Minimal surfaces are considered as alternatives to tilings in the search for simple mathematical objects that have the properties of a quasicrystal. The construction of such objects, it seems, has not been addressed before and we propose a solution in the form of a limiting process. The existence of the desired limit has not been established but is investigated numerically for the case of a quasicrystalline minimal surface with icosahedral symmetry. We find that our approximate minimal surface exhibits jumps in topology analogous to the atomic phason jumps in physical quasicrystals.