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The Japanese Pioneering Contribution to the Investigation of Modular Structures
Author(s) -
Umayahara Akihiro,
Nespolo Massimo
Publication year - 2020
Publication title -
crystal research and technology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.377
H-Index - 64
eISSN - 1521-4079
pISSN - 0232-1300
DOI - 10.1002/crat.201900045
Subject(s) - modular design , archetype , space (punctuation) , symmetry (geometry) , interface (matter) , field (mathematics) , space group , order (exchange) , pure mathematics , theoretical physics , crystallography , computer science , mathematics , geometry , physics , chemistry , optics , art , literature , x ray crystallography , bubble , finance , maximum bubble pressure method , parallel computing , diffraction , economics , operating system
Monoarchetypal modular crystal structures are built by regularly juxtaposing modules obtained from the same archetype. At the interface the coordination environment may altered, leading to a chemical modulation that accompanies the structural variability. When this 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 Order–Disorder (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 category, that he called “twinned space groups”, in which he considered only digonal operations. His approach was later expanded by Sadanaga, who recognized that Ito's “twinned space groups” are space groupoids. In this article the contribution of the Japanese school to the field is traced back, and it is shown that the fundamental concepts later developed in a more systematic way, were already present, in a nutshell, in the works of Ito and Sadanaga; finally, a slightly revised classification of monoarchetypal modular structures is proposed which better reflects the original contribution by the Japanese school.