Which Distributions of Matter Diffract ? An Initial Investigation

Abstract

We report on a preliminary investigation of the connections between quasiperiodic tilings, algebraic number theory, and cut-and-project sets. We substantially answer the question "which 1-dimensional tilings obtained by inflation rules are quasiperiodic" by showing that in general the characteristic equation of the inflation rule should have one root of absolute value greater than one and the rest of absolute value less than one. We also show that the vertices of such a tiling are contained in a cut-and-project set

Surface magnetization of aperiodic ising quantum chains

“…A condition for quasiperiodicity was established [13] in the case of aperiodic chains generated through an inflation rule. The one-dimensional lattice is quasiperiodic, i.e.…”

Order in glassy systems

“…This sequence, known as the Thue-Morse sequence, also has no δ-functions in its Fourier transform [11,12] The Rudin-Shapiro and the Thue-Morse sequences are examples of "Non Pisot sequences" [10]. Thus, the three examples have different kinds of order:…”

“…This sequence, known as the Rudin-Shapiro sequence, has a discrete Fourier transform with flat modulus [10][11][12][13]; and Rule 3':…”

Lempel-Ziv Complexity of Photonic Quasicrystals

“…This matrix does not depend on the precise form of the substitutions, only on the number of letters L or H. The eigenvalues of T contain a lot of information. Actually, as discovered by Bombieri and Taylor [65,66], if the spectrum of T contains a Pisot number as an eigenvalue, the sequence is quasiperiodic; otherwise it is not (and then is purely aperiodic). We recall that a Pisot number is a positive algebraic number (i.e., a number that is a solution of an algebraic equation) greater than one, all of whose conjugate elements (the other solutions of the defining algebraic equation) have modulus less than unity [67].…”