  Dimensionality plays an essential role in determining the nature and
properties of a physical system. For quantum systems the impact of interactions
and fluctuations is enhanced in lower dimensions, leading to a great diversity
of genuine quantum effects for reduced dimensionality. In most cases, the
dimension is fixed to some integer value. Here, we experimentally probe the
dimensional crossover from two to one dimension using strongly interacting
ultracold bosons in variable lattice potentials and compare the data to
ab-initio theory that takes into account non-homogeneous trapping and non-zero
temperature. From a precise measurement of the momentum distribution we analyze
the characteristic decay of the one-body correlation function in the two
dimensionalities and then track how the decay is modified in the crossover. A
varying two-slope structure is revealed, reflecting the fact that the particles
see their dimensionality as being one or two depending on whether they are
probed on short or long distances, respectively. Our observations demonstrate
how quantum properties in the strongly-correlated regime evolve in the
dimensional crossover as a result of the interplay between dimensionality,
interactions, and temperature.
