Circle Packings on Affine Tori

This thesis is a study of circle packings for arbitrary combinatorial tori in the geometric setting of affine tori. Certain new tools needed for this study, such as face labels instead of the usual vertex labels, are described. It is shown that to each combinatorial torus there corresponds a two real parameter family of affine packing labels. A construction of circle packings for combinatorial fundamental domains from affine packing labels is given. It is demonstrated that such circle packings have two affine side-pairing maps, and also that these side-pairing maps depend continuously on the two real parameters.