Nontechnical Abstract Thin materials can be bent, twisted, and wrinkled into a multitude of forms. The resulting shapes can be smooth like the undulating edge of a flower, or sharp like the creases and points in a crumpled ball of foil. The goal of this research is to understand when these features turn out to be smooth versus when they are sharp. This deceptively simple question is surprisingly difficult to answer, and it impacts a variety of technologies that rely on thin sheets. This project consists of a suite of model experiments where polymer films are forced into different geometries, forcing them to buckle. Because the project focuses on physical behaviors that should be scale-free, our discoveries should be applicable to material systems ranging from the very small to the very large, from graphene films to food packaging to deployable satellites. The research is coupled to a set of education and outreach projects that target a wide range of audiences. The project will train two PhD students who will work with undergraduates and high school students in the lab. The work will also engage high school teachers from Central New York, who will conduct 6-week summer research internships. Technical Abstract Despite the ubiquity of sheets in our everyday lives and their importance to a range of technologies, we nevertheless lack a general predictive framework for when a buckled sheet will have a smooth or sharp character. The overarching goal of the project is to me