Related papers: Designing aperiodic to periodic interfaces
A plethora of unconventional localization phenomena and fractal features of linear spectrum observed in quasiperiodic structures have been accompanied by a long-standing quest for the geometrical elements and structures that permit tilings…
Topological photonics provides a robust and flexible platform for controlling light, enabling functionalities such as backscattering-immune edge transport and slow-light propagation. In this work, we design and characterize photonic…
An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics…
Conventional metalenses control light by varying meta-atom geometry, a design strategy that inherently couples phase modulation to structural dimensions and exacerbates chromatic dispersion. Here, we break this paradigm by decoupling phase…
How do people come up with new sets of tiles including new tile shapes that would only tile non-periodically? This paper presents our graphical journey in tilings and provides a new set of three polyominoes named Ax for its relationship…
We give a set of tiles that enforces the sphinx tiling substitution system; the tiles are thus aperiodic.
Icosahedron and dodecahedron can be dissected into tetrahedral tiles projected from 3D-facets of the Delone polytopes representing the deep and shallow holes of the root lattice D_6. The six fundamental tiles of tetrahedra of edge lengths 1…
Mathematicians have been interested in non-periodic tilings of space for decades; however, it was the unexpected discovery of non-periodically ordered structures in intermetallic alloys which brought this subject into the limelight. These…
We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular…
This article, written for undergraduate mathematics students, provides an accessible introduction to a few key problems in tiling theory: Heesch's problem, the isohedral number problem, and the existence of an aperiodic monotile. I…
Through the glasses of didactic reduction: We consider a (periodic) tessellation $\Delta$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $\Delta$ we mean the process of cutting $M$ along corank-1…
We provide an algorithm for computing an effective basis of homology of elliptic surfaces over the complex projective line on which integration of periods can be carried out. This allows the heuristic recovery of several algebraic…
In this study, the properties of convex pentagons that can form rotationally symmetric edge-to-edge tilings are discussed. Because the rotationally symmetric tilings are formed by concave octagons that are generated by two convex pentagons…
A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as…
We show how we found substitution rules for a quasiperiodic tiling with local rotational symmetry and inflation factor 1 + sqrt(3). The base tiles are a square, a rhomb with an acute angle of 30 degrees, and equilateral triangles that are…
We report an unexpected mechanism by which an epitaxial interface can form between materials having strongly mismatched lattice constants. A simple model is proposed in which one material tilts out of the interface plane to create a…
We use the subgraph replacement method to investigate new properties of the tilings of regions on the square lattice with diagonals drawn in. In particular, we show that the centrally symmetric tilings of a generalization of the Aztec…
There are fifteen edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^3b^2$: five one-parameter families of pentagonal subdivision tilings, and ten flip modifications of three special cases of two…
A periodic lattice in Euclidean 3-space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric…