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A class of antenna arrays are introduced, which we call fractile arrays. A fractile array is defined as any array with a fractal boundary contour that tiles the plane without gaps or overlaps. It is shown that the unique geometrical features of fractiles may be exploited in order to make available a family of deterministic arrays that offer several highly desirable performance advantages over their conventional periodic planar array counterparts. Most notably, fractile arrays have no grating lobes even when the minimum spacing between elements is increased to at least one-wavelength. This has led to the development of a new design methodology for modular broadband arrays that is based on fractal tilings. Several examples of fractile arrays are considered including Peano-Gosper, terdragon, six-terdragon, and fudgeflake arrays. Efficient iterative procedures for calculating the radiation patterns of these fractile arrays to arbitrary stage of growth P are also introduced.