Binary Grassmann-Plucker relations

In some of the tests presented we made strong assumptions on the conformations allowed. However, one may not always want to accept such strong postulates. In this section, we present a special kind of rule that is valid for any geometrically realizable POF. 

As orientations are signs of determinants, it follows that the orientation function of any conformation fulfills the binary Grassmann Pliicker relations, introduced below (Herrmann Grassmann, German mathematician, 1809-1877, Julius Pliicker, German mathematician and physicist, 1801-1868). 

18 Definition (Binary Grassmann-PlUcker relation) if two quadruples of atoms and 

The same orientation for the first two quadruples mentioned may or may not differ from the same orientation for the second two quadruples. 

The last two possibilities are false for any orientation function x, as (1,2,0,1) and (0,2,0,1) both contain one repeated atom and thus these orientations are zero. We reformulate the condition by arranging each quadruple in increasing order  

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In this chapter we describe the unified generation of stereoisomers including conform-ers of a molecular structure [102,103,105]. This method has the potential to generate stereoisomers that cannot be described in terms of stereocenters, stereogenic double bonds or single bond rotations. Fundamentals such as the concept of a (partial) orientation function are discussed, and mathematical tools such as Radon partitions and binary Grassmann-Plucker relations are used to construct tests for abstract orientation functions. Some simple examples are treated in detail. 

Definition (Chirotope) An alternating, not constantly zero function n —> +,0 fulfUlingthe binary Grassmann-Plucker relations... 

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