Number of stereoisomers of tetramethylspirononane

[![2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/5xgEBm.jpg)](https://i.stack.imgur.com/5xgEBm.jpg)


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.




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)  

![(2R,3R,7R,8R)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/5gD13.png)
2. (2​*R*,3​*R*,7​*R*,8​*S*)-2,3,7,8-tetramethylspiro[4.4]nonane (8 times)  

![(2R,3R,7R,8S)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/2RHk3.png)
3. (2​*R*,3​*R*,7​*S*,8​*S*)-2,3,7,8-tetramethylspiro[4.4]nonane (4 times)  

![(2R,3R,7S,8S)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/0DWOZ.png)
4. (2​*R*,3​*S*,5​*r*,7​*R*,8​*S*)-2,3,7,8-tetramethylspiro[4.4]nonane (4 times)  

![(2R,3S,5r,7R,8S)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/K9ez5.png)
5. (2​*R*,3​*S*,5​*s*,7​*R*,8​*S*)-2,3,7,8-tetramethylspiro[4.4]nonane (4 times)  

![(2R,3S,5s,7R,8S)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/oJ25K.png)
6. (2​*R*,3​*S*,7​*S*,8​*S*)-2,3,7,8-tetramethylspiro[4.4]nonane (8 times)  

![(2R,3S,7S,8S)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/4Hmon.png)
7. (2​*S*,3​*S*,7​*S*,8​*S*)-2,3,7,8-tetramethylspiro[4.4]nonane (2 times)  

![(2S,3S,7S,8S)-2,3,7,8-tetramethylspiro[4.4]nonane](https://i.stack.imgur.com/8PEHL.png)


Therefore, there are seven different combinations.


 16

Here is another approach to @loong's solution to the seven stereoisomers of this tetramethyl spiro[4.4.0]nonane. [This question is akin to one recently asked about [tetramethylspiropentanes](https://chemistry.stackexchange.com/questions/95895/how-to-find-the-number-of-optically-active-and-optically-inactive-isomers-for-a)].  
  

[![enter image description here](https://i.stack.imgur.com/sJI7p.jpg)](https://i.stack.imgur.com/sJI7p.jpg)


Using the arbitrary numbering scheme shown above and assigning C1 as having the R-configuration, the eight permutations for C1-4 are listed on the left of the diagram. Only RSRS and RSSR have stereochemical designations at the spiro carbon (C5). Inside the black box are the eight structures. The R/S designations are in the order 1234(5). The three blue structures, **RRRS**, RRSR and RSRR, are identical and its enantiomer is RSSS, aka **SSSR**. The black RRRR isomer’s enantiomer is the SSSS isomer shown on the left. The green pair are enantiomers. The red isomer is a meso structure.  

Summing up, there are three pairs of enantiomers and one meso compound. There are seven stereoisomers.  

***ADDENDUM:*** Structure **A** (the red meso isomer RRSS) is identical to its mirror image **B** by reflection and rotation. There is no plane of symmetry.  

[![enter image description here](https://i.stack.imgur.com/LpRwT.jpg)](https://i.stack.imgur.com/LpRwT.jpg)


 13