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This paper investigates the radiation characteristics of a new type of array that is based on the family of space-filling and self-avoiding fractals known as Peano-Gosper curves. The elements of the fractal array are uniformly distributed along a Peano-Gosper curve, which leads to a planar array configuration with parallelogram cells that is bounded by a closed Koch curve. These unique properties are exploited in order to develop a design methodology for deterministic arrays that have no grating lobes even when the minimum spacing between elements is increased to at least one wavelength. This leads to a class of arrays that are relatively broad-band when compared to more conventional periodic planar arrays with square or rectangular cells and regular boundary contours. An efficient iterative procedure for calculating the radiation patterns of these Peano-Gosper fractal arrays to arbitrary stage of growth P is also introduced in this paper.