Related papers: Growth Forms of Tilings
In this paper it is proved that there exist periodic monohedral tilings and finite seeds of colored tiles, which force non-periodic coloring of the whole plane
We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of…
We examine the conjectured asymptotic shape of the three dimensional corner growth model [Olejarz et. al.,PRL, 108, 016102 (2012)] by mapping the model onto a restricted solid on solid model on a triangular lattice. By choosing appropriate…
Tip-driven growth processes underlie the development of many plants. To date, tip-driven growth processes have been modelled as an elongating path or series of segments without taking into account lateral expansion during elongation.…
The scale heights of stratification and the widths of steady solar coronal loops exhibit properties unexplained by standard theory: observed scale heights are often much greater than static theory predicts, while the nearly-constant widths…
The constraints on the models for the structure formation arising from various cosmological observations at different length scales are reviewed. The status of different models for structure formation is examined critically in the light of…
It has been well known for a long time that the height function of random lozenge tilings of large domains follow a law of large number and possible limits called dimer limit shapes are well understood. For the next order, it is expected…
Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of…
Interplay between kinetic roughening and phase ordering is studied in a growth SOS model with two kinds of particles and Ising-like interaction by Monte Carlo simulations. We found that, for a sufficiently large coupling, growth is strongly…
We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases,…
Growth and folding in one-layered model tissue sheets are studied in a stochastic, lattice-free single cell model which considers the discrete cellular structure of the tissue, and a coarse grained analytical approach. The polarity of the…
This paper is about integral zonotopes. It is proven that large zonotopes in a convex cone have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes are very close to a fixed convex set. Several…
The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by…
Crystal growth processes produce a diverse array of surface formations, primarily distinguished by their geometric shapes. While some structures strictly adhere to the underlying crystal symmetry, others exhibit universal circular or oval…
The growing sluggishness of glass-forming liquids is thought to be accompanied by growing structural order. The nature of such order, however, remains hotly debated. A decade ago, point-to-set (PTS) correlation lengths were proposed as…
We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus $g$ with a single boundary…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two…
Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of $\bbZ^2$. The model describes a crystal surface by assigning a non-negative integer height $\eta_x$ to each site $x$ in the box and 0…
Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on…