Finite motions from periodic frameworks with added symmetry

Abstract

a b s t r a c tRecent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that for finite frameworks, introducing symmetry modifies the previous general counts, and under some circumstances this symmetrized Maxwell type count can predict added finite flexibility in the structure.In this paper we combine these approaches to present new Maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells. In a number of cases, this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry. Given that many crystal structures have these added symmetries, and that their flexibility may be key to their physical and chemical properties, we present a summary of the results as a way to generate further developments of both a practical and theoretic interest.

How does symmetry impact the flexibility of proteins?

“…for symmetry-regular frameworks to have no non-trivial motion that maintains the symmetry of the structure throughout the path (see [19,38,39] for example), fairly little is known about sufficient conditions for symmetry-regular frameworks to have no non-trivial motion at all [13,40]. In fact, for finite body-bar frameworks in 3-space, the only known results are summarized in [13].…”

Fragmentary and incidental behaviour of columns, slabs and crystals

“…Any such motion will extend to a periodic motion of the infinite framework, and any periodic motion of the infinite framework will appear as a unique solution of this equation [2,10]. This matrix has 3|V O | + 2 = 3v O + 2 columns and |E O | = e O rows.…”

“…If the edge joining the equivalence class of p i and p j from V O goes from p i to (m ij t 1 + n ij t 2 ) + p j (with m ij , n ij ≥ 0, then we give the edge representative the label (i, j; m ij , n ij ). With these gains, we have a unique label for each equivalence class of edges [2,10]. an orbit multi-graph G O , m , and an orbit framework ( G O , m , p O , t 1 , t 2 ), where p O is the restriction of p to V O .…”

“…As a periodic framework, it has v O = 16 and e O = 54 = 3v O + 6-so it is periodically overbraced by 6. (As an aside, there is a re-count, using the inversive symmetry of the units of the slab, which actually brings this example down to the minimal count for periodic rigidity with inversive symmetry following the approach of Ross et al [10]. )…”

Periodic body-and-bar frameworks

“…Among other recent contributions to the subject, we mention [24] which addresses the two-dimensional case of periodic bar-and-joint frameworks and [29,32] where certain classes with 'added symmetry' are investigated. The final part of the thesis [31] includes some considerations about body-andbar frameworks "on the fixed torus".…”