The mid 1980s benefited from three major scientific announcements that altered our world view on the structure of matter and its properties. These were the discovery of quasiperiodic crystals (QCs), which are commonly referred to as quasicrystals (1984), fullerenes (1985), and high-temperature superconductivity (1986). The discovery of QC was announced to a community active in the mature science of crystallography. Crystallographers, and other scientists who studied the structure of matter and its defects, relied on a series of laws and paradigms undisputed since von Laue performed his first x-ray-diffraction experiments in 1912. One leading paradigm stated that the atomic structure of a crystal is ordered and periodic. Explanations of this paradigm, based on common sense, could be summarized as, “It is periodic because it is ordered.”
Periodicity implies a set of specific rules, among them the allowed rotational symmetries—namely one-, two-, three-, four-, and sixfold. Fivefold rotational symmetry is excluded. Past textbooks specified this, stating that fivefold rotational symmetry is impossible in periodic structures. Other books stated the impossibility of such symmetry in crystals.
Paradigms are based on experience rather than on a rigorous, scientific study process. Therefore when proven wrong, they are difficult to uproot. From 1912 to 1984, nothing shook the paradigm, “Order in crystals is periodicity.” Incommensurate structures challenged the paradigm for a while but were soon found to be a modulation of periodic crystals, saving the paradigm. However in the background, a series of articles broadened the scope of crystallography into hyperspace and prepared the mathematical platform from which the science of quasiperiodicity in crystals could take off.