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- Zachary Abel, Nadia Benbernou, Mirela Damian, Erik D. Demaine, Martin L. Demaine, Robin Y. Flatland +2 others
- SODA
- 2010

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)

- Nadia Benbernou, Patricia Cahn, Joseph O 'rourke
- 2004

We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping " volcano unfolding. " These un-foldings keep the base intact, unfold the sides outward, splayed around the base, and attach the top to the tip of some side rib. Our result… (More)

- Brad Ballinger, Nadia Benbernou, Francisco Gomez, Joseph O 'rourke, Godfried Toussaint
- 2009

The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales it claims that the complement of a chord that uses half the pitches of a scale is homometric to—i.e., has the same interval structure as—the original chord. In terms of onsets it claims that the complement of a rhythm with the same… (More)

We present data structures for triangular emptiness and reporting queries for a planar point set, where the query triangle contains the origin. The data structures use near-linear space and achieve polylogarithmic query times. 1 Introduction Simplex range searching (emptiness, reporting, counting) [1] is a fundamental problem in computational geometry.… (More)