Growth Rates in the Quaquaversal Tiling

Conway and Radin's ``quaquaversal' tiling of R ³ is known to exhibit statistical rotational symmetry in the infinite volume limit. A finite patch, however, cannot be perfectly isotropic, and we compute the rates at which the anisotropy scales with size. In a sample of volume N , tiles appear in O(N ^1/6 ) distinct orientations. However, the orientations are not uniformly populated. A small (O(N ^1/84 ) ) set of these orientations account for the majority of the tiles. Furthermore, these orientations are not uniformly distributed on SO(3) . Sample averages of functions on SO(3) seem to approach their ergodic limits as N ^-1/336 . Since even macroscopic patches of a quaquaversal tiling maintain noticeable anisotropy, a hypothetical physical quasicrystal whose structure was similar to the quaquaversal tiling could be identified by anisotropic features of its electron diffraction pattern. Received October 19, 1998, and in revised form March 11, 1999.     

Substitution tilings with statistical circular symmetry

Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all primitive substitution tilings showing tiles in infinitely many orientations.     

Some Generalizations of the Pinwheel Tiling

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite. Received April 9, 1996, and in revised form September 16, 1996.     

Expanding Polynomials and Connectedness of Self-Affine Tiles

Little is known about the connectedness of self-affine tiles in ${\Bbb R}^n$. In this note we consider this property on the self-affine tiles that are generated by consecutive collinear digit sets. By using an algebraic criterion, we call it the {\it height reducing property}, on expanding polynomials (i.e., all the roots have moduli $ > 1$), we show that all such tiles in ${\Bbb R}^n, n \leq 3$, are connected. The problem is still unsolved for higher dimensions. For this we make another investigation on this algebraic criterion. We improve a result of Garsia concerning the heights of expanding polynomials. The new result has its own interest from an algebraic point of view and also gives further insight to the connectedness problem.     

Minimum-perimeter domain assignment

For certain classes of problems defined over two-dimensional domains with grid structure, optimization problems involving the assignment of grid cells to processors present a nonlinear network model for the problem of partitioning tasks among processors so as to minimize interprocessor communication. Minimizing interprocessor communication in this context is shown to be equivalent to tiling the domain so as to minimize total tile perimeter, where each tile corresponds to the collection of tasks assigned to some processor. A tight lower bound on the perimeter of a tile as a function of its area is developed. We then show how to generate minimum-perimeter tiles. By using assignments corresponding to near-rectangular minimum-perimeter tiles, closed form solutions are developed for certain classes of domains. We conclude with computational results with parallel high-level genetic algorithms that have produced good (and sometimes provably optimal) solutions for very large perimeter minimization problems. This research was partially supported by the Air Force Office of Scientific Research under grant F49620-94-1-0036, and by the NSF under grants CCR-8907671 and CCR-9306807.     

On the existence of hyper-L triple-loop networks

Aguiló-Gost [New dense families of triple loop Networks, Discrete Math. 197/198 (1999) 15-27] has presented a new type of hyper-L tiles and used it to derive a new dense family of triple-loop networks. While Aguiló-Gost's hyper-L tile seems to be a promising tool for studying the triple-loop networks, we need to verify that the hyper-L tile producing good result is indeed the MDD of some triple-loop network. In this paper, we give necessary and sufficient conditions for the existence of Aguiló-Gost's hyper-L triple-loop networks and we correct some flaws in [F. Aguiló-Gost, New dense families of triple loop networks, Discrete Math. 197/198 (1999) 15-27].     

On two questions concerning tilings

We prove that a tiling of the plane by topological disks is locally finite at most boundary points of tiles, confirming a conjecture by Valette. This comes by way of a much more general theorem on tilings of topological vector spaces. We also investigate a question raised by Klee as to whether or not there is a tiling of separable Hilbert space by bounded convex tiles. We present evidence to support the conjecture that the answer is negative.     

Connectedness of number theoretical tilings (dual tilings generated by Pisot units of degree 3 and 4)

We study the connectedness of Pisot dual tilings. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4 which have infinitely many connected components. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, we give a complete classification of the β expansion of 1 for quartic Pisot units. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A small aperiodic set of Wang tiles

A new aperiodic tile set containing only 14 Wang tiles is presented. The construction is based on Mealy machines that multiply Beatty sequences of real numbers by rational constants.     

An aperiodic set of 13 Wang tiles

A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a recent technique developed by Kari. The tilings simulate the behavior of sequential machines that multiply real numbers in balanced representations by real constants.     

Dissecting the square into seven or nine congruent parts

We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles. This confirms a new case of a conjecture posed first by Yuan, Zamfirescu and Zamfirescu and later by Rao, Ren and Wang. Our method allows us to explore other variants of this question, for example, we also prove that no rectangle can be tiled by five or seven congruent non-rectangular polygons.     

Wild tiles in ℝ<Superscript>3</Superscript> with spherical boundaries

Wildly embedded tiles in ℝ³ with spherical boundary are discussed. The construction of the topologically complicated, crumpled cube tiles is reviewed. We construct an infinite family of wildly embedded, cellular tiles with Fox-Artin-type wild points. Finally, a condition on the set of wild points on a cellular tile is given to show that certain wild cells cannot be tiles. Several observations are recorded for further investigations. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 203–211, 2005.     

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).     

Deterministic Aperiodic Tile Sets

Wang tiles are square tiles with colored edges. We construct an aperiodic set of Wang tiles that is strongly deterministic in the sense that any two adjacent edges of a tile determine the tile uniquely. Consequently, the tiling group of this set is not hyperbolic and it acts discretely and co-compactly on a CAT(0) space. Submitted: February 1998, revised: October 1998.     

Aperiodic Tilings with One Prototile and Low Complexity Atlas Matching Rules

We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging edge-to-edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles, and we apply our method to present via these rules a single prototile that can only tile ℝ³ aperiodically, and a pair of square tiles that can only tile ℝ² aperiodically.     

A Survey on Topological Properties of Tiles Related to Number Systems

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its ‘neighbors’ is discussed.     

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, , θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.     

Topology of crystallographic tiles

We study self-affine tiles which tile the n-dimensional real vector space with respect to a crystallographic group. First we define classes of graphs that allow to determine the neighbors of a given tile algorithmically. In the case of plane tiles these graphs are used to derive a criterion for such tiles to be homeomorphic to a disk. As particular application, we will solve a problem of Gelbrich, who conjectured that certain examples of tiles which tile with respect to the ornament group p2 are homeomorphic to a disk.      

正规Tiling的邻居系

该文借助Tiling的拓扑图和边邻居图, 给出了正规Tiling 中两个Tiles 拥有相同的邻居系的一个充分必要条件, 并利用此条件证明了正规Tiling 中不可能存在三个不同的Tiles 具有相同的邻居系, 从而在一定程度上回答了Grünbaum 的一个公开问题.     

Network science faces the challenge and opportunity:exploring ``Network of Networks" and its unified theoretical framework

In the era of big data, network science is facing new challenges and opportunities. This review article focuses on discussing one of the hottest subjects of network science - ``network of networks" (NON). The main features, several typical examples and the main progress for NON are outlined, including the epidemic spreading in multilayer coupled networks. Finally the most challenging tasks for NON are proposed.