Related papers: Constructing Interlocking Assemblies with Crystall…
We show that model molecules with particular rotational symmetries can self-assemble into network structures equivalent to rhombus tilings. This assembly happens in an emergent way, in the sense that molecules spontaneously select irregular…
In this paper we demonstrate the power of a model of tile self-assembly based on active glues which can dynamically change state. We formulate the Signal-passing Tile Assembly Model (STAM), based on the model of Padilla, Liu, and Seeman to…
Assembly of spherical colloidal particles into extended structures, including linear strings, in the absence of directional interparticle bonding interactions or external perturbation could facilitate the design of new functional materials.…
Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce…
A family of stabilizer-free $P_k$ virtual elements are constructed on triangular meshes. When choosing an accurate and proper interpolation, the stabilizer of the virtual elements can be dropped while the quasi-optimality is kept. The…
The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group…
This paper introduces nucleus clustering in Voronoi tessellations of plane surfaces with applications in the geometry of digital images. A \emph{nucleus cluster} is a collection of Voronoi regions that are adjacent to a Voronoi region…
In this work, we focus on the family of shell formulations referred to as "solid shells", where the simulation of shell-type structures is performed by means of a mesh of 3D solid elements, with typically only one element through the…
We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced by the second author, Hunton and Kellendonk. To do this we completely…
Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection…
The Materials Project crystal structure database has been searched for materials possessing layered motifs in their crystal structures using a topology-scaling algorithm. The algorithm identifies and measures the sizes of bonded atomic…
Stochastic geometry provides a powerful framework for modelling complex random structures, with applications in physics, materials science, biology, and other fields. The three-dimensional microstructure of polycrystalline materials is…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
A relaxed version of Gummelt's covering rules for the aperiodic decagon is considered, which produces certain random-tiling-type structures. These structures are precisely characterized, along with their relationships to various other…
In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For…
Beams made from thin-walled elements, whilst very efficient in terms of the structural strength and stiffness to weight ratios, can be susceptible to highly complex instability phenomena. A nonlinear analytical formulation based on…
The self-assembly of functionalized (patchy) particles with directional interactions into target structures is still a challenge, despite the significant experimental advances on their synthesis. The self-assembly pathways are typically…
The central discovery of $2d$ conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire…
Hierarchical crack patterns that arise during the drying of thin films of colloidal dispersions or polymer solutions on a solid substrate are of interest both from a fundamental standpoint and in the context of the creation of transparent…
Understanding the drift motion and dynamical locking of crystalline clusters on patterned substrates is important for the diffusion and manipulation of nano- and micro-scale objects on surfaces. In a previous work, we studied the…