Find all stereoisomers of:

cyclobutane-1,2,3,4-tetracarboxylic acid

My attempt

I got four isomers, and none of them seem to be optically active as they either have a plane of symmetry or a point of symmetry:

stereoisomers of cyclobutane-1,2,3,4-tetracarboxylic acid

However, my sir keeps insisting there are five isomers. Could anybody help me identify the fifth one?

  • 6
    $\begingroup$ I don't believe there is a fifth. $\endgroup$ – Ivan Neretin Feb 20 '17 at 6:43
  • 4
    $\begingroup$ I agree with you, there are just 4 stereo isomers. If anyone of them would have been optically active then we could have said that there are 5 isomers, but as you correctly pointed out none of the stereo isomers are optically active. $\endgroup$ – Prakhar Feb 20 '17 at 13:40
  • $\begingroup$ Sometimes the teacher is wrong. $\endgroup$ – mykhal Apr 21 '17 at 13:21
  • $\begingroup$ Related: Total number of stereoisomers of truxillic acid $\endgroup$ – andselisk May 19 at 13:52

You can view 1,2,3,4-tetrasomethingcyclobutane as a square with black and/or white corners (black representing e.g. up, then white representing down configuration of the substituent). Two squares represent the same object when after applying some of the following operations on the first you get the second one:

  • rotation in the paper plane (of course)
  • mirroring AND color inversion

By drawing all 24 = 16 possible squares and using these two rules, you quickly find that they represent only 4 different objects, as seen on the following sketch.


Note that the second "controversial" operation mirroring AND color inversion is in our scenario, because of D2 symmetry of all squares, equivalent with more proper, "mirror-free" operation - rotation around some in-plane axis, considering that obverse of the square has inverse colors. (Because of the symmetry, the operation can even be reduced to color inversion only.)

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