A subset V ⊆ Fn2 is a tile if F n 2 can be covered by disjoint translates of V . In other words, V is a tile if and only if there is a subset A ⊆ Fn2 such that V + A = Fn2 uniquely (i.e., v + a = v ′ + a′ implies that v = v′ and a = a′ where v, v′ ∈ V and a, a′ ∈ A). In some problems in coding theory and hashing we are given a putative tile V , and wish to know whether or not it is a tile. In t...

Let Λ0, . . . , Λn ⊂ R be a finite collection of lattices of the same volume. When does there exist a set Ω ⊂ R that is a fundamental domain for all Λi, i = 0, . . . , n? The main result of this paper is that when the (group-theoretic) sum Λ0 + · · · + Λn of the dual lattices is direct (this means that the equation x0 + · · · + xn = 0, with xi ∈ Λi , has no nontrivial solution), then a Borel me...

We construct a class of polycubes that tile the space by translation in a latticeperiodic way and show that for this class the number of surrounding tiles cannot be bounded. The first construction is based on polycubes with an L-shape but with many distinct tilings of the space. Nevertheless, we are able to construct a class of more complicated polycubes such that each polycube tiles the space ...

We present Tiles Cards, a tool to foster end-user design thinking of smart object applications. Tiles Cards facilitate participatory design by abstracting the complexity of IoT technology into a set of interaction primitives and composition rules accessible to nonexperts. Further, it promotes creative thinking by means of a mission-based game in which teams have to fulfill design tasks under co...

Vector tiling aims at cutting vector data into smaller entities. It offers several opportunities, especially for the development of web-mapping systems, such as the possibilities to apply different styles, to access attributes or to render 3D data. Today no open and widely adopted standard exists for the implementation of web services involving vector tiles. In this paper we investigate several...

We look at sets of tiles that can tile any region of size greater than 1 on the square grid. This is not the typical tiling question, but relates closely to it and therefore can help solve other tiling problems – we give an example of this. We also present a result to a more classic tiling question with dominoes and L-shape tiles.

We discuss theoretical aspects of the self-assembly of triangular tiles, in particular, right triangular tiles and equilateral triangular tiles, and the self-assembly of hexagonal tiles. Contrary to intuition, we show that triangular tile assembly systems and square tile assembly systems cannot be simulated by each other in a non-trivial way. More precisely, there exists a square tile assembly ...

Ceramic tile industry is developing due to the technological researches in scientific area and new tiles which are not only a traditional ceramic also have many multiple functionalities have been marketed nowadays. These tiles like photocatalytic, photovoltaic, antibacterial and etc. improve the quality of life and provide lots of benefits such as self cleaning, energy production, climate contr...