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This article introduces the 2-D Polya fractal and its evolution into the modified Polya curve (MPC). The MPC is assigned a linewidth, which, together with the MPC order and shaping factor, provides optimization parameters for specific device applications. The MPCs are selected to generate useful shapes that fit within a chosen triangular envelope. These are applied as basis elements and are employed to organize families of polygonal shapes that may be used to design antennas, frequency-selective surfaces, metamaterials, ultrathin absorbers, and filters. MPCs may also be modified for implementation in 3-D geometries.