The Double Bubble problem asks the following: given two volumes, what are the two shapes admitting these volumes with the smallest perimeter, where the perimeter of the joint boundary is counted once. While past solutions focused on rotationally invariant norms, our work addresses non-isotropic scenarios found in crystalline structures and lattice-based soup bubbles. We present three results: the first provides a solution to the Double Bubble problem in the taxicab metric using elementary principles, the second shows that the solution to the discrete double bubble problem in the taxicab metric is at most two more than the ceiling function of the continuous solution, and the final result presents the solution to the Double Bubble problem in the hexagonal norm. The works consist of new geometric and discrete optimization techniques.

Joint work with Eviatar Procaccia and Rory O’Dwyer