In 1952, R.H. Bing described a wild involution of S^3, that is, a homeomorphism I:S^3--S^3 such that I^2=id and the fixed point 2-sphere of I is wildly imbedded in S^3. This talk will begin by describing the heart of Bing's construction, which involves a clever strategy of shrinking a decomposition of S^3. Recently, while Michael Freedman and I were investigating the analytic properties of Bing's involutions, we discovered a surprising new method of shrinking Bing's decomposition with properties that no previously known shrinking methods have and that defied our intuition for what is possible.