The Shape of Orthogonal Cycles in Three Dimensions

Let σ be a directed cycle whose edges have each been assigned a desired direction in 3D (East, West, North, South, Up, or Down) but no length. We say that σ is a shape cycle. We consider the following problem. Does there exist an orthogonal representation Γ of σ in 3D space such that no two edges of Γ intersect except at common endpoints and such that each edge of Γ has the direction specified in σ? If the answer is positive, we say that σ is simple. This problem arises in the context of extending orthogonal graph drawing techniques from 2D to 3D. We give a combinatorial characterization of simple shape cycles that yields linear time recognition and drawing algorithms.     

Self-assembly of gears at a fluid/air interface

This paper describes a dynamic system-a system that develops order only when dissipating energy-comprising millimeter to centimeter scale gears that self-assemble into a simple machine at a fluid/air interface. The gears are driven externally and indirectly by magnetic interactions; they are made of poly(dimethylsiloxane) (PDMS) or magnetically doped PDMS, and fabricated by soft lithography. Transfer of torque between gears can take place through three different mechanisms: mechanical interaction, hydrodynamic shear, and capillarity/overlap of menisci. Interplay between these forces allows interactions and motions that are not possible with conventional systems of gears.