) by introducing an extra variable of position—roughly equivalent to putting a lightbulb at a point x and asking how many circles illuminated by that lightbulb have radius larger than r. The count will fluctuate, depending on exactly where the bulb is placed. But it fluctuates in a very predictable way. For instance, the count is unchanged if you move the bulb to the location of any of its kaleidoscopic reflections.
This property makes the “lightbulb counting function” a very special kind of function, one which is invariant under the same symmetries as the Apollonian gasket itself. It can be broken down into a spectrum of similarly symmetric functions, just as a sound wave can be decomposed into a fundamental frequency and a series of overtones. From this spectrum, you can in theory find out everything you want to know about the lightbulb counting function, including its value at any particular location of the lightbulb.
For a musical instrument, the fundamental frequency or lowest overtone is the most important one. Similarly, it turned out that the first symmetric function was all that Kontorovich and Oh needed to figure out what happens to N(r) as r approaches 0.
In this way, a simple problem in geometry connects up with some of the most fundamental concepts of modern mathematics. Functions that have a kaleidoscopic set of symmetries are rare and wonderful. Kontorovich calls them “the Holy Grail of number theory.” Such functions were, for instance, used by Andrew Wiles in his proof of Fermat’s Last Theorem. An interesting new kaleidoscope is enough to keep mathematicians happy for years.