Symmetry properties of Penrose type tilings

The Penrose tiling is directly related to the atomic structure of certain decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It is known that the numbers 1, $-\tau$, $(-\tau)^2$, $(-\tau)^3$, ..., where $\tau =(1+\sqrt{5})/2$, are scaling factors of the Penrose tiling. We show that the set of scaling factors is much larger, and for most of them the number of the corresponding inflation centers is infinite.Comment: Paper submitted to Phil. Mag. (for Proceedings of Quasicrystals: The Silver Jubilee, Tel Aviv, 14-19 October, 2007

Conceptual mechanization studies for a horizon definition spacecraft attitude control subsystem, phase A, part II, 10 October 1966 - 29 May 1967

Attitude control subsystem for spin stabilized spacecraft for mapping earths infrared horizon radiance profiles in 15 micron carbon dioxide absorption ban

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to an amplitude instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows to predict the stability and instability strength

Chiral Quasicrystalline Order and Dodecahedral Geometry in Exceptional Families of Viruses

On the example of exceptional families of viruses we i) show the existence of a completely new type of matter organization in nanoparticles, in which the regions with a chiral pentagonal quasicrystalline order of protein positions are arranged in a structure commensurate with the spherical topology and dodecahedral geometry, ii) generalize the classical theory of quasicrystals (QCs) to explain this organization, and iii) establish the relation between local chiral QC order and nonzero curvature of the dodecahedral capsid faces.Comment: 8 pages, 3 figure

Modelling quasicrystals at positive temperature

We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This model has uncountably many ground state configurations, all of which are nonperiodic. The question addressed in this paper is whether nonperiodicity persists at low but positive temperature. We present arguments, mostly numerical, that this is indeed the case. In particular, we define an appropriate order parameter, prove that it is identically zero at high temperatures, and show by Monte Carlo simulation that it is nonzero at low temperatures

Local Complexity of Delone Sets and Crystallinity

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package

A symmetry group of a Thue-Morse quasicrystal

We present a method of coding general self-similar structures. In particular, we construct a symmetry group of a one-dimensional Thue-Morse quasicrystal, i.e., of a nonperiodic ground state of a certain translation-invariant, exponentially decaying interaction.Comment: 6 pages, Late

Diffusive limits on the Penrose tiling

In this paper random walks on the Penrose lattice are investigated. Heat kernel estimates and the invariance principle are shown

Symmetry Breaking in the Double-Well Hermitian Matrix Models

We study symmetry breaking in $Z_2$ symmetric large $N$ matrix models. In the planar approximation for both the symmetric double-well $\phi^4$ model and the symmetric Penner model, we find there is an infinite family of broken symmetry solutions characterized by different sets of recursion coefficients $R_n$ and $S_n$ that all lead to identical free energies and eigenvalue densities. These solutions can be parameterized by an arbitrary angle $\theta(x)$, for each value of $x = n/N < 1$. In the double scaling limit, this class reduces to a smaller family of solutions with distinct free energies already at the torus level. For the double-well $\phi^4$ theory the double scaling string equations are parameterized by a conserved angular momentum parameter in the range $0 \le l < \infty$ and a single arbitrary $U(1)$ phase angle.Comment: 23 pages and 4 figures, Preprint No. CERN-TH.6611/92, Brown HET-863, HUTP -- 92/A035, LPTHE-Orsay: 92/2