461 research outputs found
Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n
We exploit the fact that two-dimensional facets of the Voronoi and Delone
cells of the root lattice A_n in n-dimensional space are the identical
rhombuses and equilateral triangles respectively.The prototiles obtained from
orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root
lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections
lead to various rhombuses and several triangles respectively some of which have
been extensively discussed in the literature in different contexts. For
example, rhombuses of the Voronoi cell of the root lattice A_4 projects onto
only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly
the Delone cells tiling the same root lattice projects onto two isosceles
Robinson triangles which also lead to Penrose tilings with kites and darts. We
point out that the Coxeter element of order h=n+1 and the dihedral subgroup of
order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the
Coxeter plane. After setting the general scheme we give examples leading to
tilings with 4-fold, 5-fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with
rhombic and triangular tilings of the plane which are useful in modelling the
quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face
centered cubic (f.c.c.) lattice described by the root lattice A_(3)whose
Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a
square lattice with an h=4 fold symmetry.Comment: 22 pages, 17 figure
Triangular dissections, aperiodic tilings and Jones algebras
The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph
of type determines planar fractal sets obtained by infinite dissections
of a given triangle. All triangles appearing in the dissection process have
angles that are multiples of There are usually several possible
infinite dissections compatible with a given but a given one makes use of
triangle types if is even. Jones algebra with index (values of the discrete range) act naturally on vector spaces
associated with those fractal sets. Triangles of a given type are always
congruent at each step of the dissection process. In the particular case ,
there are isometric and the whole structure lead, after proper inflation, to
aperiodic Penrose tilings. The ``tilings'' associated with other values of the
index are discussed and shown to be encoded by equivalence classes of infinite
sequences (with appropriate constraints) using digits (if is even)
and generalizing the Fibonacci numbers.Comment: 14 pages. Revised version. 18 Postcript figures, a 500 kb uuencoded
file called images.uu available by mosaic or gopher from
gopher://cpt.univ-mrs.fr/11/preprints/94/fundamental-interactions/94-P.302
Tiling Billards on Triangle Tilings, and Interval Exchange Transformations
We consider the dynamics of light rays in triangle tilings where triangles
are transparent and adjacent triangles have equal but opposite indices of
refraction. We find that the behavior of a trajectory on a triangle tiling is
described by an orientation-reversing three-interval exchange transformation on
the circle, and that the behavior of all the trajectories on a given triangle
tiling is described by a polygon exchange transformation. We show that, for a
particular choice of triangle tiling, certain trajectories approach the Rauzy
fractal, under rescaling.Comment: 31 pages, 19 figures, 2 appendices. Comments welcome
Statistical mechanics of glass transition in lattice molecule models
Lattice molecule models are proposed in order to study statistical mechanics
of glass transition in finite dimensions. Molecules in the models are
represented by hard Wang tiles and their density is controlled by a chemical
potential. An infinite series of irregular ground states are constructed
theoretically. By defining a glass order parameter as a collection of the
overlap with each ground state, a thermodynamic transition to a glass phase is
found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 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
- …