16,564 research outputs found
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
Symmetric Assembly Puzzles are Hard, Beyond a Few Pieces
We study the complexity of symmetric assembly puzzles: given a collection of
simple polygons, can we translate, rotate, and possibly flip them so that their
interior-disjoint union is line symmetric? On the negative side, we show that
the problem is strongly NP-complete even if the pieces are all polyominos. On
the positive side, we show that the problem can be solved in polynomial time if
the number of pieces is a fixed constant
Modeling for Control of Symmetric Aerial Vehicles Subjected to Aerodynamic Forces
This paper participates in the development of a unified approach to the
control of aerial vehicles with extended flight envelopes. More precisely,
modeling for control purposes of a class of thrust-propelled aerial vehicles
subjected to lift and drag aerodynamic forces is addressed assuming a
rotational symmetry of the vehicle's shape about the thrust force axis. A
condition upon aerodynamic characteristics that allows one to recast the
control problem into the simpler case of a spherical vehicle is pointed out.
Beside showing how to adapt nonlinear controllers developed for this latter
case, the paper extends a previous work by the authors in two directions.
First, the 3D case is addressed whereas only motions in a single vertical plane
was considered. Secondly, the family of models of aerodynamic forces for which
the aforementioned transformation holds is enlarged.Comment: 7 pages, 4 figure
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