10,405 research outputs found
Aperiodic colorings and tilings of Coxeter groups
We construct a limit aperiodic coloring of hyperbolic groups. Also we
construct limit strongly aperiodic strictly balanced tilings of the Davis
complex for all Coxeter groups
Tube formulas and complex dimensions of self-similar tilings
We use the self-similar tilings constructed by the second author in
"Canonical self-affine tilings by iterated function systems" to define a
generating function for the geometry of a self-similar set in Euclidean space.
This tubular zeta function encodes scaling and curvature properties related to
the complement of the fractal set, and the associated system of mappings. This
allows one to obtain the complex dimensions of the self-similar tiling as the
poles of the tubular zeta function and hence develop a tube formula for
self-similar tilings in \. The resulting power series in
is a fractal extension of Steiner's classical tube formula for
convex bodies K \ci \bRd. Our sum has coefficients related to the curvatures
of the tiling, and contains terms for each integer , just as
Steiner's does. However, our formula also contains terms for each complex
dimension. This provides further justification for the term "complex
dimension". It also extends several aspects of the theory of fractal strings to
higher dimensions and sheds new light on the tube formula for fractals strings
obtained in "Fractal Geometry and Complex Dimensions" by the first author and
Machiel van Frankenhuijsen.Comment: 41 pages, 6 figures, incorporates referee comments and references to
new result
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
Matching complexes of polygonal line tilings
The matching complex of a simple graph is a simplicial complex consisting
of the matchings on . Jeli\'c Milutinovi\'c et al. studied the matching
complexes of the polygonal line tilings, and they gave a lower bound for the
connectivity of the matching complexes of polygonal line tilings. In this
paper, we determine the homotopy types of the matching complexes of polygonal
line tilings recursively, and determine their connectivities.Comment: 16 page
Supersymmetry, lattice fermions, independence complexes and cohomology theory
We analyze the quantum ground state structure of a specific model of
itinerant, strongly interacting lattice fermions. The interactions are tuned to
make the model supersymmetric. Due to this, quantum ground states are in
one-to-one correspondence with cohomology classes of the so-called independence
complex of the lattice. Our main result is a complete description of the
cohomology, and thereby of the quantum ground states, for a two-dimensional
square lattice with periodic boundary conditions. Our work builds on results by
J. Jonsson, who determined the Euler characteristic (Witten index) via a
correspondence with rhombus tilings of the plane. We prove a theorem, first
conjectured by P. Fendley, which relates dimensions of the cohomology at grade
n to the number of rhombus tilings with n rhombi.Comment: 40 pages, 28 figure
Higher rank inner products, Voronoi tilings and metric degenerations of tori
We introduce higher rank inner products on real and complex vector spaces and
study their corresponding Voronoi tilings. We use the framework to describe
metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings
of discrete sets.Comment: 65 pages, 12 figures. Comments welcome
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