A self a ne tile in R is a set T of positive Lebesgue measure satisfying A T d D T d where A is an expanding n n real matrix with j det A j m an integer and D fd dmg R n a set of m digits Self a ne tiles arise in many contexts including radix expansions fractal geometry and the construction of compactly sup ported orthonormal wavelet bases of L R They are also studied as interesting tiles In th...

About 300 years ago Sébastien Truchet systematically studied patterns that could be formed from square tiles that were divided by a diagonal into a white triangle and a black triangle. Other pattern creators have been inspired by him to make Truchet-like tilings composed of circular arcs and other motifs. These tilings are all based on Euclidean tessellations, usually the tiling by squares. In ...

A promising way to gain more insights into how people read and understand a map is by analyzing their eye gaze while working with a map. However, eye gaze logging on interactive maps is challenging due to of various reasons. One of the biggest challenges comes from the interaction with a map that can drastically change the visual representation of the depicted content, making it difficult to de...

The physical properties of plaster bandages are a very important factor in achieving the basic functions of immobilization (maintaining bone fragments in the best possible position), which directly affects the speed and quality of fracture healing. This paper compares the differences between the physical properties of plaster bandages (mass, specific weight, drying rate, elasticity and strength...

An integral self-affine tile is the solution of a set equation AT = ⋃d∈D(T +d), where A is an n× n integer matrix and D is a finite subset of Z. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Qn×n. We define rational self-affine tiles as compact subsets of the open subring R ×∏pKp of the adèle ring AK , where ...

Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. A systematic study of self-assembly as a mathematical process has been initiated. The individual components are modelled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific “glue”, and two adjacent tiles will sti...

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.

Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Recent experiments in self-assembly demonstrate its potential for the parallel creation of a large number of nanostructures, including possibly computers. A systematic study of self-assembly as a mathematical process has been initiated by L. Adleman and E. Win...

We give a new construction of strongly aperiodic set of tiles in H, exhibiting a kind of hierarchical structure, simplifying the central framework of Margenstern’s proof that the Domino problem is undecidable in the hyperbolic plane [13]. Ludwig Danzer once asked whether, in the hyperbolic plane, where there are no similarities, there could be any notion of hierarchical tiling—an idea which pla...