LAUSR

Modular crystal structures are built by regularly juxtaposing 0-, 1-or 2-periodic modules. When the modules are obtained from the same archetype, the resulting structure is known as monoarchetypal and has been called a cell-twin. At the interface between two modules the coordination environment may be altered with respect to the archetype, leading to a chemical modulation that accompanies the structural variability. When such a modification of the coordination does not occur the result is a series of polytypes, whose building modules are not limited to layers, although the latter represent the most common examples. The symmetry theory of polytypes was developed by the OD school, which however considered only layers as building modules, and only those polytypes in which pairs of layers are geometrically equivalent. Well before the OD school, Ito in Japan recognized the need of a wider algebraic structure to deal with polytypes, which he called "polysynthetic structures", today known as "Ito twins". He called this algebraic structure "twinned space groups", because of the geometric similarity between layers in modular structures and individuals in twins; further, from his empirical analysis of examples from minerals he considered only digonal operations, i.e. operations whose linear part has order two. His approach was later expanded by Sadanaga to include other types of operations and in particular operations other than digonal. Sadanaga recognized that Ito's "twinned space groups" are space groupoids and proposed a classification in which polytypes were exclusive of Ito twins, contrary to the official definition. In this article we trace back the contribution of the Japanese school to the field, which is mostly unknown because published in rare and hardly accessible journals and partly in Japanese; we show that, despite some confusion in the language, the fundamental concepts later developed in a more systematic way, were already present, in a nutshell, in the works of Ito and Sadanaga; finally, we propose a slightly revised classification of monoarchetypal modular structures which better reflects the original contribution by the Japanese school.