We aim to give a classification of Euclidian bijective polygonal piecewise isometries with a finite number of compact polygonal atoms. We rely on a specific type of triangulation process that enables us to describe a notion of combinatorial type similar to its one-dimensional counterpart for interval exchange maps. Moreover, it is possible to handle all the possible piecewise isometries, given two combinatorial types. We show that most of the examples treated in the literature of piecewise isometries can be retrieved by systematic computations. The computations yield new examples with apparently interesting behavior, but they still have to be studied in more detail. We also exhibit a new class of maps, the piecewise similarities, which fit nicely in this framework and whose behavior is shown to be highly nontrivial.
"A Tentative Classification of Bijective Polygonal Piecewise Isometries." Experiment. Math. 16 (1) 77 - 100, 2007.