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Analytical phantoms have closed form Fourier transform expressions and are used to simulate MRI acquisitions. Existing three-dimensional (3D) analytical phantoms are unable to accurately model shapes of biomedical interest. The goal of this study was to demonstrate that polyhedral analytical phantoms have closed form Fourier transform expressions and can accurately represent 3D biomedical shapes.The Fourier transform of a polyhedron was implemented and its accuracy in representing faceted and smooth surfaces was characterized. Realistic anthropomorphic polyhedral brain and torso phantoms were constructed and their use in simulated 3D and two-dimensional (2D) MRI acquisitions was described.Using polyhedra, the Fourier transform of faceted shapes can be computed to within machine precision. Smooth surfaces can be approximated with increasing accuracy by increasing the number of facets in the polyhedron; the additional accumulated numerical imprecision of the Fourier transform of polyhedra with many faces remained small. Simulations of 3D and 2D brain and 2D torso cine acquisitions produced realistic reconstructions free of high frequency edge aliasing compared with equivalent voxelized/rasterized phantoms.Analytical polyhedral phantoms are easy to construct and can accurately simulate shapes of biomedical interest. Magn Reson Med 76:663–678, 2016. © 2015 Wiley Periodicals, Inc.