7 research outputs found
Perfectly quilted rectangular snake tilings
AbstractWe introduce a particular form of snake tilings to define picture languages, and relate the obtained family to the recognizable picture languages (as defined by Wang tiles). The correspondence for substitution tilings is even closer, and hence is applicable to the Hilbert curve
Consistency of multidimensional combinatorial substitutions
Multidimensional combinatorial substitutions are rules that replace symbols
by finite patterns of symbols in . We focus on the case where the
patterns are not necessarily rectangular, which requires a specific description
of the way they are glued together in the image by a substitution. Two problems
can arise when defining a substitution in such a way: it can fail to be
consistent, and the patterns in an image by the substitution might overlap.
We prove that it is undecidable whether a two-dimensional substitution is
consistent or overlapping, and we provide practical algorithms to decide these
properties in some particular cases.Comment: 13 pages, v2 includes corrections to match the published versio
Domino Snake Problems on Groups
In this article we study domino snake problems on finitely generated groups.
We provide general properties of these problems and introduce new tools for
their study. The first is the use of symbolic dynamics to understand the set of
all possible snakes. Using this approach we solve many variations of the
infinite snake problem including the geodesic snake problem for certain classes
of groups. Next, we introduce a notion of embedding that allows us to reduce
the decidability of snake problems from one group to another. This notion
enable us to establish the undecidability of the infinite snake and ouroboros
problems on nilpotent groups for any generating set, given that we add a
well-chosen element. Finally, we make use of monadic second order logic to
prove that domino snake problems are decidable on virtually free groups for all
generating sets.Comment: Accepted to FCT 2023. Comments welcome
Geometrical Tile Design for Complex Neighborhoods
Recent research has showed that tile systems are one of the most suitable theoretical frameworks for the spatial study and modeling of self-assembly processes, such as the formation of DNA and protein oligomeric structures. A Wang tile is a unit square, with glues on its edges, attaching to other tiles and forming larger and larger structures. Although quite intuitive, the idea of glues placed on the edges of a tile is not always natural for simulating the interactions occurring in some real systems. For example, when considering protein self-assembly, the shape of a protein is the main determinant of its functions and its interactions with other proteins. Our goal is to use geometric tiles, i.e., square tiles with geometrical protrusions on their edges, for simulating tiled paths (zippers) with complex neighborhoods, by ribbons of geometric tiles with simple, local neighborhoods. This paper is a step toward solving the general case of an arbitrary neighborhood, by proposing geometric tile designs that solve the case of a “tall” von Neumann neighborhood, the case of the f-shaped neighborhood, and the case of a 3 × 5 “filled” rectangular neighborhood. The techniques can be combined and generalized to solve the problem in the case of any neighborhood, centered at the tile of reference, and included in a 3 × (2k + 1) rectangle
On the tileability of polygons with colored dominoes
Analysis of Algorithm