  Many soft and biological materials display so-called 'soft glassy' dynamics;
their constituents undergo anomalous random motions and complex cooperative
rearrangements. A recent simulation model of one soft glassy material, a
coarsening foam, suggested that the random motions of its bubbles are due to
the system configuration moving over a fractal energy landscape in
high-dimensional space. Here we show that the salient geometrical features of
such high-dimensional fractal landscapes can be explored and reliably
quantified, using empirical trajectory data from many degrees of freedom, in a
model-free manner. For a mayonnaise-like dense emulsion, analysis of the
observed trajectories of oil droplets quantitatively reproduces the
high-dimensional fractal geometry of the configuration path and its associated
energy minima generated using a computational model. That geometry in turn
drives the droplets' complex random motion observed in real space. Our results
indicate that experimental studies can elucidate whether the similar dynamics
in different soft and biological materials may also be due to fractal landscape
dynamics.
