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String-node nets and meshes. (English) Zbl 1384.52021

Summary: New classes of infinite bond-node structures are introduced, namely string-node nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of a (countable) line segment net by a countable scaling group. A linear mesh has strings that are straight lines and nodes given by the intersection points of these lines. Classes of meshes, such as the regular meshes in \(\mathbb R^2\) and \(\mathbb R^3\), are defined and classified. String-length preserving motions are also determined for a number of fundamental examples, and contrasting flexing and rigidity properties are obtained with respect to noncrossing motions in the space of smooth meshes.

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
74N05 Crystals in solids
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References:

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