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Programmable matter is a material whose properties can be programmed to achieve specific shapes or stiffnesses upon command. This concept requires constituent elements to interact and rearrange intelligently in order to meet the goal. This paper considers achieving programmable sheets that can form themselves in different shapes autonomously by folding.(More)
We introduce the problem of <i>shape replication</i> in the Wang tile self-assembly model. Given an input shape, we consider the problem of designing a self-assembly system which will replicate that shape into either a specific number of copies, or an unbounded number of copies. Motivated by practical DNA implementations of Wang tiles, we consider a model(More)
We present a universal crease pattern—known in geometry as the tetrakis tiling and in origami as box pleating—that can fold into any object made up of unit cubes joined face-to-face (polycubes). More precisely, there is one universal finite crease pattern for each number n of unit cubes that need to be folded. This result contrasts previous universality(More)
This paper considers planning and control algorithms that enable a programmable sheet to realize different shapes by autonomous folding. Prior work on self-reconfiguring machines has considered modular systems in which independent units coordinate with their neighbors to realize a desired shape. A key limitation in these prior systems is the typically many(More)
— Modular robots consist of many small units that attach together and can perform local motions. By combining these motions, we can achieve a reconfiguration of the global shape. The term modular comes from the idea of grouping together a fixed number of units into a module, which behaves as a larger individual component. Recently, a fair amount of research(More)
The empty space around n disjoint line segments in the plane can be partitioned into n + 1 convex faces by extending the segments in some order. The dual graph of such a partition is the plane graph whose vertices correspond to the n + 1 convex faces, and every segment end-point corresponds to an edge between the two incident faces on opposite sides of the(More)
We address the question: How many edge guards are needed to guard an orthogonal polyhedron of e edges, r of which are reflex? It was previously established [3] that e/12 are sometimes necessary and e/6 always suffice. In contrast to the closed edge guards used for these bounds, we introduce a new model, open edge guards (excluding the endpoints of the(More)
A set of n disjoint line segments in the plane and a permutation π of the 2n segment endpoints define a partition of the plane into convex faces: extend the segments beyond their endpoints one-by-one in the order given by π until they hit another segment, a previous extension, or infinity. If no three segment endpoints are collinear, then every permutation(More)