Let's start by proving the the one dimensional version of the Fundamental Theorem of Projective Geometry. We call it one dimensional because it concerns projectivities between lines in in P³.
Next we'll prove Pappus's Theorem, for which we'll need a neat little construction.
We're now ready to prove Pappus's Theorem!
Finally, let's prove the two dimensional version of the Fundamental Theorem of Projective Geometry. We call it two dimensional because it concerns projectivities between planes in P³.
In order to complete the proof, we'll need a generalization of the Three Fixed Points Theorem...namely, the Four Fixed Points Lemma!
Great, we've proved some interesting results about projectivities between planes. But how does this relate to our original question, about the possible shadows of a square?