Abstract
Given any polyhedron in R3, we can cut it open along its edges, flatten it out, and obtain a polygon in the plane R2. In this project, we explored the opposite process, an open question that was first posed about 70 years ago: given a polygon in R2, what is the folding procedure to reconstruct the polyhedron in R3? We focused on a special case, where we are given a polygon with n vertices and we try to find its companion shape (another polygon with n vertices) so that, glued together and folded, we obtain a polyhedron where all n cone angles are equal. The case where n=4 (i.e., the quadrilaterals) was explored in great detail by studying the shapes of all tetrahedra with equal cone angles and how parallelograms together with their companion shapes are folded into tetrahedra. We then applied this general theory to other R2 developments, in particular the case of a polygon with n vertices, as n goes to infinity, where the curvature distribution approximates harmonic measure on a shape in the plane. We explored an iterative process of constructing the “harmonic caps”. It was our conjecture that in the R2 plane, the iteration will produce rounder shapes and ultimately limit on a perfect circle.