Combinatorial excursions in moduli space

Given an abstract triangulation of a torus, there is a unique point in moduli space which supports a circle packing for this triangulation. We will describe combinatorial deformations analogous to the process of conformal welding. These combinatorial deformations allow us to travel in moduli space from any packable torus to a point arbitrarily close to any other torus we choose. We also provide two proofs of Toki's result that any torus can be transformed into any other by a conformal welding and compute the maps necessary to accomplish the welding.