Ciprian S. Borcea

Periodic frameworks and flexibility

We formulate a concise deformation theory for periodic bar-and-joint frameworks in Rd and illustrate our algebraic–geometric approach on frameworks related to various crystalline structures. Particular attention is given to periodic frameworks modelled on silica, zeolites and perovskites. For frameworks akin to tectosilicates, which are made of one-skeleta of d-dimensional simplices, with each vertex common to exactly two simplices, we prove the existence of a space of periodicity-preserving infinitesimal flexes of dimension at least . However, these infinitesimal flexes need not come from genuine flexibility, as shown by rigid examples. The changes implicated in passing from a given lattice of periods to a sublattice of periods are illustrated with frameworks modelled on perovskites.

The Number of Embeddings of Minimally Rigid Graphs

AbstractnRigid frameworks in some Euclidean space are embedded graphsnhaving a unique local realization (up to Euclidean motions) fornthe given edge lengths, although globally they may have several.nWe study the number of distinct planar embeddings of minimallynrigid graphs with

Frameworks with crystallographic symmetry

n

Line Transversals to Disjoint Balls

vertices. We show that, modulo planar rigidnmotions, this number is at most

Common Tangents to Spheres in R 3

{{2n-4}choose {n-2}} approxn4^n

On the number of embeddings of minimally rigid graphs

. We also exhibit several families which realize lower boundsnof the order of

Nodal quintic threefolds and nodal octic surfaces

2^n

Periodic Body-And-Bar Frameworks

,

Liftings and Stresses for Planar Periodic Frameworks

2.21^n

Exact workspace boundary by extremal reaches

and