Self-assembly is the spontaneous and reversible organization of individual building blocks into ordered structures. In this contribution, we will focus on two design strategies for targeting aperiodic structures with self-assembly. The first strategy employs anisotropic shape to enforce local icosahedral order resulting in geometric frustration. An example is the observation of a dodecagonal quasicrystal in simulations of hard regular tetrahedra. The second strategy investigates the role of competing distances for interacting point particles generating a tunable modulated superstructure lattice. Although originally motivated by advances in the materials sciences of nanoparticles and polymers, the findings now allow a better understanding of the origins of aperiodic order in nature. Related aspects that will be discussed are packing problems of polyhedral shapes and the role of configurational entropy.
Delone sets are infinite point sets which are uniformly discrete (points are not arbitrarily close to each other) and relatively dense (any sufficiently large ball contains at least one element of the point set). Examples of Delone sets include point lattices and aperiodic point sets derived from quasicrystals. Two Delone sets D, E are called bi-Lipschitz equivalent, if there is a bijection f : D -> E, such that f is Lipschitz continuous in both directions. D and E are furthermore called wobbling equivalent, if the distance between x and f(x) is bounded by a common constant for all x in D. This talk presents old and new results on bi-Lipschitz equivalence and wobbling equivalence of periodic and aperiodic Delone sets. For instance, it has been known for many years that the set of the vertices in a Penrose tiling is wobbling equivalent to an appropriate point lattice. Hence it is bi-Lipschitz equivalent to the square lattice Z^2. This talk shows how to generalize these results to a wide class of Delone sets. This is joint work with Alexey Garber (Moscow)
In [1] Boca, Cobeli and Zaharescu gave a very simple representation of the first
consecutive spacing distribution when looking at the visible points of the square
lattice $\ZZ^2$. Here one considers the lattice points which are "visible" from the
origin. This amounts to selecting those points which satisfy $\gcd(x, y)=1$ for their
coordinate $(x,y)$. Now place a circle of radius $R$ at the origin and project all
points inside onto this circle, effectively reducing the polar coordinate of the
point to the angle information. Then sort all these angles and measure the difference
between neighbouring ones. In [1] it was proved (even in a more general setup) that
there is a limit distribution of the differences when $R$ tends to infinity.
One might now ask the question, if the limit distribution somehow encodes information
about the degree of order of the input point set. Or phrased differently: How much
does the distribution vary when exchanging the original point set with something
else? E.g. it is known that the set of Poisson distributed points in the plane yields
the exponential distribution (representing the most "chaotic" set). We take a first
look at the numerical results when using the vertex set of aperiodic tilings in the
plane as input (e.g. Ammann-Beenker or rhombic Penrose).
[1] Boca, F.P., Cobeli, C., Zaharescu, A.: Distribution of lattice points visible
from the origin (DOI)
The Delone covering approach devised by P. Kramer [1] applied to the icosahedral
tiling T*(2F) [2] leads to an incomplete covering of the tiling. This approach
however, is based entirely on the projection technique. By including also the
inflation symmetry of this tiling, new ideas how to extend the three Delone cells in
order to cover the whole structure are presented.
[1] P. Kramer: Quasicrystals: atomic coverings and windows are dual projects
J. Phys. A: Math. Gen. 32 (1999) 5781-5793
[2] Z. Papadopolos and G. Kasner: The efficency of Delone Coverings of the
canonical tilings T*(A4) and T*(2F) : in Coverings of Discrete Quasiperiodic
Sets, Springer Tracts on Modern Physics 180, pp. 165-184
In the paper 'Quasiperiodicity and Randomness in Tilings of the Plane' by C. Godrèche and J. M. Luck from 1988 the concept of random substitutions was introduced by a brief discussion on the random Fibonacci substitution F. In this talk I will introduce a generalisation F' of F and will elaborate on the ergodicity of the measure defined on the hull of F'. Time permitting, I will also justify some calculations in the Godrèche and Luck paper, concerning the absolutely continuous part of the diffraction measure of F and F', respectively.
By studying the so-called trace-map, and in particular its non-unstable set, one can get a good description of the spectrum of a class of one-dimensional aperiodic Schrödinger operators, which is given by a subset of it. This subset is viewed as the limit of a decreasing sequence of finite unions of intervals (bands of the periodic approximants). By representing it with a Cantor set of sequences of symbols, one can get some of its geometrical properties, such as bounds on the Haussdorf dimension for instance. To investigate the dynamical properties of these operators, such as the time-evolution of an initially localized wave-packet, a better description of the spectral measure is needed. As a starting point, the coding function has to be characterized more quantitatively. We will recall briefly the coding procedure, and a way to get an efficient numerical inversion of it. Then, we will present some scaling properties of the spectrum and the wavefunctions.
Rhombic quasicrystals with n-fold symmetry are generalizations of the well-known
Penrose tilings. Generated by the cut-and-project method with zero offset they
possess sites, called flowers or stars, with 2n- or n-fold point symmetry. The
flowers are very predominant in fivefold quasicrystals and play an important role for
example in the stabilization of colloidal quasicrystals [1,2]. The frequency of the
stars and all other tile arrangements is encoded in the projection of the unit cube
complementary to the tiling. We have determined the number of stars directly up to
n=11 with the qhull convex hull program. For larger n this procedure becomes
impossible due to the large dimension and the number of vertices of the polytope.
Only statistical methods can be used to determine the number of stars for higher n.
We observe that if n increases, the number of stars decreases more than
exponentially. Thus we argue that the tiling patches with very high n (up to 30)
which many groups study experimentally should not be called quasicrystals since they
do not form a representative sample due to their small size.
[1] J. Mikhael, M. Schmiedeberg, S. Rausch, J. Roth, H. Stark, and C. Bechinger, PNAS
107, 7214 (2010).
[2] M. Schmiedeberg, J. Mikhael, S. Rausch, J. Roth, L. Helden, C. Bechinger, and H.
Stark, Eur. Phys. J. E 32, 25 (2010).
In this talk, we look at the complexity of some aperiodic structures that are \textit{not} generated by a substitution rule: A Kolakoski sequence is a sequence that is equal to its own runlength sequence, usually over the two number alphabet $\{1,2\}$, e.g., $12211212212211\ldots$. In this case (and more generally whenever the two-symbol alphabet consists of one odd and one even integer), only upper and lower bounds are known for many properties like the complexity. Another example arises if we consider which vertices in the Ammann-Beenker tiling are visible: What is the complexity of patches (and their frequencies) we get after such a construction? (These questions were inspired by Pleasants' treatment and calculation of the entropy of the visible lattice points.) We will give an overview of the difficulties and current status in finding answers to these problems.
The dipolar spin ice have attracted much attention because of their intriguing ground state ordering
and non-equilibrium, elementary excitations known as emerging magnetic monopoles [1]. Until now the spin-ice
properties of periodic, infinite lattices has been investigated [2]. We present a theoretical study of magnetic dipolar
spin ice on aperiodic lattices of finite dimensions. We consider an octagonal tiling as well as a new type of frustrated
spin network with pentagonal loops and long-range quasiperiodic structural order [3]. Especial attention is paid to
the evolution and the distribution of excitations with magnetic charges as a function of magnetic field and magnetic
potential. It is demonstrated that depending on the micromagnetic reversal mechanism in individual particles charge
ordered states or accumulation of magnetic charges can be observed.
1. E. Mengotti et al., Nature Physics 7, 68 (2011). 2. A. Schumann, P. Szary, E. Y. Vedmedenko, and H. Zabel, New
J. Phys., accepted. 3. A. Jaganatthan, B. Motz, and E. Y. Vedmedenko, Phil. Mag. 93, 2765 (2010).