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The application of the Shannon entropy to study the relationship betweeninformation and structures has yielded insights into molecular and materialsystems. However, the difficulty in directly observing and manipulating atomsand molecules hampers the ability of these systems to serve as model systemsfor further exploring the links between information and structures. Here, weuse, as a model experimental system, hundreds of spinning magnetic micro-disksself-organizing at the air-water interface to generate various spatiotemporalpatterns with varying degrees of orders. Using the neighbor distance as theinformation-bearing variable, we demonstrate the links among information,structure, and interactions. Most importantly, we establish a direct linkbetween information and structure without using explicit knowledge ofinteractions. Finally, we show that the Shannon entropy by neighbor distancesis a powerful observable in characterizing structural changes. Our findings arerelevant for analyzing natural self-organizing systems and for designingcollective robots.

Order and Information in the Patterns of Spinning Magnetic Micro-disks at the Air-water Interface