  For a fixed irrational number $\alpha$ and $n\in \mathbb{N}$, we look at the
shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where
$f(i) = \alpha i \mod 1$. Our primary result is that the boundary of the
Schensted shape is approximated by a piecewise linear function with at most two
slopes. This piecewise linear function is explicitly described in terms of the
continued fraction expansion for $\alpha$. Our results generalize those of Boyd
and Steele, who studied longest monotone subsequences. Our proofs are based on
a careful analysis of monotone sets in two-dimensional lattices.
