organic chemistry - Number of stereoisomers of tetramethylspirononane

The above compound – 2,3,7,8-tetramethylspiro[4.4]nonane – clearly exhibits both optical and geometrical isomerism. The question is, how many stereoisomers does it have in all?

As the number of possibilities are many, I couldn't really visualise the molecule's all possible stereoisomers properly. Could someone please help me count the total number of stereoisomers?

I figured that the carbon common to both rings is sp³ hybridised (tetrahedral geometry), and the two rings don't lie in the same plane.

Answer

The compound 2,3,7,8-tetramethylspiro[4.4]nonane has five potential stereogenic or pseudoasymmetric centres: the four C atoms with the methyl substituents and the spiro atom. In total there are $2^5=32$ possible configurations. However, many of these combinations are actually identical compounds.

A simple and obviously exhaustive (but also exhausting) approach is to draw all 32 configurations and name them. (This approach may also be used to check the results if someone finds a more elegant solution.) The resulting names for the 32 combinations are:

1. (2​R,3​R,7​R,8​​R)-2,3,7,8-tetramethylspiro[4.4]nonane (2 times)
   


2. (2​R,3​R,7​R,8​S)-2,3,7,8-tetramethylspiro[4.4]nonane (8 times)
   


3. (2​R,3​R,7​S,8​S)-2,3,7,8-tetramethylspiro[4.4]nonane (4 times)
   



4. (2​R,3​S,5​r,7​R,8​S)-2,3,7,8-tetramethylspiro[4.4]nonane (4 times)
   


5. (2​R,3​S,5​s,7​R,8​S)-2,3,7,8-tetramethylspiro[4.4]nonane (4 times)
   


6. (2​R,3​S,7​S,8​S)-2,3,7,8-tetramethylspiro[4.4]nonane (8 times)
   


7. (2​S,3​S,7​S,8​S)-2,3,7,8-tetramethylspiro[4.4]nonane (2 times)
   

Therefore, there are seven different combinations.