Physical Sciences and Mathematics Commons^™

Generalized And Higher Dimensional Apollonian Packings, Daniel Lautzenheiser

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this thesis, we show that circle, sphere, and higher dimensional sphere packings may

be realized as subsets of the boundary of hyperbolic space, subject to certain symmetry

conditions based on a discrete group of motions of the hyperbolic space. This leads to

developing and applying counting methods which admit rigorous upper and lower bounds on

the Hausdorff (or Besikovitch) dimension of the residual set of several generalized Apollonian

circle packings. We find that this dimension (which also coincides with the critical exponent

of a zeta-type function) of each packing is strictly greater than that of the Apollonian

packing, supporting …