Why is pic related only possible in 3 or more dimensions? Is there any more theoretical formulation of this problem that might help me understand what's going on? Is there a 3D version of this problem? If not, why is the jump to 3 dimensions special?

Am I schizo, or is this somehow related to the way the jump from the 2nd hyperoperation (multiplication) to the 3rd (exponentiation) is special? (It's special because 1) the commutative property suddenly disappears and 2) the larger set of numbers we have to define to accommodate its inverse, as we did with naturals -> integers -> rationals, is suddenly uncountably infinite.)

Feel free to post your hilarious and original solutions to le ebin autism test, as any bumps will likely help me get a serious answer.