1,920 research outputs found
Periodicity in tilings
Tilings and tiling systems are an abstract concept that arise both as a
computational model and as a dynamical system. In this paper, we characterize
the sets of periods that a tiling system can produce. We prove that up to a
slight recoding, they correspond exactly to languages in the complexity classes
\nspace{n} and \cne
From Aztec diamonds to pyramids: steep tilings
We introduce a family of domino tilings that includes tilings of the Aztec
diamond and pyramid partitions as special cases. These tilings live in a strip
of of the form for some integer , and are parametrized by a binary word that
encodes some periodicity conditions at infinity. Aztec diamond and pyramid
partitions correspond respectively to and to the limit case
. For each word and for different types of boundary
conditions, we obtain a nice product formula for the generating function of the
associated tilings with respect to the number of flips, that admits a natural
multivariate generalization. The main tools are a bijective correspondence with
sequences of interlaced partitions and the vertex operator formalism (which we
slightly extend in order to handle Littlewood-type identities). In
probabilistic terms our tilings map to Schur processes of different types
(standard, Pfaffian and periodic). We also introduce a more general model that
interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6,
new improved proof of Proposition 11
Slopes of Tilings
We study here slopes of periodicity of tilings. A tiling is of slope if it is
periodic along direction but has no other direction of periodicity. We
characterize in this paper the set of slopes we can achieve with tilings, and
prove they coincide with recursively enumerable sets of rationals.Comment: Journ\'ees Automates Cellulaires 2010, Turku : Finland (2010
Structural aspects of tilings
In this paper, we study the structure of the set of tilings produced by any
given tile-set. For better understanding this structure, we address the set of
finite patterns that each tiling contains. This set of patterns can be analyzed
in two different contexts: the first one is combinatorial and the other
topological. These two approaches have independent merits and, once combined,
provide somehow surprising results. The particular case where the set of
produced tilings is countable is deeply investigated while we prove that the
uncountable case may have a completely different structure. We introduce a
pattern preorder and also make use of Cantor-Bendixson rank. Our first main
result is that a tile-set that produces only periodic tilings produces only a
finite number of them. Our second main result exhibits a tiling with exactly
one vector of periodicity in the countable case.Comment: 11 page
Rhombic Tilings and Primordia Fronts of Phyllotaxis
We introduce and study properties of phyllotactic and rhombic tilings on the
cylin- der. These are discrete sets of points that generalize cylindrical
lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical
system S that models plant pattern formation by stacking disks of equal radius
on the cylinder. This system has the advantage of allowing several disks at the
same level, and thus multi-jugate config- urations. We provide partial results
toward proving that the attractor for S is entirely composed of rhombic tilings
and is a strongly normally attracting branched manifold and conjecture that
this attractor persists topologically in nearby systems. A key tool in
understanding the geometry of tilings and the dynamics of S is the concept of
pri- mordia front, which is a closed ring of tangent disks around the cylinder.
We show how fronts determine the dynamics, including transitions of parastichy
numbers, and might explain the Fibonacci number of petals often encountered in
compositae.Comment: 33 pages, 10 picture
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