According to Wikipedia, "the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups." However, it doesn't appear that this has anything to do with the symmetries of the circle, which surprised me, because that seems like the most natural generalization of the finite-order dihedral groups, which are sets of symmetries of regular polygons.
Put another way, since regular polygons "approach" becoming a circle as the number of vertices, $n\to\infty$, it seems like $D_\infty$ should be the symmetries of the circle.
So what kinds of symmetries does $D_\infty$ represent, and why was this group chosen for extension of the finite dihedral groups over the symmetries of the circle? (Or are they related in some way I'm just not grasping?)