A design procedure is presented that improves the energy efficiency and saves catalyst material of a polymer electrolyte membrane fuel cell (PEMFC). The method is demonstrated for a single cell in a stack and uses the theorem of equipartition of entropy production to maximize energy efficiency. The gas supply and water outlet systems, designed to produce entropy uniformly, have a fractal structure inspired by the human lung. The tree-like gas distributor engraved in the bipolar plates may eliminate the need for porous transport layers. Mathematical solutions are given for the optimal height, macroporosity, and nanoporous column width of the electro-catalytic layer beneath the gas supply system. It is shown that the optimal macroporosity of the catalytic layer is equal to 1/2 for the model chosen and that the optimal height of the catalytic layer depends upon the coefficient for first-order reaction kinetics at the cathode, the diffusion constant for oxygen in the gas phase, and the oxygen concentration of the inlet flow. An upper bound can be used for the column width. The general results are illustrated using the standard E-TEK Elat/Std/DS/V2 gas diffusion electrode with 0.5 mg of Pt/cm2 membrane area and 20% Pt/C on Vulcan XC-72 as a support material. It is indicated that the amount of catalyst can be reduced by a factor of 4−8, while the energy efficiency can be increased by 10−20% at high current densities.