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 broadband 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 will also be presented.