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This paper investigates the performance of planar inductors based on space filling curves, a family of fractals with the property of completely filling a bounded area. Fractal-based inductor design is a method for obtaining a very long trace lengths-and thus inductance densities - in 2-D space as a replacement for the serpentines currently used in one layer inductors. Because of the intricate course created by a fractal curve, these types of inductors are particularly well suited for stretchable electronics, where a tortuous path relieves mechanical stress and creates a more compliant structure. Inductors based on seven common space filling curves, all bounded within a one square millimeter area, were both simulated and measured experimentally and found to vary between 3.0 and 7.1 nH. Lower order fractals were found to give comparable performance to serpentine inductors with similar inductance density. More complicated fractals, after more than two iterations, were found to have lower inductance density than similar resistance serpentines. Mechanical simulations demonstrate a reduction in stress by a factor of 10 or more compared with the loop and serpentine designs.