One form of 2d periodicity is encapsulated by the definitions of 2d side-sharing tandems and runs. A 2d side-sharing tandem consists of two adjacent non-overlapping occurrences of the same rectangular block, and a side-sharing run is a maximally extended chain of side-sharing tandems. Furthering our understanding of 2d periodicity has long been an important goal, with motivation in the fields of image matching and multi-dimensional compression schemes. Much research has been accomplished on exact 2d periodicity, however, there have been few results on approximate 2d periodicity. In this paper we introduce several versions of approximate side-sharing tandems with k mismatches along with efficient algorithms for locating them in a rectangular array.