Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

Abstract

<jats:p>Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.</jats:p>

Generating Functions Related to the Fibonacci Substitution

“…The Fibonacci Substitution is one instance of a substitu-tion tiling that can be viewed this way. The Fibonacci Substitution is an aperiodic substitution tiling that has the letters 'a' and 'b' as its tiles [2]. Its inflation rule is as follows, This inflation rule repeatedly concatenates the string of letters, providing a finite subword for each iteration.…”

Highly symmetric aperiodic structures -INVITED

“…In this brief note, we shall discuss some paradigmatic planar structures, focussing on their symmetry and diffraction. For mathematical and crystallographic background on the diffraction of aperiodic structures we refer to [7][8][9] and references therein.…”

Dodecahedral structures with Mosseri–Sadoc tiles

“…Aperiodic tiling in general (Baake & Grimm, 2013;Baake & Grimm, 2020) and the icosahedral quasicrystallography in particular (Di Vincenzo & Steinhardt, 1991;Janot, 1993;Senechal, 1995) constitute the main theme of research for many scientists from diverse fields of interest. The subject is mathematically intriguing as it requires the aperiodic tiling of the space by some prototiles.…”