A Tisket, a Tasket, an Apollonian Gasket
Fractals made of circles do funny things to mathematicians
Kontorovich learned about the Apollonian kaleidoscope from his mentor, Peter Sarnak of Princeton University, who learned about it from Lagarias, who learned about it from Wilks and Mallows. For Sarnak, the Apollonian gasket is wonderful because it has neither too few nor too many mirrors. If there were too few, you would not get enough information from the spectral decomposition. If there were too many, then previously known methods, such as the ones Wiles used, would already answer all your questions.
Because Apollonian gaskets fall right in the middle, they generate a host of unsolved number-theoretic problems. For example, which numbers actually appear as bends in a given gasket? These numbers must satisfy certain “congruence restrictions.” For example, in the bugeye gasket, the only legal bends have a remainder of 2, 3, 6 or 11 when divided by 12. So far, it seems that every number that satisfies this congruence restriction does indeed appear in the figure somewhere. (The reader may find it amusing to hunt for 2, 3, 6, 11, 14, 15, 18, 23, etc.) “Computation indicates that every number occurs, but we can’t prove that even 1 percent of them actually occur!” says Ron Graham of the University of California at San Diego. For other Apollonian gaskets, such as the “coins” gasket in the fifth figure, there are some absentees—numbers that obey the congruence restrictions but don’t appear in the gasket. Sarnak believes, however, that the number of absentees is always finite, and beyond a certain point any number that obeys the congruence restrictions does appear somewhere in the gasket. At this point, though, he is far from proving this conjecture—the necessary math just doesn’t exist yet.
And even if all the problems concerning the classic Apollonian gaskets are solved, there are still gaskets galore for mathematicians to work on. As mentioned before, they could study random Apollonian gaskets. Another modification is the gasket shown in the last figure, where each pore is filled by three circles instead of one. Mallows and Gerhard Guettler have shown that such gaskets behave similarly to the original Apollonian gaskets—if the first six bends are integers, then all the rest of the bends are as well. Ambitious readers might want to work out the “Descartes formula” and the “baby Descartes formula” for these configurations, and investigate whether there are congruence restrictions on the bends.
Perhaps you, too, will be inspired to write a poem or paint a tablet in honor of Apollonius’ ingenious legagy. “For me, what’s attractive about Apollonian gaskets is that even my 14-year-old daughter finds them interesting,” says Sarnak. “It’s truly a god-given problem—or perhaps a Greek-given problem.”