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An Apollonian Opportunity

To the Editors:

In Dana Mackenzie’s interesting column “A Tisket, a Tasket, an Apollonian Gasket” (January–February), Peter Sarnak remarked on the present-day inability of mathematics to prove or explain certain conjectures, including his own. Those conjectures concern the number series of the bends in Apollonian Gaskets. “The necessary mathematics has not been invented yet,” Sarnak said.

It is interesting to remember something stated more than 200 years ago by Carl Friedrich Gauss. In his one-page proof of the long-unproven Wilson’s prime number theorem, first published by Edward Waring, Gauss noted that “neither of them was able to prove the theorem, and Waring confessed that the demonstration seemed more difficult because no notation can be devised to express a prime number. But in our opinion truths of this kind should be drawn from notions rather than from notations.”

Sarnak seems to have ignored Gauss’s advice. That, unwittingly, may dissuade those who might otherwise attempt to prove those unsolved theorems.

Bernard H. Soffer
Pacific Palisades, CA

To the Editors:

Considering that the geometry of the circle involves irrational numbers such as pi and square roots, I was struck by the seemingly infinite array of integers in the Apollonian gaskets described by Dana Mackenzie. One view of this is that each pair of mutually tangent circles has two infinite series of tangent circles spiraling into crevices between them. There are an infinite number of these mutually tangent pairs, each with a pair of infinite series. Some series appear more than once and some are part of other series.

I have found a linear relation that is somewhat different than Mackenzie’s by choosing two tangent circles (say with curvatures a and b) from any four mutually tangent circles. Of the remaining two circles, call the curvature of the larger d0 and the curvature of the smaller d1. Thus d0 is the starting term in a series of curvatures and d1 is the second term. Other curvatures are determined by the linear formula obtained by subtracting Descartes’s equation written for a, b, dn-2, dn-1 from that for a, b, dn-1, dn. The resulting equation, dn = 2(a + b + dn-1) - dn-2, can be used to determine the successive values of dn by a process of iteration. Because Descartes’s equation is a quadratic, the difference of the differences between consecutive terms is a constant and equal to 2(a + b) in each series. This seems to apply to the irrational roots of Descartes’s equation, also.

Ronald Csuha
New York, NY

Dr. Mackenzie responds:

I see no conflict between Sarnak’s quote and Gauss’s admonition. Sarnak would certainly agree that new notions, not new notations, are needed to prove his “local-to-global principle” for Apollonian packings.

I am glad to report that Elena Fuchs (Sarnak’s student) and Jean Bourgain have proven a “positive density theorem.” This relates to the question Ron Graham asked, about whether even 1 percent of the numbers that could occur in an Apollonian gasket actually do occur. Fuchs has shown that the answer is yes, provided 1 percent is replaced by a sufficiently small (but positive) number. Interestingly, her approach was to use carefully selected subsets of the Apollonian gasket, an approach not too dissimilar from what Ronald Csuha proposes. Instead of the sequence of all circles tangent to two fixed circles, she looks at the somewhat more complicated sequence of circles tangent to a single fixed circle. The preprint will be posted at the open-access site

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