  Napoleon's theorem in elementary geometry describes how certain linear
operations on plane polygons of arbitrary shape always produce regular
polygons. More generally, certain triangulations of a polygon that tiles R^2
admit deformations which keep fixed the symmetry group of the tiling. This
gives rise to isolation phenomena in cusped hyperbolic 3-manifolds, where
hyperbolic Dehn surgeries on some collection of cusps leaves the geometric
structure at some other collection of cusps unchanged.
