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 five-fold symmetry, for crystals have been discovered. Medieval Islamic decorators as well as Albrecht Dürer, Johannes Kepler, Roger Penrose, Mackay himself, and other pioneer crystallographers raised important contributions to the theoretical discovery of crystalline possibilities long before or independently of the discovery of their actual existence. In this paper, 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.

中文翻译：

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进一步阐明 Mackay 对结晶纯可能性的理论发现的哲学意义

早在 1981 年，也就是 Shechtman 发现真正的准晶体的大约 1 年前，Alan L. Mackay 在一篇开创性的论文中讨论了将晶体学扩展到现代阶段的第一步。在这个阶段，发现了晶体结构和秩序的新可能性，例如五重对称结构。中世纪的伊斯兰装饰家以及阿尔布雷希特·丢勒、约翰内斯·开普勒、罗杰·彭罗斯、麦凯本人和其他先驱晶体学家，早在或独立于发现晶体可能性之前，就对晶体可能性的理论发现做出了重要贡献。在本文中，我进一步讨论了 Mackay 理论发现的哲学意义以及他对扩展纯几何晶体学、生物晶体学、