  A theory is developed of the intricately fingered patterns of flux domains
observed in the intermediate state of thin type-I superconductors. The patterns
are shown to arise from the competition between the long-range Biot-Savart
interactions of the Meissner currents encircling each region and the
superconducting-normal surface energy. The energy of a set of such domains is
expressed as a nonlocal functional of the positions of their boundaries, and a
simple gradient flow in configuration space yields branched flux domains
qualitatively like those seen in experiment. Connections with pattern formation
in amphiphilic monolayers and magnetic fluids are emphasized.
