Mathematicians have long been asking the question: Can a given convex polyhedron can be unfolded into a polygon and then refolded into any other convex polyhedron? One facet of this question investigates the space of polyhedra that can be realized from folding a given polygon. While convex polygons are relatively well understood, there are still many open questions regarding the foldings of non-convex polygons. We analyze these folded realizations and their volumes derived from the polygonal family of `L-shapes,' parallelograms with another parallelogram removed from a corner. We investigate questions of maximal volume, diagonal flipping, and topological connectedness and discuss the family of polyhedra that share a L-shape polygonal net.

Author Bio

Emily Dinan just finished her first year at the University of Washington. She is interested in studying probability and geometric analysis.

Alice Nadeau just finished her second year of graduate school at the University of Minnesota. She is studying dynamical systems as they relate to the climate.

Issac Odegard is starting his first year of graduate school in the fall at the University of North Dakota and will be student teaching. He is interested in studying algebra.