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. 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 deterministic square (hexagonal) tile assemb...

We prove the computational intractability of rotating and placing n square tiles into a 1×n array such that adjacent tiles are compatible—either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the...

Understanding the neural circuitry of the retina requires us to map the connectivity of individual neurons in large neuronal tissue sections and analyze signal communication across processes from the electron microscopy images. One of the major bottlenecks in the critical path is the image mosaicing process where 2D slices are assembled from scanned microscopy image tiles. The problem of assemb...

A variety of two-color fractal tilings (f-tilings) are described, in which no two adjacent tiles have the same color. Two-colorable examples from f-tilings that have been described previously are identified, and two techniques are used for converting f-tilings that are not two-colorable into new f-tilings that can be so colored. In the first of these, tiles are combined in order to change the v...

This paper deals with sets V of tiles (compact, convex sets) in R. Tiles are a generalization of pixels or voxels (in R or R); they can have arbitrary shapes and are allowed to overlap. The union of all the tiles of V is denoted by U(V). The neighborhood N-p(P) of a tile P is the union of the tiles of V that intersect V. P is called simple if deletion of P from V does not change the topology (i...

Suppose $f\in L^1(\Bbb{R}^d)$, $\Lambda\subset\Bbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with weight. We prove there exists lattice $L\subset\Bbb{R}^d$ $f+L$ also tiles, possibly different As corollary, together result Kolountzakis, it implies any convex polygon multi-tiles the plane by translations admits multi-tiling, multiplicity.

THIS NOTE is prompted by the acquisition by the Wellcome Institute of the History of Medicine of an exceedingly rare pharmaceutical tile, decorated with the Arms of the Society of Apothecaries (see fig. 1), and also because of growing interest in tiles bearing the Arms of the Society, an interest that has raised more questions than it has answered.' One particular point at issue is whether the ...

Anechoic chambers which are used for emission and immunity testing require expensive ferrite tiles on their inner surfaces. This paper describes a method to reduce the number of required ferrite tiles, whilst ensuring a reliable and specified test region. In this method, the positions of some ferrite tiles are found optimally to keep the performance of the anechoic chamber as high as possible. ...