Mathematical aspects of quasicrystals

Abstract

A two-dimensional model of a quasi-crystal is the Penrose tiling (1974), which is an aperiodic “disjoint” covering of the plane generated by two rhombi displaystyle R_{36^{circ }} and displaystyle R_{72^{circ }} with equal side lengths. It is crucial that the areas’ ratio is irrational φ=areaR72∘areaR36∘=1+52(goldenratio)(φ2-φ-1=0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\displaystyle \\varphi = \\frac{\ extrm{area} \\,R_{72^{\\circ }}}{\ extrm{area}\\, R_{36^{\\circ }}} = \\frac{1+\\sqrt{5}}{2}\\, (\\mathrm{golden\\, ratio})\\, (\\varphi ^2 - \\varphi - 1 = 0), \\end{aligned}$$\\end{document}which in turn reveals a local five-fold symmetry, forbidden for crystals. Recent advances on “Wang tiles”, that is square tiles that cover the plane but cannot do it in a periodic fashion, are due to Jeandel and Rao (An aperiodic set of 11 Wang tiles, Advances in Combinatorics, pp 1–37, 2021), giving a definitive answer to the problem raised by Hao Wang in 1961. Other recent applications to variational problems in Homogenization are also mentioned (Braides et al. in C R Acad Sci Paris 347(11–12):697–700, 2009).