>>>"Why is it that a 2 dimensional number is represented by the square root of minus one as its one-dimensional representitive?"
Maybe I'm just ignorant, but the notion of numbers having dimensions is news to me. What does this even mean? Where do you get this information? It's very odd, but as I said maybe I'm just oblivious to whatever cutting edge science you're getting into.
It's very difficult to talk about these things without getting ourselves tangled within "lay" speak, and I think it'd have been clearer if you had labeled the parts of your scenario's framework as 'space,' 'geometry,' 'spatial planes,' or something along these lines (instead of 'dimensions'). Still, I enjoy mind games that require me to visualize things, so I'm giving it a shot:
I think that there is a structural problem in your example: you are discussing numbers putatively limited to one or two dimensions, but your entire example requires three spatial dimensions. Think about it -- could you see a "square" from the 2D plane in which the square exists? ("Suppose I look down at it so I only see the top edge".) Or, if we revert to "an alternative 2-D," from which we can see 1D of the original square, this 2D from which we currently observe must intersect the 2D in which the square was originally conceived. In other words, our own visual bias aside, 3 spatial planes are presumed in your example (two intersecting, perpendicular planes of 2D = 3D).
As far as the remainder of your computations, it looks a lot like you're pumping numbers into a matrix specifically designed to produce the results you seek. A little creativity goes a long way, and the integers 1, 2, and 3 hold more than nominal utility for the facile number-cruncher. But then again, I'm no mathematician.
Still, I'm intrigued by what you mean by a "2 dimensional number."