Related papers: Domino tilings beyond 2D
P. Di Francesco first introduced the "Aztec triangle" in his study of the relationship between the twenty-vertex model and domino tilings. He conjectured an exact formula for the number of tilings of the Aztec triangle, and it has since…
High-dimensional data and high-dimensional representations of reality are inherent features of modern Artificial Intelligence systems and applications of machine learning. The well-known phenomenon of the "curse of dimensionality" states:…
In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the…
Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models,…
The field of two-dimensional topological semimetals, which emerged at the intersection of two-dimensional materials and topological materials, have been rapidly developing in recent years. In this article, we briefly review the progress in…
Given a periodic placement of copies of a tromino (either L or I), we prove co-RE-completeness (and hence undecidability) of deciding whether it can be completed to a plane tiling. By contrast, the problem becomes decidable if the initial…
We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec…
Finding an efficient optimal partial tiling algorithm is still an open problem. We have worked on a special case, the tiling of Manhattan polyominoes with dominoes, for which we give an algorithm linear in the number of columns. Some…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
We establish a lower bound on the forcing numbers of domino tilings computable in polynomial time based on height functions. This lower bound is sharp for a 2n by 2n square as well as other cases.
In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the…
This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given…
This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These…
The possibility that spacetime is extended beyond the familiar 3+1-dimensions has intrigued physicists for a century. Indeed, the consequences of a dimensionally richer spacetime would be profound. Recently, new theories with higher…
In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a…
In the past three decades, the study of rhombus tilings and domino tilings of various plane regions has been a thriving subfield of enumerative combinatorics. Physicists classify such work as the study of dimer covers of finite graphs. In…
We investigate the generalization of the Costas property in 3 or more dimensions, and we seek an appropriate definition; the 2 main complications are a) that the number of ``dots'' this multidimensional structure should have is not obvious,…
The advent of modern technology, permitting the measurement of thousands of characteristics simultaneously, has given rise to floods of data characterized by many large or even huge datasets. This new paradigm presents extraordinary…
Despite all attempts, exact solutions for black rings in more than five dimensions remain elusive. In this paper we clarify some of the reasons for that, in particular we show that a peculiar symmetry of the five--dimensional black ring -…
There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant.…