Related papers: Aperiodic tilings and entropy

Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that…

We review recent advances in Klein and anti-Klein tunneling in one- and two-dimensional materials. Using a general tight-binding framework applied to multiple periodic systems, we establish the criteria for the emergence of Klein tunneling…

Magnetic topological insulators (MTIs) are narrow gap semiconductor materials that combine non-trivial band topology and magnetic order. Unlike their nonmagnetic counterparts, MTIs may have some of the surfaces gapped due to breaking the…

We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry…

We prove that for each positive integer g, there exists a complete minimal surface of genus g that is properly embedded in three-dimensional euclidean space and that is asymptotic to the helicoid.

In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…

We explore the entanglement entropy of CFT systems in contact with large bath system, such that the complete system lives on the boundary of $AdS_{d+1}$ spacetime. We are interested in finding the HEE of a bath (system-B) in contact with a…

We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation-preserving images of the tile. The winner remains Sommerville's type 4v.

We prove some uniqueness results for conics of minimal area that enclose a compact, full-dimensional subset of the elliptic plane. The minimal enclosing conic is unique if its center or axes are prescribed. Moreover, we provide sufficient…

Architected materials achieve unique mechanical properties through precisely engineered microstructures that minimize material usage. However, a key challenge of low-density materials is balancing high stiffness with stable deformability up…

In this paper we study algorithms for tiling problems. We show that the conditions $(T1)$ and $(T2)$ of Coven and Meyerowitz, conjectured to be necessary and sufficient for a finite set $A$ to tile the integers, can be checked in time…

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present…

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of…

Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular (color) pattern. For k >= 1, k-PATS is a variant of PATS that restricts input patterns to those with at most…

We construct the first example of a lattice on an irreducible Euclidean building that is not residually finite. Conjecturally, the normal subgroup theorem extends to this lattice making it virtually simple.

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from e.g. kites and darts, squares and equilateral triangles, rhombi or shield shaped tiles and can have a variety of different symmetries. However, almost all…

We find explicit optimal vertex, edge and face coulourings for the chair tiling, the Ammann--Beenker tiling, the rational pinwheel tiling and the pinwheel tiling.

Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropy-minimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it.

A challenge of molecular self-assembly is to understand how to design particles that self-assemble into a desired structure and not any of a potentially large number of undesired structures. Here we use simulation to show that a strategy of…

We consider the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long wavelength fluctuations in a broad class of one-dimensional substitution tilings. We present a simple argument that predicts the…