  We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map
$\Delta_k$. The map $\Delta_1$ is the pentagram map and $\Delta_k$ is a
generalization. $\Delta_k$ does not preserve convexity, but we prove that
$\Delta_k$ preserves a subset $B_k$ of certain star-shaped polygons which we
call $k$-birds. The action of $\Delta_k$ on $B_k$ seems similar to the action
of $\Delta_1$ on the space of convex polygons. We show that some classic
geometric results about $\Delta_1$ generalize to this setting.
