Related papers: Strict Self-Assembly of Fractals using Multiple Ha…
Patchy colloids are promising candidates for building blocks in directed self-assembly. To be successful the surface patterns need to both be simple enough to be synthesized, while feature-rich enough to cause the colloids to self-assemble…
Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic…
Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for all $N \in \mathbb{N}$, there is a tile set that uniquely self-assembles into an $N \times N$ square shape at temperature 1 with optimal…
The present paper focuses on the order-disorder transition of an Ising model on a self-similar lattice. We present a detailed numerical study, based on the Monte Carlo method in conjunction with the finite size scaling method, of the…
Three-dimensional topological insulators support gapless Dirac fermion surface states whose rich topological properties result from the interplay of symmetries and dimensionality. Their topological properties have been extensively studied…
We investigate the power of the Wang tile self-assembly model at temperature 1, a threshold value that permits attachment between any two tiles that share even a single bond. When restricted to deterministic assembly in the plane, no…
We investigate a fundamental question regarding a benchmark class of shapes in one of the simplest, yet most widely utilized abstract models of algorithmic tile self-assembly. Specifically, we study the directed tile complexity of a $k…
Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which…
The fast multipole method (FMM) performs fast approximate kernel summation to a specified tolerance $\epsilon$ by using a hierarchical division of the domain, which groups source and receiver points into regions that satisfy local…
We investigate the topological phases of the Haldane model on the Sierpi\'nski gasket. As a consequence of the fractal geometry, multiple fractal gaps arise. Additionally, a flat band appears, and due to a complex next-nearest neighbour…
We present a series of results related to mathematical models of self-assembling tiles and the impacts that three diverse properties have on their dynamics. We expand upon a series of prior results which showed that (1) the abstract Tile…
We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and…
The ability to design and synthesize ever more complicated colloidal particles opens the possibility of self-assembling a zoo of complex structures, including those with one or more self-limited length scales. An undesirable feature of…
Self-assembly is the mechanism that controls the formation of well defined structures from disordered pre-existing parts. Despite the importance of self-assembly as a manufacturing method and the increasingly large number of experimental…
New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our…
Algorithmic self-assembly, a generalization of crystal growth processes, has been proposed as a mechanism for autonomous DNA computation and for bottom-up fabrication of complex nanostructures. A `program' for growing a desired structure…
Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure:…
Self-assembly is one of the most promising strategies for making functional materials at the nanoscale, yet new design principles for making self-limiting architectures, rather than spatially unlimited periodic lattice structures, are…
We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bi-cromatic packing known up to now.
A robust route for the biased production of single-handed chiral structures has been found in generating non-spherical, multi-component double emulsions using microfluidics. The specific type of handedness is determined by the final packing…