We have designed and constructed DNA complexes in the form of triangles. We have created hexagonal planar tilings from these triangles via self-assembly. Unlike previously reported structures self-assembled from DNA, our structures appear to involve bending of double helices. Bending helices may be a useful design option in the creation of self-assembled DNA structures. It has been suggested that DNA self-assembly may lead to novel materials and efficient computational devices.

We designed a molecular complex, the double-double crossover, consisting of four DNA double helices connected by six reciprocal exchanges. Atomic force micrographs suggest that double-double crossover complexes self-assemble into high-density, doubly connected, two-dimensional, planar structures. Such structures may be suitable as substrates for the deposition of nanomaterials in the creation of high-density electrical and quantum devices. We speculate about a modified double-double crossover complex that might self-assemble into high-density, doubly connected, three-dimensional structures.

Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical

Abstract. Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Recent experiments in self-assembly demonstrate its potential for the parallel creation of a large number of nanostructures, including possibly computers. A systematic study of self-assembly as a mathematical process has been initiated by L. Adleman and E. Winfree. The individual components are modeled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific "glue," and two adjacent tiles will stick iff they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional "structures" such as squares and rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called a ribbon: a non-self-crossing rectilinear sequence of tiles on the plane, in which successive tiles are adjacent along an edge and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. While the problem can be proved undecidable using existing techniques if the ribbon is required to start with a given "seed" tile, our result settles the "unseeded" case, an open problem formerly known as the "unlimited infinite snake problem." The proof is based on a construction, due to R. Robinson, of a special set of tiles that allow only aperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the "strong plane-filling property"-a variation of the "plane-filling property" previously defined by J. Kari. A construction of "sandwich" tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable tiling problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A "motif" construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable. An immediate consequence of our result is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set. 1. Introduction. Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Atoms bind to each other by chemical bonds to form molecules, molecules may form crystals or macromolecules, and cells interact to form biological organisms. Recently it has been suggested that complex self-assembly processes will ultimately be used in circuit fabrication, nanorobotics, DNA computation, and amorphous computing. Indeed, in electronics, engineering, medicine, material science, manufacturing, and other disciplines, there is a continuou...

We consider the problem of fault-tolerance in nanoscale algorithmic self-assembly. We employ a standard variant of Winfree's abstract Tile Assembly Model (aTAM), the two-handed aTAM, in which square "tiles" -a model of molecules constructed from DNA for the purpose of engineering self-assembled nanostructuresaggregate according to specific binding sites of varying strengths, and in which large aggregations of tiles may attach to each other, in contrast to the seeded aTAM, in which tiles aggregate one at a time to a single speciallydesignated "seed" assembly. We focus on a major cause of errors in tile-based self-assembly: that of unintended growth due to "weak" strength-1 bonds, which if allowed to persist, may be stabilized by subsequent attachment of neighboring tiles in the sense that at least energy 2 is now required to break apart the resulting assembly; i.e., the errant assembly is stable at temperature 2.We study a common self-assembly benchmark problem, that of assembling an n × n square using O(log n) unique tile types, under the two-handed model of self-assembly. Our main result achieves a much stronger notion of fault-tolerance than those achieved previously. Arbitrary strength-1 growth is allowed; however, any assembly that grows sufficiently to become stable at temperature 2 is guaranteed to assemble into the correct final assembly of an n × n square. In other words, errors due to insufficient attachment, which is the cause of errors studied in earlier papers on fault-tolerance, are prevented absolutely in our main construction, rather than only with high probability and for sufficiently small structures, as in previous faulttolerance studies.