Non-periodic Tilings Research Articles

A two-vertex theorem for normal tilings

We regard a smooth, d=2-dimensional manifold mathcal {M} and its normal tiling M, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by bar{v}^{star } and we prove that if M is periodic then bar{v}^{star } ge 2. We show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with bar{v}^{star }=0.

A Family of Non-Periodic Tilings of the Plane by Right Golden Triangles

We study a family of substitution tilings with similar right triangles of two sizes which is obtained using the substitution rule introduced in Danzer and van Ophuysen (Res. Bull. Panjab Univ. Sci. 50(1–4), 137–175 (2000)). In that paper, it is proved this family of tilings can be obtained from a local rule using decorated tiles. That is, that this family is sofic. In the present paper, we provide an alternative proof of this fact. We use more decorated tiles than Danzer and van Ophuysen (22 in place of 10). However, our decoration of supertiles is more intuitive and our local rule is simpler.

Additions to Penrose nonperiodic tiling: A simple methodological approach

Additions to Penrose nonperiodic tiling: A simple methodological approach

Forcing nonperiodic tilings with one tile using a seed

The so-called “einstein problem” (a pun playing with the famous scientist’s name and the German term “ein Stein” for “one stone”) asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules. In this paper a class of spiral tilings (and one non-spiral example) is linked to a weaker form of the einstein problem where one or several seed tiles are used. Furthermore, the classical types of matching rules are listed and some new types are discussed.

New members for the class of penrose nonperiodic tilings of the plane

New members for the class of penrose nonperiodic tilings of the plane

An aperiodic monotile that forces nonperiodicity through dendrites

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 rules that apply only to adjacent tiles. The first is inspired by the Socolar–Taylor monotile, but can be realised by shape alone. The second is a dendrite rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our dendrite rule initiates a new method to produce tilings of the plane.

Your Friendly Neighborhood Voderberg Tile

SummaryWe present a generalization of the Voderberg tile, which, in addition to admitting periodic and nonperiodic spiral tilings of the plane, has the property that just two copies can surround 1 or 2 copies of the tile. We construct a generalization of this tile that admits periodic and nonperiodic spiral tilings of the plane while enjoying the property that any number of copies of the tile can be surrounded by just 2 copies. In doing so, we solve two open problems posed in the classic book Tilings and Patterns by Grünbaum and Shephard.

Non-periodic Tiling of Procedural Noise Functions

Procedural noise functions have many applications in computer graphics, ranging from texture synthesis to atmospheric effect simulation or to landscape geometry specification. Noise can either be precomputed and stored into a texture, or evaluated directly at application runtime. This choice offers a tradeoff between image variance, memory consumption and performance. Advanced tiling algorithms can be used to decrease visual repetition. Wang tiles allow a plane to be tiled in a non-periodic way, using a relatively small set of textures. Tiles can be arranged in a single texture map to enable the GPU to use hardware filtering. In this paper, we present modifications to several popular procedural noise functions that directly produce texture maps containing the smallest complete Wang tile set. The findings presented in this paper enable non-periodic tiling of these noise functions and textures based on them, both at runtime and as a preprocessing step. These findings also allow decreasing repetition of noise-based effects in computer-generated images at a small performance cost, while maintaining or even reducing the memory consumption.

Strongly nonperiodic hyperbolic tilings using single vertex configuration

A strongly nonperiodic tiling is defined as a tiling that does not admit infinite cyclic symmetry. The purpose of this article is to construct, up to isomorphism, uncountably many strongly nonperiodic hyperbolic tilings with a single vertex configuration by a hyperbolic rhombus tile. We use a tile found by Margulis and Mozes [5], which admits tilings, but no tiling with a compact fundamental domain.

A measure of regularity for polygonal mosaics in biological systems.

BackgroundThe quantification of the spatial order of biological patterns or mosaics provides useful information as many properties are determined by the spatial distribution of their constituent elements. These are usually characterised by methods based on nearest neighbours distances, by the number of sides of cells, or by angles defined by the adjacent cells.MethodsA measure of regularity in polygonal mosaics of different kinds in biological systems is proposed. It is based on the condition of eutacticity, expressed in terms of eutactic stars, which is closely related to regularity of polytopes. Thus it constitutes a natural measure of regularity. The proposed measure is tested with numerical and real data. Numerically is tested with a hexagonal lattice that is distorted progressively and with a non-periodic regular tiling. With real data, the distribution of oak trees in forests from three locations in the State of Querétaro, Mexico, and the spiral pattern of florets in a flowering plant are characterised.ResultsThe proposed measure performs well and as expected while tested with a numerical experiment, as well as when applied to a known non-periodic tiling of the plane. Concerning real data, the measure is sensitive to the degree of perturbation observed in the distribution of oak trees and detects high regularity in a phyllotactic pattern studied.ConclusionsThe measure here proposed has a clear geometrical meaning, establishing what regularity means, and constitute an advantageous general purposes alternative to analyse spatial distributions, capable to indicate the degree of regularity of a mosaic or an array of points.

Phason space analysis and structure modelling of 100 Å-scale dodecagonal quasicrystal in Mn-based alloy

The dodecagonal quasicrystal classified into the five-dimensional space group P126/mmc, recently discovered in a Mn–Cr–Ni–Si alloy, has been analysed using atomic-resolution spherical aberration-corrected electron microscopy, i.e. high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) and conventional transmission electron microscopy. By observing along the 12-fold axis, non-periodic tiling consisting of an equilateral triangle and a square has been revealed, of which common edge length is a = 4.560 Å. These tiles tend to form a network of dodecagons of which size is a ≈ 17 Å in diameter. The tiling was interpreted as an aggregate of 100 Å-scale oriented domains of high- and low-quality quasicrystals with small crystallites appearing at their boundaries. The quasicrystal domains exhibited a densely filled circular acceptance region in the phason space. This is the first observation of the acceptance region in an actual dodecagonal quasicrystal. Atomic structure model consistent with the electron microscopy images is a standard Frank-Kasper decoration of the triangle and square tiles that can be inferred from the crystal structures of Zr4Al3 and Cr3Si. Four kinds of layers located at z = 0, ±1/4 and 1/2 are stacked periodically along the 12-fold axis, and the atoms at z = 0 and 1/2 form hexagonal anti-prisms consistently with the 126-screw axis. The validity of this structure model was examined by means of powder X-ray diffraction.

On Non-Periodic Tilings of the Real Line by a Function

It is known that a positive, compactly supported function $f \in L^1(\mathbb R)$ can tile by translations only if the translation set is a finite union of periodic sets. We prove that this is not the case if $f$ is allowed to have unbounded support. On the other hand we also show that if the translation set has finite local complexity, then it must be periodic, even if the support of $f$ is unbounded.

APERIODIC TILING USING P SYSTEM

Tile Pasting P System is a computational model, based on the P System model, to generate two-dimensional tiling patterns using pasting rules at the edges of the regular polygons. Computational mechanism plays an important role in understanding the various complexities involved in the formation of complex patterns. In this paper we study the construction of non-periodic tiling patterns using corner tiles and aperiodic Wang tiles using the computational model Tile Pasting P System. We show that the Tile Pasting P System requires a minimum of four membranes to generate the non-periodic Wang tiling.

When Periodicities Enforce Aperiodicity

Aperiodic tilings are non-periodic tilings defined by local rules. They are widely used to model quasicrystals, and a central question is to understand which of the non-periodic tilings are actually aperiodic. Among tilings, those by rhombi can be easily seen as approximations of surfaces in higher dimensional spaces. In particular, those which approximate irrational planes are non-periodic. But which ones are also aperiodic? This paper introduces the notion of subperiod, which links algebraic properties of a plane with geometric properties of the tilings that approximate it. A necessary and sufficient condition is obtained for tilings that can be seen in the four dimensional Euclidean space. This result is then applied to some examples in higher codimensions, notably tilings with n-fold rotational symmetry.

Ammann Tilings in Symplectic Geometry

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 constructed highly singular symplectic spaces and we show that they are diffeomorphic but not symplectomorphic. These spaces inherit from the tiling its very interesting symmetries.

Local Rules for Computable Planar Tilings

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 is to characterize, among a class of non-periodic tilings, the aperiodic ones. In this paper, we answer this question for the well-studied class of non-periodic tilings obtained by digitizing irrational vector spaces. Namely, we prove that such tilings are aperiodic if and only if the digitized vector spaces are computable.

Non-periodic Tilings of ℝ n by Crosses

An n-dimensional cross consists of 2n+1 unit cubes: the “central” cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of ℝ n by crosses have been constructed by several authors for all n∈N. No non-periodic tiling of ℝ n by crosses has been found so far. We prove that if 2n+1 is not a prime, then the total number of non-periodic Z-tilings of ℝ n by crosses is $2^{\aleph _{0}}$ while the total number of periodic Z-tilings is only ℵ0. In a sharp contrast to this result we show that any two tilings of ℝ n ,n=2,3, by crosses are congruent. We conjecture that this is the case not only for n=2,3, but for all n where 2n+1 is a prime.

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpinski gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.

Ultrasound therapy transducers with space-filling non-periodic arrays

Ultrasound transducers designed for therapeutic purposes such as tissue ablation, histotripsy, or drug delivery require large apertures for adequate spatial localization while providing sufficient power and steerability without the presence of secondary grating lobes. In addition, it is highly preferred to minimize the total number of channels and to maintain simplicity in electrical matching network design. To this end, we propose array designs that are both space-filling and non-periodic in the placement of the elements. Such array designs can be generated using the mathematical concept of non-periodic or aperiodic tiling (tessellation) and can lead to reduced grating lobes while maintaining full surface area coverage to deliver maximum power. For illustration, we designed two 2-D space-filling therapeutic arrays with 128 elements arranged on a spherical shell. One was based on the two-shape Penrose rhombus tiling, and the other was based on a single rectangular shape arranged non-periodically. The steerability performance of these arrays was studied using acoustic field simulations. For comparison, we also studied two other arrays, one with circular elements distributed randomly, and the other a periodic array with square elements. Results showed that the two space-filling non-periodic arrays were able to steer to treat a volume of 16 x 16 x 20 mm while ensuring that the grating lobes were under -10 dB compared with the main lobe. The rectangular non-periodic array was able to generate two and half times higher power than the random circles array. The rectangular array was then fabricated by patterning the array using laser scribing methods and its steerability performance was validated using hydrophone measurements. This work demonstrates that the concept of space-filling aperiodic/non-periodic tiling can be used to generate therapy arrays that are able to provide higher power for the same total transducer area compared with random arrays while maintaining acceptable grating lobe levels.

Integer Cech cohomology of a class of n-dimensional substitutions

A class of non-periodic tilings in n-dimensions is considered. They are based on one-dimensional substitution tilings that force the border, a property preserved in the construction for higher dimensions. This fact allows to compute the integerČech cohomology of the tiling spaces in an efficient way. Several examples are analyzed, some of them with PV numbers as inflation factors, and they have finitely or infinitely generated torsion-free cohomologies. Copyright © 2010 John Wiley & Sons, Ltd.