相关论文: Aperiodic Tilings on the Computer
In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly…
Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging…
A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of…
We describe a computer algorithm that searches for substitution rules on a set of triangles, the angles of which are all integer multiples of {\pi}/n. We find new substitution rules admitting 7-fold rotational symmetry at many different…
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…
Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show…
We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct…
We show that translational tiling problems in a quotient of $\mathbb{Z}^d$ can be effectively reduced or ``simulated'' by translational tiling problems in $\mathbb{Z}^d$. In particular, for any $d \in \mathbb{N}$, $k < d$ and…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
Quadrilateral layouts on surfaces are valuable in texture mapping, and essential in generation of quadrilateral meshes and in fitting splines. Previous work has characterized such layouts as a special metric on a surface or as a meromorphic…
A method for creating metasurfaces using a standing wave, formed in a dielectric, is proposed. Such metasurfaces are formed from metal suspensions, deposited on a dielectric plate, placed in a metal frame-screen. A series of parameters for…
The chapter contains a detailed presentation of the surface integral theory for modelling light diffraction by surface-relief diffraction gratings having a one-dimensional periodicity. Several different approaches are presented, leading…
An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers…
The main goal of this paper is to define a 1-1 correspondence between between substitution tilings constructed by inflation and the arithmetic of positional representation in the underlying real vector space. It introduces a generalization…
Kirigami, the art of introducing cuts in thin sheets to enable articulation and deployment, has become an inspiration for a novel class of mechanical metamaterials with unusual properties. Here we complement the use of periodic tiling…
The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to…
A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be…
By reformulating Wang tiles with tensors, we propose a natural generalization to the probabilistic and quantum setting. In this new framework, we introduce notions of tilings and periodicity directly extending their classical counterparts.…
We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…