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We present a novel approach to reconstruction-based super-resolution that uses aperiodic pixel tilings, such as a Penrose tiling or a biological retina, for improved performance. To this aim, we develop a new variant of the well-known error back projection super-resolution algorithm that makes use of the exact detector model in its back projection operator for better accuracy. Pixels in our model can vary in shape and size, and there may be gaps between adjacent pixels. The algorithm applies equally well to periodic or aperiodic pixel tilings. We present analysis and extensive tests using synthetic and real images to show that our approach using aperiodic layouts substantially outperforms existing reconstruction-based algorithms for regular pixel arrays. We close with a discussion of the feasibility of manufacturing CMOS or CCD chips with pixels arranged in Penrose tilings.