Thin sheets exhibit a daunting array of patterns. A key difficulty in their analysis is that while we have many examples, we have no classification of the possible "patterns."

I have explored an alternative viewpoint in a series of recent projects with Peter Bella, Hoai-Minh Nguyen, and others. Our goal is to identify the *scaling law* of the minimum elastic energy (with respect to the sheet thickness, and the other parameters of the problem). Success requires proving upper bounds and lower bounds that scale the same way. The upper bounds are usually easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. This approach has been used successfully in several examples of tension-induced wrinkling, involving

(a) the wrinkling of an annular sheet, loaded in tension on both boundaries (with P Bella) (b) the cascade of wrinkles seen at the edge of a confined floating sheet (with H-M Nguyen) (c) the cascade of folds seen in a hanging drape (with P Bella). It has also been used in an example of compression-induced wrinkling, involving (d) the herringbone pattern seen in a compressed thin film bonded to a compliant substrate (with H-M Nguyen).

My talk will discuss some of these examples.