  Through the glasses of didactic reduction: We consider a (periodic)
tessellation $\Delta$ of either Euclidean or hyperbolic $n$-space $M$. By a
piecewise isometric rearrangement of $\Delta$ we mean the process of cutting
$M$ along corank-1 tile-faces into finitely many convex polyhedral pieces, and
rearranging the pieces to a new tight covering of the tessellation $\Delta$.
Such a rearrangement defines a permutation of the (centers of the) tiles of
$\Delta$, and we are interested in the group $PI(\Delta)$ of all piecewise
isometric rearrangements of $\Delta$.
  In this paper we offer: a) An illustration of piecewise isometric
rearrangements in the visually attractive hyperbolic plane, b) an explanation
how this is related to Richard Thompson's groups, c) a chapter on the structure
of the group pei$(\mathbb Z^n)$ of all piecewise Euclidean rearrangements of
the standard tessellation of $\mathbb R^n$ by unit-cubes, and d) results on the
finiteness properties of some subgroups of pei$(\mathbb Z^n)$.
