Geometric and Topological Thinking in Organic Chemistry

Visualirable Elastic Concrete FI G1V.E) F] R"(V,EI M b 4 ) 1 1 SPACES FORMS FORMS F I G U R E S M O L E C U L A R YxY S T R U C T U R E Fig. 7. A whematic reprc\ciil.iii~iii 01 ihe rnicrrclationship of topological spaces. graphical forms, geometric forms, and geometric figures. The embedding of any of these objects in conventional chemical paradigms produces a model 01' molecular structure. R , Euclidean space (n= 1. 2, 3). For other abbreviations, see Fig. 6 and text. tures. We shall now discuss the nature of this mapping process. 4.4. Transformation from Mathematical Objects to Chemical Objects The transformation from mathematical objects (graphs, forms, figures) of topological geometry and Euclidean geometry may be viewed as the embedding of the mathematical objects into a chemical paradigm (see Fig. 7) to produce models for molecular structures. The paradigm provides the basis for the mapping process C that transforms mathematical objects into chemical objects. The historical pathway to modern chemical structure is related to that shown in Figure 7 except that Euclidean geometric figures (configurational stereochemistry) were accepted as the models for molecular structure before topological geometric forms (conformational stereochemistry).ii51 The historical pathway is summarized in Figure 8 for cycloAbstract VisualizableVisualizable (numberr . reprerentatton