1,616 research outputs found
The Topology of Tile Invariants
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 of tiles
and a set of regions tileable by is isomorphic to a quotient of the second
homology group of a 2-complex built from . In this topological setting we
derive some well-known tile invariants, one of which we apply to the solution
of a tiling question involving modified rectangles.Comment: 25 pages, 24 figure
Fast domino tileability
Domino tileability is a classical problem in Discrete Geometry, famously
solved by Thurston for simply connected regions in nearly linear time in the
area. In this paper, we improve upon Thurston's height function approach to a
nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39
The asymptotic determinant of the discrete Laplacian
We compute the asymptotic determinant of the discrete Laplacian on a
simply-connected rectilinear region in R^2. As an application of this result,
we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.Comment: 36 pages, 4 figures, to appear in Acta Mathematic
Ribbon Tilings and Multidimensional Height Functions
We fix and say a square in the two-dimensional grid indexed by
has color if . A {\it ribbon tile} of order is a
connected polyomino containing exactly one square of each color. We show that
the set of order- ribbon tilings of a simply connected region is in
one-to-one correspondence with a set of {\it height functions} from the
vertices of to satisfying certain difference restrictions.
It is also in one-to-one correspondence with the set of acyclic orientations of
a certain partially oriented graph.
Using these facts, we describe a linear (in the area of ) algorithm for
determining whether can be tiled with ribbon tiles of order and
producing such a tiling when one exists. We also resolve a conjecture of Pak by
showing that any pair of order- ribbon tilings of can be connected by a
sequence of local replacement moves. Some of our results are generalizations of
known results for order-2 ribbon tilings (a.k.a. domino tilings). We also
discuss applications of multidimensional height functions to a broader class of
polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new
references, a new illustration, and a few cosmetic changes). To appear in
Transactions of the American Mathematical Societ
Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices
Step by step completion of a left-to-right tiling of a rectangular floor with
tiles of a single shape starts from one edge of the floor, considers the
possible ways of inserting a tile at the leftmost uncovered square, passes
through a sequence of rugged shapes of the front line between covered and
uncovered regions of the floor, and finishes with a straight front line at the
opposite edge. We count the tilings by mapping the front shapes to nodes in a
digraph, then counting closed walks on that digraph with the transfer matrix
method.
Generating functions are detailed for tiles of shape 1 x 3, 1 x 4 and 2 x 3
and modestly wide floors. Equivalent results are shown for the 3-dimensional
analog of filling bricks of shape 1x 1 x 2, 1 x 1 x 3, 1 x 1 x 4, 1 x 2 x 2 or
1 x 2 x 3 into rectangular containers of small cross sections.Comment: 21 pages, 21 figure
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