Anisotropic isoperimetric double tilings of the plane

We study periodic partitions of the plane into two distinct cells minimizing the anisotropic ${\mathrm{\ell }}_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,{\mathrm{\ell }}_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the ${\mathrm{\ell }}_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a vertex. This configuration is unique unless the two cells are assigned the same area. Finally, we prove that the limiting (non periodic) partitions obtained by sending one volume to infinity are locally ${\mathrm{\ell }}_1$-isoperimetric.