[Super Newb]

Dear fellow chemist,

I was attending a class on stereochemistry and we were discussing on the stereochemistry of cis-1,4-dimethylcyclohexane. I was told that there are two planes of symmetry as shown below. But, I am left unconvinced as I feel that the second plane of symmetry which cuts the two C-C bonds orthogonally may not give rise to symmetry as one of the methyl group is pointing up equatorially while the other one is pointing up axially in a chair conformation.

If they are right about symmetry–– that there are indeed two planes of symmetry for cis-1,4-dimethylcyclohexane, can I also say that it is true that (1R,2R,3S,4S)-1,2,3,4-tetramethylcyclohexane is also an achiral molecule as shown below?

[AWK]

cyclohexane ring is not flat. Moreover, it can show two different limiting conformations (and many intermediate ones between them).
https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/sterism2.htm

[Super Newb]

Dear AWK,

I am cognizant of the various intermediate conformation of cyclohexane. However, I am still not too certain about the possibility of having two line of symmetry to determine the chirality/achirality of the molecule.

Thanks!

[AWK]

Your first drawing (left upper) is correct for chair conformation (boat conformation is impossible because of steric hindrance). One plane of symmetry is sufficient for the lack of optical activity.
You second upper drawing is wrong for chair conformation. It would be correct for flat ring but this is impossible for cyclohexane.

[orgopete]

I think the problem here is with conformation versus configuration. Even ethane can be drawn as two non-symmetrical conformations, staggered v eclipsed, yet is symmetrical. cis-1,4-dimethylcyclohexane will exist with one axial and one equatorial conformation (as will cis-1,2-dimethylcylcohexane). Drawing them flat with wedges and hashes removes the conformational information and allows one to decide simply on symmetry. It isn't that cyclohexane does not have a conformation, but that the hashed and wedged structures do not contain that information. They are simply cis and trans. When considered from that perspective, I see two planes of symmetry.