Fragmentary and incidental behaviour of columns, slabs and crystals

Abstract

Between the study of small finite frameworks and infinite incidentally periodic frameworks, we find the real materials which are large, but finite, fragments that fit into the infinite periodic frameworks. To understand these materials, we seek insights from both (i) their analysis as large frameworks with associated geometric and combinatorial properties (including the geometric repetitions) and (ii) embedding them into appropriate infinite periodic structures with motions that may break the periodic structure. A review of real materials identifies a number of examples with a local appearance of ‘unit cells’ which repeat under isometries but perhaps in unusual forms. These examples also refocus attention on several new classes of infinite ‘periodic’ frameworks: (i) columns—three-dimensional structures generated with one repeating isometry and (ii) slabs—three-dimensional structures with two independent repeating translations. With this larger vision of structures to be studied, we find some patterns and partial results that suggest new conjectures as well as many additional open questions. These invite a search for new examples and additional theorems.

The rigidity of periodic body–bar frameworks on the three-dimensional fixed torus

“…The paper of Whiteley [25] addresses the question of 'fragments' of periodic frameworks. That is, he considers finite pieces of periodic structures.…”

Symmetry Adapted Assur Decompositions

“…Unusually, these two combine to give all of the information about the framework without either symmetry being forced. When we stack two layers of this example, we gain insight into examples explored with cruder tools in [29]. For example, when stacking two copies, turned to have fixed vertices in one and fixed edges in the other, this double analysis provides clearer insights.…”

Rigidity through a Projective Lens

“…We mention that a class of examples in that paper, called 'towers', were sufficiently transparent to provide initial examples of how infinitesimal motions of one part of a framework would transmit through to infinitesimal motions elsewhere: a form of mathematical allostery [126]. They also provided illustrative examples for motions of finite and infinite tubes, which also occur in biology of proteins [144]. Under appropriate conditions, we can contract such a shared edge to find a smaller infinitesimally rigid framework [108].…”