  Schubert varieties in finite dimensional flag manifolds G/P are a
well-studied family of projective varieties indexed by elements of the
corresponding Weyl group W. In particular, there are many tests for smoothness
and rational smoothness of these varieties. One key result due to
Lakshmibai-Sandhya is that in type A the smooth Schubert varieties are
precisely those that are indexed by permutations that avoid the patterns 4231
and 3412. Recently, there has been a flurry of research related to the infinite
dimensional analogs of flag manifolds corresponding with G being a Kac-Moody
group and W being an affine Weyl group or parabolic quotient. In this paper we
study the case when W is the affine Weyl group of type A or the affine
permutations. We develop the notion of pattern avoidance for affine
permutations. Our main result is a characterization of the rationally smooth
Schubert varieties corresponding to affine permutations in terms of the
patterns 4231 and 3412 and the twisted spiral permutations.
