Related papers: Self-Assembly with Geometric Tiles
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…
This paper considers the shape formation problem within the 3D hybrid model, where a single agent with a strictly limited viewing range and the computational capacity of a deterministic finite automaton manipulates passive tiles through…
Multi-component self-assembly mixtures offer the possibility of encoding multiple target structures with the same set of interacting components. Selective retrieval of one of the stored structures has been attempted by preparing an initial…
Microstructural geometry plays a critical role in the response of heterogeneous materials. Consequently, methods for generating microstructural samples are increasingly crucial to advanced numerical analyses. We extend Sonon et al.'s…
An \emph{auspicious tatami mat arrangement} is a tiling of a rectilinear region with two types of tiles, $1 \times 2$ tiles (dimers) and $1 \times 1$ tiles (monomers). The tiles must cover the region and satisfy the constraint that no four…
The traditional goal of inverse self-assembly is to design interactions that drive particles toward a desired target structure. However, achieving successful self-assembly also requires tuning the thermodynamic conditions under which the…
An oritatami system (OS) is a theoretical model of self-assembly via co-transcriptional folding. It consists of a growing chain of beads which can form bonds with each other as they are transcribed. During the transcription process, the…
Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use…
We generalize a previously established ab initio approach-originally developed for hexagonal close-packed (hcp) metals-to accommodate solids with both internal and external degrees of freedom. This extension enables the thermodynamic and…
The design of isophoric phased arrays composed of two-sized square-shaped tiles that fully cover rectangular apertures is dealt with. The number and the positions of the tiles within the array aperture are optimized to fit desired…
The number of complete tilings of m X n floors for tiles of shape 1 X 2, 1 X 3, 1 X 4 and 2 X 3 is computed numerically for floors up to width m=9 and variable floor lengths n. Counts are obtained for two classes, for fixed tile stack…
Computationally, low-resolution coarse-grained models provide the most viable means for simulating the large length and time scales associated with mesoscopic phenomena. Moreover, since lyotropic phases in solution may contain high solvent…
We present fast simulation methods for the self-assembly of complex shapes in two dimensions. The shapes are modeled via a general boundary curve and interact via a standard volume term promoting overlap and an interpenetration penalty. To…
We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer…
The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for…
We describe a novel geometric methodology for analyzing free-energy and kinetics of assembly driven by short-range pair-potentials in an implicit solvent, and provides illustrations of its unique capabilities. An atlas is a labeled…
A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent…
In the self-assembly process which drives the formation of cellular membranes, micelles, and capsids, a collection of separated subunits spontaneously binds together to form functional and more ordered structures. In this work, we study the…
Furniture assembly is a crucial yet challenging task for robots, requiring precise dual-arm coordination where one arm manipulates parts while the other provides collaborative support and stabilization. To accomplish this task more…
Users frequently seek to fabricate objects whose outer surfaces consist of regions with different surface attributes, such as color or material. Manufacturing such objects in a single piece is often challenging or even impossible. The…