The Rebirth of Crystallography

Exactly 20 years have passed since the surprising discovery of the first quasicrystal by Shechtman [1]. Centuries of prior research in crystallography had led to the development of a beautiful and well-established science of crystals [2]. Modern crystallography was born in the 17 th century thanks to the brilliant idea – – attributed to such great scientists as Kepler and Hooke– –that the observed properties of crystals were the result of internal order of " atomic " units. Unfortunately, by the 19 th century, when Haüy began formulating the mathematical theory of crystallography , Kepler's insightful drawings of aperiodic tilings with decagonal symmetry were long forgotten [3, 4]. It became universally accepted, though never proven, that the internal order of crystals was achieved through a periodic filling of space. Crystallography treated " order " and " peri-odicity " synonymously, both serving equally to define the notion of a " crystal ". With that came the so-called " crystallographic restriction " , stating that crystals cannot have certain forbidden symmetries, such as 5-fold rotations. The periodic nature of crystals was " confirmed " with the discovery of x-ray crystallography and numerous other experimental techniques throughout the 20 th century. The periodicity of crystals became the underlying paradigm not only for crystallography itself, but also for other disciplines such as materials science and solid state physics, whose most basic tools, like the Brillouin zone, relied on periodicity. By 1982 crystallography was a mature science, standing firmly on its theoretical paradigms, and offering sophisticated experimental tools for the structural study of materials. Shechtman's discovery of a crystal with 5-fold symmetry shook the science of crystals by its foundations. Crystallography was reborn. On the occasion of the 10 th anniversary of its announcement , Cahn [5] described the discovery of quasi-crystals as a Kuhnian scientific revolution [6]. In my view, crystallography is now in the midst of the most exciting stage of this revolution. The old established paradigms, most importantly that of the periodicity of crystals, are being overthrown. The initial skepticism of the scientific community is being replaced by a growing acceptance. New notions and paradigms are being tested and carefully adopted. New theories of quasicrystals, aperiodic tilings, and symmetry are being developed. Experimental techniques are undergoing fundamental modifications to encompass aperiodic crystals. All this intense activity is being pursued by hundreds of scientists worldwide, ranging from pure mathematicians and crystallographers to physicists …