207 research outputs found
Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings
This paper provides a bridge between the classical tiling theory and the
complex neighborhood self-assembling situations that exist in practice. The
neighborhood of a position in the plane is the set of coordinates which are
considered adjacent to it. This includes classical neighborhoods of size four,
as well as arbitrarily complex neighborhoods. A generalized tile system
consists of a set of tiles, a neighborhood, and a relation which dictates which
are the "admissible" neighboring tiles of a given tile. Thus, in correctly
formed assemblies, tiles are assigned positions of the plane in accordance to
this relation. We prove that any validly tiled path defined in a given but
arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of
microtiles. A ribbon is a special kind of polyomino, consisting of a
non-self-crossing sequence of tiles on the plane, in which successive tiles
stick along their adjacent edge. Finally, we extend this construction to the
case of traditional tilings, proving that we can simulate
arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving
some of their essential properties.Comment: Submitted to Theoretical Computer Scienc
Polyomino convolutions and tiling problems
We define a convolution operation on the set of polyominoes and use it to
obtain a criterion for a given polyomino not to tile the plane (rotations and
translations allowed). We apply the criterion to several families of
polyominoes, and show that the criterion detects some cases that are not
detectable by generalized coloring arguments.Comment: 8 pages, 8 figures. To appear in \emph{J. of Combin. Theory Ser. A
Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry
We describe computer algorithms that produce the complete set of isohedral
tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains
and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups
of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral
tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or
polyiamonds as fundamental domains. We display the algorithms' output and give
enumeration tables for small values of n. This expands on our earlier works
(Fukuda et al 2006, 2008)
Fixed Parameter Undecidability for Wang Tilesets
Deciding if a given set of Wang tiles admits a tiling of the plane is
decidable if the number of Wang tiles (or the number of colors) is bounded, for
a trivial reason, as there are only finitely many such tilesets. We prove
however that the tiling problem remains undecidable if the difference between
the number of tiles and the number of colors is bounded by 43.
One of the main new tool is the concept of Wang bars, which are equivalently
inflated Wang tiles or thin polyominoes.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
An Optimal Algorithm for Tiling the Plane with a Translated Polyomino
We give a -time algorithm for determining whether translations of a
polyomino with edges can tile the plane. The algorithm is also a
-time algorithm for enumerating all such tilings that are also regular,
and we prove that at most such tilings exist.Comment: In proceedings of ISAAC 201
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