What does it show?
The projective plane is the surface obtained by adding to the plane a point at infinity for every family of parallel lines; it is the object of projective geometry, born in the seventeenth century from Girard Desargues’s work on perspective. It cannot be placed in three-dimensional space without crossing itself; the cross-cap in the model is an image of it that crosses itself along a line segment.
A basic fact of topology is this: when a disc is removed from the projective plane, what remains is always a Möbius strip. In the model you can change the size of the disc and cut it away; the remaining part, whatever its size, is a strip with one edge and one side.
From his seminars of the mid-1960s onwards Lacan read this cut as the separation of the objet a from the divided subject ($). The disc is the object that falls, that separates from the surface; the Möbius strip is the subject structured by this fall. In the thirteenth seminar, L’objet de la psychanalyse, Lacan used the same structure to think the object gaze through perspective and Velázquez’s Las Meninas.
Related essay: The Object Gaze