Why does 1,2-dimethylcyclohexane only possess three stereoisomers? Why does 1,2-dimethylcyclohexane in chair form have only three stereoisomers?


I understand that the trans isomer doesn't possess plane of symmetry and thus it has two stereoisomers. But my book says that the cis isomer has does not have two  stereoisomers. Why is this so?



The structure of cis-1,2-dimethylcyclohexane is shown above. Its mirror image is non-superimposable and cannot be obtained by rotation about single single bonds or by a ring flip. Can anyone explain?Thanks
Quoting , ORGANIC CHEMISTRY second edition by JOSEPH M. HORNBACK, UNIVERSITY OF DENVER

1,2-dimethylcyclohexane has two chirality centers.


The cis-diastereomer has a plane of symmetry bisecting the ring bond between
the two methyl groups  and therefore a meso compound.It is not chiral and does not rotate plane-polarized light.The trans-diastereomer exists as a pair of enantiomers.

The chair conformation of cis-1,2-dimethylcyclohexane  shows that it is chiral.
But due to  ring-flip , another  conformation ,its  enantiomer of the original conformation is also formed.These conformers interconvert rapidly at room temperature, therefore cis-1,2-dimethylcyclohexane is not chiral.

Further I quote

"stereochemical analyses such as the preceding one can be done with drawings using planar rings" pg.235.

A molecule cannot be optically active if its chiral conformations are in
equilibrium with their mirror images. We consider such a molecule to
be achiral. PG- 193 - ORGANIC CHEMISTRY  .EIGHTH EDITION
L.G. WADE , JR . WHITMAN COLLEGE