The Philosophical Significance of Alan Mackay’s Theoretical Discovery of Quasicrystals

Dan Shechtman was the first to discover an actual quasicrystal (on April 8, 1982). As early as 1981, about 1 year before Shechtman’s discovery of an actual quasicrystal, Alan L. Mackay discussed, in a seminal paper, the first steps for the expansion of crystallography toward its modern phase. In this phase, new possibilities of structures and order (such as the structures of fivefold symmetry) for crystals have been discovered. Medieval Islamic artists as well as Albrecht Dürer, Johannes Kepler, Roger Penrose, Mackay himself, and other pioneer crystallographers raised important contributions to the theoretical discovery of pure crystalline possibilities long before or independently of the discovery of their actual existence. Shechtman, however, was not the first to discover the individual pure possibility of this novel structure (the theoretical discovery), which had been excluded from the range of the possibilities of crystals (as it had been fixed by both theoretical and empirical means at the beginning of the twentieth century). Penrose and Mackay, in particular, had contributed to the discovery of the individual pure possibilities of quasicrystals, which are merely structural, and, like purely mathematical entities, they do not exist spatiotemporally and causally, whereas actual quasicrystals exist only spatiotemporally and causally. The individual pure possibilities of quasicrystals do not depend on their actualities, and without these possibilities, those actualities would have been theoretically groundless, meaningless, and could not be correctly identified, if at all. Hence, Mackay’s contribution to the meaning and theoretical basis of the discovery of actual quasicrystals is indispensable. In this Chapter, I discuss further the philosophical significance of Mackay’s theoretical discovery and his contribution to the expansion of pure geometrical crystallography, biological crystallography, and generalized crystallography.