Explore an advanced mathematical lecture examining isoperimetric problems for planar tilings that incorporate unequal repeating cells. Delve into general existence and regularity results while focusing on the classification of planar isoperimetric double tilings featuring two repeating cells. Learn how to explicitly compute the associated energy profile and gain a complete understanding of phase transitions in this context. Discover the theoretical foundations and practical applications of this specialized area of mathematics through research conducted in collaboration with M. Novaga and E. Paolini. This 31-minute presentation was delivered as part of the Thematic Programme on "Free Boundary Problems" at the Erwin Schrödinger International Institute for Mathematics and Physics, providing insights into cutting-edge research in geometric analysis and variational problems.