New Geometric Algorithms for Fully Connected Staged Self-Assembly

We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in O(log^2 n) stages, for various scale factors and temperature {\tau} = 2 as well as {\tau} = 1. Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.Comment: 21 pages, 14 figures; full version of conference paper in DNA2

Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly

In this paper we explore the power of geometry to overcome the limitations of non-cooperative self-assembly. We define a generalization of the abstract Tile Assembly Model (aTAM), such that a tile system consists of a collection of polyomino tiles, the Polyomino Tile Assembly Model (polyTAM), and investigate the computational powers of polyTAM systems at temperature 1, where attachment among tiles occurs without glue cooperation (i.e., without the enforcement that more than one tile already existing in an assembly must contribute to the binding of a new tile). Systems composed of the unit-square tiles of the aTAM at temperature 1 are believed to be incapable of Turing universal computation (while cooperative systems, with temperature \u3e 1, are able). As our main result, we prove that for any polyomino P of size 3 or greater, there exists a temperature-1 polyTAM system containing only shape-P tiles that is computationally universal. Our proof leverages the geometric properties of these larger (relative to the aTAM) tiles and their abilities to effectively utilize geometric blocking of particular growth paths of assemblies, while allowing others to complete. In order to prove the computational powers of polyTAM systems, we also prove a number of geometric properties held by all polyominoes of size ≥ 3. To round out our main result, we provide strong evidence that size-1 (i.e. aTAM tiles) and size-2 polyomino systems are unlikely to be computationally universal by showing that such systems are incapable of geometric bitreading, which is a technique common to all currently known temperature-1 computationally universal systems. We further show that larger polyominoes with a limited number of binding positions are unlikely to be computationally universal, as they are only as powerful as temperature-1 aTAM systems. Finally, we connect our work with other work on domino self-assembly to show that temperature-1 assembly with at least 2 distinct shapes, regardless of the shapes or their sizes, allows for universal computation

Geometrical Tile Design for Complex Neighborhoods

Recent research has showed that tile systems are one of the most suitable theoretical frameworks for the spatial study and modeling of self-assembly processes, such as the formation of DNA and protein oligomeric structures. A Wang tile is a unit square, with glues on its edges, attaching to other tiles and forming larger and larger structures. Although quite intuitive, the idea of glues placed on the edges of a tile is not always natural for simulating the interactions occurring in some real systems. For example, when considering protein self-assembly, the shape of a protein is the main determinant of its functions and its interactions with other proteins. Our goal is to use geometric tiles, i.e., square tiles with geometrical protrusions on their edges, for simulating tiled paths (zippers) with complex neighborhoods, by ribbons of geometric tiles with simple, local neighborhoods. This paper is a step toward solving the general case of an arbitrary neighborhood, by proposing geometric tile designs that solve the case of a “tall” von Neumann neighborhood, the case of the f-shaped neighborhood, and the case of a 3 × 5 “filled” rectangular neighborhood. The techniques can be combined and generalized to solve the problem in the case of any neighborhood, centered at the tile of reference, and included in a 3 × (2k + 1) rectangle

Self assembling magnetic tiles

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.Includes bibliographical references (p. 27).Self assembly is an emerging technology in the field of manufacturing. Inspired by nature's ability to self assembly proteins from amino acids, this thesis attempts to demonstrate self assembly on the macro-scale. The primary focus of the thesis was to improve the design of magnetic tile self assembly. By constructing a flexible chain embedded with permanent magnets, self assembly is achieved through magnetic interaction. Theory has shown that such a chain is capable of self assembling into any 3D shape without self-intersection. The 3D shape created by the chain is predetermined by the sequence of the tiles. For this thesis, two chains were manufactured, each self assembling into one distinct shape. One chain self assembled into a sphere while the other self assembled into a '3-leaf clover'. An important characteristic shared by the two chains is that they both were constructed from 48 tiles that had the same proportion of north-pole and south-pole facing magnets. The difference between the two 3D shapes created is a direct result of the magnet tile sequencing, only. To connect the tiles, two different types of connectors were designed: one rigid and one flexible.(cont.) The rigid connector design was able to stabilize the chain geometry; however some joints displayed excessive rotational friction. Additionally, the chain was not robust and was easily broken if dropped. When the chain was manufactured using flexible connectors, the amount of friction in the joints was significantly reduced. However, the chain lost geometric stability since the flexible connectors could not overcome some torsion forces created by the magnets. Ultimately, this thesis provided supporting data for the theoretical arguments concerning the ability of a flexible chain to self assemble into arbitrary 3D shapes. By predetermining a sequence of magnetic tiles, it can be known with certainty what shape the chain will assume. This thesis furthered the understanding of the mechanisms of self assembly, providing groundwork for the eventual application on the nano-scale.by Jessica A. Rabl.S.B

Contrasting Geometric Variations of Mathematical Models of Self-assembling Systems

Self-assembly is the process by which complex systems are formed and behave due to the interactions of relatively simple units. In this thesis, we explore multiple augmentations of well known models of self-assembly to gain a better understanding of the roles that geometry and space play in their dynamics. We begin in the abstract Tile Assembly Model (aTAM) with some examples and a brief survey of previous results to provide a foundation. We then introduce the Geometric Thermodynamic Binding Network model, a model that focuses on the thermodynamic stability of its systems while utilizing geometrically rigid components (dissimilar to other thermodynamic models). We show that this model is computationally universal, an ability conjectured to be impossible in similar models with non-rigid components. We continue by introducing the Flexible Tile Assembly Model, a generalization of the 2D aTAM that allows bonds between tiles to flex and assemblies to therefore reconfigure. We show how systems in this model can deterministically assemble shapes that adhere to a number of certain restrictions. Finally, we introduce the Spatial abstract Tile Assembly Model, a variation of the 3D aTAM that restricts tiles from attaching without a diffusion path. We show that this model is intrinsically universal, a property of computational models to simulate themselves which has been shown for the 3D aTAM and other similar models. We conclude this thesis with a summary of the presented results, a brief impact analysis, and potential directions for future research within this area