We give a light talk on optimality of shapes in geometry and physics. First, we recollect classical geometric results that the disk has the largest area (respectively, the smallest perimeter) among all domains of a given perimeter (respectively, area). Second, we recall that the circular drum has the lowest fundamental tone among all drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. In parallel, we discuss the analogous optimality of square among all rectangles in geometry and physics. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.