Enzymind Labs

The boundary between the black and white regions in the figure below would touch every point on a square if the generating algorithm were repeated ad infinitum.

As a topological torus (top-left edge abutting bottom-right, and top-right to bottom left), the black fractal pattern is identical to the white fractal 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.

The boundary is like other space-filling curves, such as the Peano Curve, Hilbert Curve, and Z-order curve, and can be generated by an L-System (Lindenmayer System, a form of iterative string replacement). These curves, and some other space-filling curves, can also divide a square and can be closed on a topological torus. With the Peano and Z-order curves the two regions on a topological torus can also be identical in all respects except for a 180 degree rotation. I suspect there are many 2-D Lindenmayer System figures that have this property. But, curiously, it appears to me that the Hilbert curve can't be closed such that there are two identical regions -- on a square, cylinder or torus.