Are we limited to use only xy, yz and zx planes to check for planes of symmetry of a molecule? Because I see a plane of symmetry in the following molecule through the two shown H atoms or OH groups. I am aware of the 3D structure shown by this projection. And since this molecule is Chiral, and has enantiomeric mirror image, this existence of a plane of symmetry appears contradictory.
3$\begingroup$ This question would be more suited to Chemistry? $\endgroup$– joseph hFeb 16, 2021 at 7:10
2$\begingroup$ Mother Nature uses whatever axes she wants, and is not limited to our frame of reference, so, yes. $\endgroup$– Jon CusterFeb 16, 2021 at 14:31
$\begingroup$ Isn't this the meso isomer of this chemical? I'll admit that I'm rusty on what 3D structure is understood by this diagram, but if the main carbon chain is "zig-zagging" towards & away from the viewer, then the molecule is superposable on itself via a point reflection about the center of mass (and therefore via a plane reflection as well.) $\endgroup$– Michael SeifertFeb 16, 2021 at 18:36
$\begingroup$ @MichaelSeifert No, this is not a meso, and no, it has neither point nor plane symmetry. $\endgroup$– Ivan NeretinFeb 16, 2021 at 23:23
$\begingroup$ I think both answers so far fail to address properly the utter otherworldliness of the premise in question. You have a plane of symmetry (approximate, of course); are you only allowed to walk along the xy, yz and zx planes of the coordinate system which is official in your jurisdiction? I guess no. Well, neither are molecules. $\endgroup$– Ivan NeretinFeb 16, 2021 at 23:43
I don't see planes of symmetry through H atoms or OH groups. Actually, these planes are not present. It is easy to check this by looking at the position of the CH3 group after reflection in one of these planes: you find CH3 at a completely different position (none of the original positions is mapped and hence you can distinguish the starting configuration from the final one).
@gryphus's answer is correct for the statement that there is no mirror plane in the structure of the question. Independent that this is a late comment to the answer, the reader of such a Fischer projection should not hesitate to construct a model of the underlying structure (here obviously with quite some non favourable steric interactions) to get the molecule a 3D object:
It then becomes easier to recognize that regardless how one rotates the substitutents around the two sterogenic centers marked (S,S), there will be no conformation including a mirror plane.
To answer the general part of the question: no, we are not limited to the Cartesian planes and axes. The obvious examples are the ammonia and the methane milecules: the former has several reflection planes, positioned at $120^\circ$, whereas the latter has multiple reflection planes and also the rotation axes (one can easily google both).
Regarding the specific molecule mentioned in the question: not only does it not have the indicated symmetry plane, as discussed by @gryphus, but it is generally a bad practice to treat whole groups (OH and CH3 in this case) as single points, as well as to consider a 3D molecule as planar.