Syntopy

Krause, H. and E.I. Grussendorf. 1978. Syntopy of Bowen s disease and lewisite scar. Hautarzt 29 490 93. 

Molecular symmetry and quasisymmetry, using the syntopy model and related approaches" ... 

V. ANOTHER FUZZY SYMMETRY APPROACH SYNTOPY AND SYNTOPY GROUPS... 

Syntopy and syntopy groups were introduced in an early approach to a fuzzy set representation of approximate symmetry, where imperfect symmetry is regarded as fuzzy symmetry. Whereas any symmetry is a discrete property within a metric space, it is natural to consider a fuzzy set approach for a continuous extension of the discrete symmetry concept to quasisymmetric objects, such as some almost symmetric molecular structures. The syntopy approaches take into account the nonlocalized, quantum-mechanical, fuzzy nature of nuclear arrangements of molecules. 

The discontinuous symmetry changes and the binary nature of the presence or absence of symmetry elements hinders the application of point group symmetry methods for general molecular structures. In the syntopy approach, based on fuzzy set theory, the discrete concept of point symmetry is replaced by a continuous concept and is applicable to cases of almost symmetric or quasisymmetric molecular arrangements. When replacing symmetry with syntopy, some of the advantages of the group... 

All possible energy-based syntopies generated by the various potential energy surfaces of the given stoichiometry are interrelated through the fundamental syntopy of the nuclear configuration space M. 

A fuzzy set generalization of nuclear point symmetry in terms of these two syntopy models is applicable to all nuclear arrangements. Using appropriate membership functions, syntopy provides a measure of symmetry resemblance of actual, general nuclear configurations to ideal, fully symmetric nuclear configurations. 

Within the syntopy models discussed, the concept of point symmetry is generalized from a restricted family of symmetric nuclear arrangements to all possible nuclear arrangements by using fuzzy sets and the symmetry resemblance between actual molecular arrangements to those having some ideal, exact symmetry. 

Various generalizations of syntopy are possible by replacing the configuration space distance by some alternative similarity measures. In fact, any of the similarity measures and symmetry deficiency measures may serve as a suitable parameter in the definition of fuzzy syntopy membership functions. 

SYMMETRY AND APPROXIMATE SYMMETRY, SYMMETRY DEFICIENCY MEASURES, SYNTOPY, AND SYMMORPHY... 

Imperfect symmetry can be viewed in the broader context of molecular distortions. In this book the emphasis is on descriptions and measures based on topological arguments [54,55,106,108,158,240,243,247,248,345,444,445], or on the related concepts of syntopy [252,394,395] and symmorphy [43,108] discussed briefly in subsequent parts of this chapter. For a variety of alternative approaches and important additional insight, the reader may consult references [58,446-449]. 

There have been two types of syntopy proposed, one based on energetic [252,394], the other on purely geometric [395] resemblance of nuclear arrangements to those of specific point symmetries. 

Within the syntopy model, the essential algebraic structure of point symmetry groups is retained (in fact, this structure is extended), and the elements of syntopy groups are derived from ordinary point symmetry operators [252,394,395]. There are, however, alternative approaches for the generalization of symmetry, where fundamentally different algebraic structures are used. 

J. Maruani and P. G. Mezey, Compt. Rend., 305, Ser. II, 1051 (1987). The Concept of Syntopy A Continuous Extension of the Symmetry Concept for Quasi-Symmetric Structures Using Fuzzy Set Theory. [Erratum, ibid., 306, Set. II, 1141 (1988).]... 

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