  The Douglas--Rachford algorithm is a popular algorithm for solving both
convex and nonconvex feasibility problems. While its behaviour is settled in
the convex inconsistent case, the general nonconvex inconsistent case is far
from being fully understood. In this paper, we focus on the most simple
nonconvex inconsistent case: when one set is a hyperplane and the other a
doubleton (i.e., a two-point set). We present a characterization of cycling in
this case which --- somewhat surprisingly --- depends on whether the ratio of
the distance of the points to the hyperplane is rational or not. Furthermore,
we provide closed-form expressions as well as several concrete examples which
illustrate the dynamical richness of this algorithm.
