We use a simple fragmentation model to describe the statistical behavior of
the Voronoi cell patterns generated by a set of points in 1D and in 2D. In
particular, we are interested in the distribution of sizes of these Voronoi
cells. Our model is completely defined by two probability distributions in 1D
and again in 2D, the probability to add a new point inside an existing cell and
the probability that this new point is at a particular position relative to the
preexisting point inside this cell. In 1D the first distribution depends on a
single parameter while the second distribution is defined through a
fragmentation kernel; in 2D both distributions depend on a single parameter.
The fragmentation kernel and the control parameters are closely related to the
physical properties of the specific system under study. We use our model to
describe the Voronoi cell patterns of several systems. Specifically, we study
the island nucleation with irreversible attachment, the 1D car parking problem,
the formation of second-level administrative divisions, and the pattern formed
by the Paris M\'etro stations.Comment: 12 pages, 9 figure
