629 research outputs found
An aperiodic hexagonal tile
We show that a single prototile can fill space uniformly but not admit a
periodic tiling. A two-dimensional, hexagonal prototile with markings that
enforce local matching rules is proven to be aperiodic by two independent
methods. The space--filling tiling that can be built from copies of the
prototile has the structure of a union of honeycombs with lattice constants of
, where sets the scale of the most dense lattice and takes all
positive integer values. There are two local isomorphism classes consistent
with the matching rules and there is a nontrivial relation between these
tilings and a previous construction by Penrose. Alternative forms of the
prototile enforce the local matching rules by shape alone, one using a
prototile that is not a connected region and the other using a
three--dimensional prototile.Comment: 32 pages, 24 figures; submitted to Journal of Combinatorial Theory
Series A. Version 2 is a major revision. Parts of Version 1 have been
expanded and parts have been moved to a separate article (arXiv:1003.4279
Discrete models of force chain networks
A fundamental property of any material is its response to a localized stress
applied at a boundary. For granular materials consisting of hard, cohesionless
particles, not even the general form of the stress response is known. Directed
force chain networks (DFCNs) provide a theoretical framework for addressing
this issue, and analysis of simplified DFCN models reveal both rich
mathematical structure and surprising properties. We review some basic elements
of DFCN models and present a class of homogeneous solutions for cases in which
force chains are restricted to lie on a discrete set of directions.Comment: 17 pages, 6 figures, dcds-B.cls; Minor corrections to version 2, but
including an important factor of 2; Submitted to Discrete and Continuous
Dynamical Systems B for special issue honoring David Schaeffe
Controlling spatiotemporal dynamics with time-delay feedback
We suggest a spatially local feedback mechanism for stabilizing periodic
orbits in spatially extended systems. Our method, which is based on a
comparison between present and past states of the system, does not require the
external generation of an ideal reference state and can suppress both absolute
and convective instabilities. As an example, we analyze the complex
Ginzburg-Landau equation in one dimension, showing how the time-delay feedback
enlarges the stability domain for travelling waves.Comment: 4 pages REVTeX + postscript file with 3 figure
Forcing nonperiodicity with a single tile
An aperiodic prototile is a shape for which infinitely many copies can be
arranged to fill Euclidean space completely with no overlaps, but not in a
periodic pattern. Tiling theorists refer to such a prototile as an "einstein"
(a German pun on "one stone"). The possible existence of an einstein has been
pondered ever since Berger's discovery of large set of prototiles that in
combination can tile the plane only in a nonperiodic way. In this article we
review and clarify some features of a prototile we recently introduced that is
an einstein according to a reasonable definition. [This abstract does not
appear in the published article.]Comment: 18 pages, 10 figures. This article has been substantially revised and
accepted for publication in the Mathematical Intelligencer and is scheduled
to appear in Vol 33. Citations to and quotations from this work should
reference that publication. If you cite this work, please check that the
published form contains precisely the material to which you intend to refe
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