On the Mathematics of Flat Origamis

Abstract

Origami is the art of folding pieces of paper into works of sculpture without the aid of scissors or glue. Modern advancements in the complexity of origami (e.g., the work of Montroll and Maekawa) reveal a rich geometric structure governing the possibilities of paperfolding. In this paper we initiate a mathematical study of this "origami geometry" and explore the possibilities of a graph theoretic model. In particular, we study the properties of origami models which fold flat (i.e., can be pressed in a book without crumpling). Necessary and sufficient conditions are given for an origami model to locally fold flat, and the problems encountered in trying to extend these results globally are discussed. 1 Introduction Origami is the art of folding paper, without the aid of scissors or glue, into pieces of sculpture. Even when we restrict ourselves to folding a single square of paper, the possible results offer rich variety and complexity (see [3], [8], [10], or [11]). Things explode furth...