  The application of the Shannon entropy to study the relationship between
information and structures has yielded insights into molecular and material
systems. However, the difficulty in directly observing and manipulating atoms
and molecules hampers the ability of these systems to serve as model systems
for further exploring the links between information and structures. Here, we
use, as a model experimental system, hundreds of spinning magnetic micro-disks
self-organizing at the air-water interface to generate various spatiotemporal
patterns with varying degrees of orders. Using the neighbor distance as the
information-bearing variable, we demonstrate the links among information,
structure, and interactions. Most importantly, we establish a direct link
between information and structure without using explicit knowledge of
interactions. Finally, we show that the Shannon entropy by neighbor distances
is a powerful observable in characterizing structural changes. Our findings are
relevant for analyzing natural self-organizing systems and for designing
collective robots.
