168 Essays in Biochemistry 



cannot lead to a differentiation of the carbon atoms labeled C and 

 C" as these have become randomly distributed over the two positions. 



Since even a single intermediate which does not permit the differ- 

 entiation of identical substituents suffices to bring about this result 

 in a long chain of reactions, it seems of particular importance to recog- 

 nize the structural characteristics which prevent selective reactions of 

 identical groups. To our knowledge this question has not been an- 

 swered previously. Rules 1 and 3 describe structures which permit 

 differentiation. Since we have shown that neither criterion covers all 

 situations where this may occur, these rules clearly are not con- 

 vertible and hence give no reliable information about structures which 

 do not permit differentiation. Rule 2 tried to answer this question, 

 but the criterion was found to have exceptions. One may conclude, 

 therefore, that these three rules are no more than partial solutions of 

 the problem. As one should expect to find the general principle that 

 prevents differentiation of identical substituents in some structural 

 regularity, we shall attempt to link rule 4 to molecular symmetry. 



Mathematical analysis has shown that two rigid objects, so related 

 that the distance between any two points in one of them equals the 

 distance of the corresponding points in the other, can be brought into 

 coincidence by a combination of at most three operations, a trans- 

 lational motion, a rotation around an axis, and a reflection in a plane 

 perpendicular to this axis. If the two objects have one point in 

 common, the rotation and the reflection will always suffice. 9,10 A finite 

 rigid object is said to possess symmetry if two or more indistinguishable 

 arrangements exist that can be interconverted by these two types of 

 operations. 11 We therefore can distinguish two kinds of symmetry: 

 If a rotation suffices to produce another indistinguishable arrangement, 

 the object is congruent with itself in more than one way and the axis 

 of rotation is termed a simple axis of symmetry. If the conversion 

 to another indistinguishable arrangement is possible by a reflection or 

 by a rotation and reflection, the object is congruent with its mirror 

 image and the axis of rotation is termed an alternating axis of 

 symmetry.* 



Organic chemistry has concerned itself almost exclusively with sym- 

 metry of the second kind. It is quite clear, however, that this mirror- 

 image symmetry has no bearing on our problem, since the operations 

 considered in rule 4 are motions and not reflections and any super- 

 position which cannot be achieved without a reflection is of no concern. 



* See footnote on p. 161. 



