Suppose we have a ladder of length $1$, and it's sliding down the $y-$axis. We know that the curve enveloped by it is an astroid:
However, what if we iterate this process? We call this astroid curve $t_1$ and consider an identical ladder sliding down $t_1$ in the same way(with two endpoints on $t_1$ instead of the $x$ and $y$ axis), and call the curve enveloped by that $t_2$. Then we slide a ladder down $t_2$, and so on. I speculate that this will eventually fill up the entire first quadrant.
Is there a way to describe the curve obtained on each iteration?