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...

We present a variant of Ammann tiles consisting of two similar rectilinear hexagons with edge subdivision, which can tile the plane but only in non periodic ways. A special matching rule, ghost marking, plays a key role in the proof. We shall show that the set of tiles in Figure 1 is aperiodic, that is, it tiles the plane but only in non periodic way.

In 1966, Golomb showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a ‘notched cube’) that tiles a larger copy...

Głównym walorem estetycznym w ukończonych obiektach i budynkach mieszkalnych stały się praktyczne, ale także nowoczesne designerskie koncepcje związane z wykorzystaniem płytek ceramicznych. Na rynku budowlanym obserwuje obecnie określone potrzeby oczekiwania potencjalnych klientów. Stąd też celem rozważań jest analiza ceramicznych Polsce. początku dokonano analizy procesów zachodzących na oraz ...

We study the times to grow structures within the tile self-assembly model proposed by Winfree, and the possible shapes that can be achieved. Our earlier work was confined to the growth of rectangular structures, in which the rates of attachment of border tiles and rule tiles were the same. By varying the relative rates one can engineer interesting new shapes, which have been observed in the lab...

From black and white linoleum on the kitchen floor to magnificent Islamic mosaic to the intricate prints of M.C. Escher, tilings are an essential component of the decorative arts, and the complex mathematical structure behind them has intrigued scientists, mathematicians, and enthusiasts for centuries. Johannes Kepler, most famous for his laws of planetary motion, was known to have been interes...

© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 5541 wileyonlinelibrary.com structures with homogeneous composition, but less effi cient at generating hollow, 3D structures, especially when they comprise multiple materials with different mechanical or physical properties. To fabricate 3D structures that incorporate elastomeric, infl atable components for use in the actuation of soft robots an...

In this study, properties of newly developed reclaimed tiles in a harmful environment were investigated. A portion of clay used to manufacture tiles was replaced with sewage sludge ash (SSA) and waste glass to produce the new reclaimed tiles. To investigate the effects of SSA and waste glass on the properties of the tiles, different specimens were blended and placed in acid-alkali solutions. Th...