100 research outputs found
Matching rules for quasicrystals : the composition-decomposition method
A general method is presented which proves that an appropriately chosen set of matching rules for a quasiperiodic tiling enforces quasiperiodicity. This method, which is based on self-similarity, is formulated in general terms to make it applicable to many different situations. The method is then illustrated with two examples, one of which is a new set of matching rules for a dodecagonal tiling
Cohomology of One-dimensional Mixed Substitution Tiling Spaces
We compute the Cech cohomology with integer coefficients of one-dimensional
tiling spaces arising from not just one, but several different substitutions,
all acting on the same set of tiles. These calculations involve the
introduction of a universal version of the Anderson-Putnam complex. We show
that, under a certain condition on the substitutions, the projective limit of
this universal Anderson-Putnam complex is isomorphic to the tiling space, and
we introduce a simplified universal Anderson-Putnam complex that can be used to
compute Cech cohomology. We then use this simplified complex to place bounds on
the rank of the first cohomology group of a one-dimensional substitution tiling
space in terms of the number of tiles.Comment: 26 pages, 4 figure
Dynamics and topology of the Hat family of tilings
The recently discovered Hat tiling admits a 4-dimensional family of shape
deformations, including the 1-parameter family already known to yield alternate
monotiles. The continuous hulls resulting from these tilings are all
topologically conjugate dynamical systems, and hence have the same dynamics and
topology. We construct and analyze a self-similar element of this family called
the CAP tiling, and we use it to derive properties of the entire family. The
CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and
comes from a natural cut-and-project scheme with 2-dimensional Euclidean
internal space. All other members of the Hat family, in particular the original
version constructed from 30-60-90 right triangles, are obtained via small
modifications of the projection from this cut-and-project scheme.Comment: 30 pages, 10 figures, 3 tables; slightly expanded and improved
version with additional explanations and result
Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings
The two main techniques for the generation of quasipcriodic tilings.,de Bruijn's grid method and the projection formalism, are generalised. A very broad class or quasi periodic tilings is obtained in this way. The two generalised methods are Shown to be equivalent. The standard calculation of Fourier spectra is extended to the whole general class of tilings. Various upplications are discussed
Cluster Model of Decagonal Tilings
A relaxed version of Gummelt's covering rules for the aperiodic decagon is
considered, which produces certain random-tiling-type structures. These
structures are precisely characterized, along with their relationships to
various other random tiling ensembles. The relaxed covering rule has a natural
realization in terms of a vertex cluster in the Penrose pentagon tiling. Using
Monte Carlo simulations, it is shown that the structures obtained by maximizing
the density of this cluster are the same as those produced by the corresponding
covering rules. The entropy density of the covering ensemble is determined
using the entropic sampling algorithm. If the model is extended by an
additional coupling between neighboring clusters, perfectly ordered structures
are obtained, like those produced by Gummelt's perfect covering rules.Comment: 10 pages, 20 figures, RevTeX; minor changes; to be published in Phys.
Rev.
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