RANDOM DODECAGONAL TILING - A SIMULATION
B. Rubinstein
Department of Physics, Ben-Gurion University
of the Negev, POB 653, IL-84105 Be'er-Sheba, Israel
e-mail: boazrb@bgumail.bgu.ac.il
Keywords: Quasicrystal, tilings, random
tilings, dodecagon, vertex
Quasicrystals are structures having long range order but no
translational symmetry. Building a perfect quasicrystal seems to
require long range information, and thus the way these structures
form is not fully understood yet. Quasicrystals vary in their
level of perfection. Some are close to perfect quasiperiodic
tilings (mathematically achieved, for instance, as cuts from
higher space periodic structures). Others are more like random
tilings, tilings made by randomly attaching tiles (edge to edge)
with appropriate matching rules. Dodecagonal symmetry appears in
Ni-Cr alloys [1], and in V-Ni and V-Ni-Si alloys [2]. The Ni-Cr alloys are well described
by appropriate quasiperiodic tilings proposed by Gahler [3], and
independently by Niizeki and Mitani [4]. Trying to mimic
the formation of a two dimensional dodecagonal quasicrystal, I
wrote a simulation program. The algorithm used was solidification
from a melt consisting of squares and triangles. When no
restrictions were imposed no long range order was observed, and
even locally the structures were usually very different from
those seen in nature. A tendency to phase separation was also
observed (fig. 1). Adding a simple constraint on the tiling,
forcing an edge picked at random to be shared by a square and a
triangle, resulted in a dramatic improvement (fig. 2). It caused
the resulting tilings to have a strong preference for local
patterns occurring in nature (for instance in V-N and V-Ni-Si).
When counting the number of vertices of each of the types formed
(the type of a vertex is defined by the arrangement of tiles
around it), it was found that their statistics resembled that of Stampfli's hierarchical random
construction (SHRC) [5]. Moreover, this same constraint
brought the ratio of squares to triangles to approach that of a
perfect SHRC (within less then 1.5% of it after building 17000
vertices!). These observations point out that SHRC may be a good
candidate for describing random dodecagonal quasicrystals.
fig. 1. Phase separation in a solidification
from a melt, the ratio of squares to
triangles is 30.5/4.
fig. 2. The tiling resulting from imposing the
constraint that an edge picked at random be shared by a square
and a triangle.