Abstract. We consider beacons, an analog of geographical greedy routing, motivated by sensor network applications. A beacon b is a point that can be activated to create a 'magnetic pull' towards itself everywhere in a polygonal domain P . We explore the properties of beacons and their effect on points in polygons, as well as demonstrate polynomial-time algorithms to compute a variety of structures defined by the action of beacons on P . We establish a polynomial-time algorithm for routing from a point s to a point t using a discrete set of candidate beacons, as well as a 2-approximation and a PTAS for routing between beacons placed without restriction in P .

When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability.