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**by Peter Sarnak (Princeton University and Institute for Advanced Study)**

**Award:** Lester R. Ford

**Year of Award:** 2012

**Publication Information:** *The American Mathematical Monthly*, vol. 118, no. 4, April 2011, pp. 291-306

**Summary** (From the Prizes and Awards booklet, MathFest 2012)

The paper begins by considering three coins—a nickel, a dime, and a quarter. A theorem of Apollonius says that another coin can be placed in the region that they bound so that all four coins are mutually tangent. Actually, Apollonius's theorem says more: given any three mutually tangent circles, there are two circles tangent to all three. This paper is about the radii of these circles, investigated through the curvature (reciprocal of the radius). Descartes established a beautiful relation among the five curvatures, and his result implies that the radii of all further circles lie in an extension field of the rationals. (It is generated by just one square root obtained from the three original radii).

Another consequence of Descartes' result is that when the three original curvatures are all integers and one other elementary condition is satisfied, then all of the subsequent curvatures are integers. This is the point at which this article takes off – it leads to connections with several other areas of mathematics, and the author acquaints the reader with several of these. They include algebra through the Apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime.

**About the Author:** (From the Prizes and Awards booklet, MathFest 2012)

**Peter Sarnak** is a Professor of Mathematics at Princeton University and the Institute for Advanced Study, Princeton. He received a B.S. degree from the University of Witwatersrand (Johannesburg) and a Ph.D. from Stanford University. His mathematical interests are wide-ranging and his research focuses on problems in number theory, automorphic forms, geometric analysis and related combinatorics, and mathematical physics.

Publication Date:

Wednesday, August 8, 2012