Deceptions in quasicrystal growth

Abstract

We discuss a new general phenomenon pertaining to tiling models of quasicrystal growth. It is known that with Penrose tiles no (deterministic) local matching rules exist which guarantee defect-free tiling for regions of arbitrary large size. We prove that this property holds quite generally: namely, that the emergence of defects in quasicrystal growth is unavoidable for all aperiodic tiling models in the plane with local matching rules, and for many models in R 3 satisfying certain conditions.

Biological Quasicrystals and the Golden Mean: Relevance to Quantum Computation

“…The Penrose tiling consists of two basic tiles, the kite and the dart (Figure 1). Along the edges of each are bumps and dents which encode local matching rules (Dworkin and Shieh, 1995). A key aspect of such aperiodic tilings is that they do not possess translational symmetry, unlike regular or periodic tilings.…”

“…However, these studies of quasicrystal growth are for three-dimensional quasicrystals and the biological quasicrystals of interest in this study are two-dimensional. While it appears that local rules can dictate the structure of threedimensional quasicrystals, in two-dimensional instances non-local rules are required (Dworkin and Shieh, 1995;van Ophuysen, 1998).…”

Growth of a One Dimensional Quasiperiodic Covering with Locally Determined Decorations

“…Therefore, a "deception" of length 13 [14] is possible even if we use a rule that checks for the arrangement of 11 tiles (and allow only correct configurations of length 12). Here, deception means a legal (satisfying the growth rule), but incorrect, configuration [5]. Since the growth process does not allow tiles to be removed, a deception (which is not a part of a Fibonacci tiling) cannot grow to a Fibonacci tiling, so we need a growth rule that allows no deceptions of any size.…”

“…A major criticism for the former scenario is that non-local information is likely needed for a perfect quasiperiodic structure growth. Such criticisms were mainly based on Penrose's observation that no local growth rules can produce a perfect quasicrystalline tiling in two-dimensional (2D) space [5,10]. Recently, we provided a local growth algorithm to produce a perfect quasicrystalline structure in 3D and showed that non-local information is not necessary for growth with the perfect quasiperiodic order [11].…”

Growing Perfect Decagonal Quasicrystals by Local Rules

“…In the former, the growth occurs at any surface site with the same attaching probability, while it occurs with different attaching probabilities in the latter. In 1988, Penrose proved that a PPT cannot be grown by local rules with uniform growth by showing that "deceptions" are unavoidable [5] where a deception is a legal patch which cannot be found in a PPT [5,6]. In the same year, Onoda et al introduced a preferential growth algorithm which can avoid deception by local rules called "vertex rules" [3].…”