A tourist enjoying the sunset at the harbor of a Mediterranean village will likely notice fishing nets spread out on the quay walls for drying. If the observer has a sense for aesthetics of structure, he or she will recognize that the beauty of the nets lies in their regularity and simplicity. Except for where they need mending, the nets consist of knots, or cross-links, connected by rope segments of uniform length, each knot connected to three or four neighbors.

Topologically, such a net is a two-dimensional object. Stretched out flat on the ground, its regularity—or periodicity—is apparent. Gathered up into a compact, 3D state, the periodicity is hidden and the net’s beauty is diminished. Not only that, the net transitions from a more functional state to a less functional one.

Over the past several decades, chemists have been working to realize fishing nets on the molecular scale. Much the...

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