Nematic cells with surface anchoring fields exhibiting quasicrystalline symmetry can be fabricated using linear photopolymerizable polymers and suitable optics. Using the Lebwohl-Lasher model we carry out Monte Carlo simulations of a nematic cell where the top and bottom surfaces have an anchoring field with the symmetry of a periodic approximant to a Penrose lattice. We show that the anchoring field has point topological defects, nearly all with topological charge +/-1 . Our simulations, which assume infinitely strong anchoring of the nematic to the anchoring field, show that half-integer disclination lines emerge from the point defects and either traverse the thickness of the cell for small cell thicknesses, hug the patterned surfaces for large thicknesses, or combine these behaviors for intermediate thicknesses. We show that estimates of the values of the thicknesses separating these three behaviors can be obtained using the properties of Penrose tilings.