  Motivated by questions from Ehrhart theory, we present new results on
discrete equidecomposability. Two rational polygons $P$ and $Q$ are said to be
discretely equidecomposable if there exists a piecewise affine-unimodular
bijection (equivalently, a piecewise affine-linear bijection that preserves the
integer lattice $\mathbb{Z} \times \mathbb{Z}$) from $P$ to $Q$. In this paper,
we primarily study a particular version of this notion which we call rational
finite discrete equidecomposability. We construct triangles that are Ehrhart
equivalent but not rationally finitely discretely equidecomposable, thus
providing a partial negative answer to a question of Haase--McAllister on
whether Ehrhart equivalence implies discrete equidecomposability. Surprisingly,
if we delete an edge from each of these triangles, there exists an infinite
rational discrete equidecomposability relation between them. Our final section
addresses the topic of infinite equidecomposability with concrete examples and
a potential setting for further investigation of this phenomenon.
