Cantarella, Sullivan and the speaker have shown that, in the class of
unit thickness curves in R^3, one can minimize the length functional
in each isotopy class. In fact, a somewhat stronger result follows
from their arguments: length is a proper function on this class
(modulo translations of R^3). This means, for example, that one can
also maximize length among curves with length bounded above, such as
curves which lie in a box of finite volume, and in particular, any
such box has a maximally dense packing of unit thickness curves. In
this talk we will explore what is known and speculate on what is
unknown about the geometry of (local) maxima or critical
configurations for this "rope in a box" problem.