Related papers: Reflections on Tiles (in Self-Assembly)
In this paper we first show that every non-zero $\tau$-rigid $A$-module induces at least one stratifying system in the module category of $A$. Moreover, we show that each of these stratifying systems can be seen as a signed…
We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A {\it ribbon tile} of order $n$ is a connected polyomino containing exactly one square of each color. We show that the set…
The paper considers ribbon tilings of large regions and their per-tile entropy (the logarithm of the number of tilings divided by the number of tiles). For tilings of general regions by ribbon tiles of length $n$, we give an upper bound on…
We show there exist UV-complete field-theoretic models in general dimension, including $2+1$, with the spontaneous breaking of a global symmetry, which persists to the arbitrarily high temperatures. Our example is a conformal vector model…
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…
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…
Our earlier article proved that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. We re-prove this…
The problem of rectangle tiling binary arrays is defined as follows. Given an $n \times n$ array $A$ of zeros and ones and a natural number $p$, our task is to partition $A$ into at most $p$ rectangular tiles, so that the maximal weight of…
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…
Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain. Following our recently developed analytical theory for the athermal limit, here and in the…
Let $R$ be an arbitrary subset of a commutative ring. We introduce a combinatorial model for the set of tame frieze patterns with entries in $R$ based on a notion of irreducibility of frieze patterns. When $R$ is a ring, then a frieze…
We present assembly-theory, a Rust package for computing assembly indices of covalently bonded molecular structures. This is a key complexity measure of assembly theory, a recent theoretical framework quantifying selection across diverse…
Several old and recent classes of picture grammars, that variously extend context-free string grammars in two dimensions, are based on rules that rewrite arrays of pixels. Such grammars can be unified and extended using a tiling based…
We construct explicit tableau-level maps between indecomposable projective modules for the type A 0-Hecke algebra that assemble into canonical split short exact sequences lifting the basic ribbon product rule in NSym via concatenation and…
A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that…
We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1,…
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…
Inverse methods of statistical mechanics have facilitated the discovery of pair potentials that stabilize a wide variety of targeted lattices at zero temperature. However, such methods are complicated by the need to compare, within the…
Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence;…
This work builds on the previous introduction [1] of a coupled experimental-computational system devised to fully characterize the thermal behavior of complex 3D submicron electronic devices. The new system replaces the laser-based surface…