The boundary between black and white would touch every point on the square if the algorithm were iterated ad infinitum.
On a topological torus (top-left edge abutting bottom-right, and top-right to bottom left), the black pattern is identical to the white pattern, but for a 180 degree rotation and small shift. The boundary between the black and white regions is then connected, forming a closed curve (on a 2-D surface embedded in a 3-D space) that separates the two regions.
Constructed with PSP, in a manner analogous to a construction of the Hilbert Curve
and other L-System