Bounded Apollonian circle packings (ACP’s) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In [S1], Sarnak proves that there are inf...

An Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well, making the packings of great interest from a number...

“Nature” is one of the most challenging concepts in philosophy, and notoriously difficult to define. In ancient Greece, two strategies for coming to terms with nature were developed. On the one hand, nature was seen as a perfect geometrical order, analysable with the help of geometry and deductive reasoning. On the other hand, a more Dionysian view emerged, stressing nature’s unpredictability, ...

In the middle part of his Brouillon Project on conics, Girard Desargues develops theory traversale, a notion that generalizes Apollonian diameter and allows to give unified treatment three kinds conics. We showed elsewhere it leads complete projective polarity for The present article, which shall close our study Project, is devoted last text, in puts traversal into practice by giving very elega...

Jeffrey C. Lagarias* ([email protected]), Dept. of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043. From Apollonian circle packings to Fibonacci numbers. Apollonian circle packings are infinite packings of circles, constructed recursively from a initial configuration of four mutually touching circles by adding circles externally tangent to triples of such circl...

We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and spheres in 3d. As expected, the densest packing is achieved with power-law size distributions. We also test the method on homogeneous and on empirical real ...