Hofstadter Butterfly and a Hidden Apollonian Gasket

Statistical Regularity of Apollonian Gaskets

Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic the...

Energy and Laplacian on the Sierpiński Gasket

This is an expository paper which includes several topics related to the Dirichlet form analysis on the Sierpiński gasket. We discuss the analog of the classical Laplacian; approximation by harmonic functions that gives a notion of a gradient; directional energies and an equipartition of energy; analysis with respect to the energy measure; harmonic coordinates; and non self-similar Dirichlet fo...

Letter to J . Lagarias about Integral Apollonian Packings June , 2007 from

Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

A Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. Part I shoewed there is a natural group action on Desc...

The sensual Apollonian circle packing

The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i], inspired by, and intimately related to, the ‘sensual qu...