Related papers: Tiling by rectangles and alternating current
We prove that one can cover the $1 \times b$ rectangle by equal squares on both sides in one layer iff $b = p \pm \sqrt{p^2 - r^2} $, where $p \ge r \ge 0$ and $p,q \in \mathbb{Q}$.
We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it…
We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in…
A grid method using tiling by fundamental domain of simple 2D lattices is presented. It refer to a previous work done by Stampfli in $1986$ using two tilings by regular hexagons, one rotate by $\pi/2$ relatively to the other. This allows to…
Electrical currents at the surface or edge of a topological insulator are intrinsically spin-polarized. We show that such surface/edge currents can be used to switch the orientation of a molecular magnet weakly coupled to the surface or…
We characterize the polytopes in $\mathbb{R}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in…
In this paper, we study 2-port networks and introduce new concepts of voltage drop and $\Pi$-equivalence. The main result is that each planar network is $\Pi$-equivalent to a network with no more than 5 edges. This implies that if an…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…
We show that the following problem is undecidable: given two polygonal prototiles, determine whether the plane can be tiled with rotated and translated copies of them. This improves a result of Demaine and Langerman [SoCG 2025], who showed…
The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$…
These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling.…
Linear and non-linear surface waves on a ferrofluid cylinder surrounding a current-carrying wire are investigated. Suppressing the Rayleigh-Plateau instability of the fluid column by the magnetic field of a sufficiently large current in the…
Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…
We consider three-dimensional domino tilings of cylinders $\mathcal{D} \times [0,N] \subset \mathbb{R}^3$, where $\mathcal{D} \subset \mathbb{R}^2$ is a balanced quadriculated disk and $N \in \mathbb{N}$. A flip is a local move in the space…
We investigate CMC-surfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMC-surface, we develop an…
We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. We have two main results. The first is a…
In a recent paper, Byun presented nice formulas for the enumeration of lozenge tilings of certain hexagonal regions with intrusions. This paper attempts to generalise some of Byun's investigations.
Suppose $f\in L^1(\mathbb{R}^d)$, $\Lambda\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles,…