Paper Folding: Extensions of Gallivan’s Formula

In 2002, Britney Gallivan derived a formula estimating the length of paper lost when it is folded repeatedly in half along the same side. This placed a theoretical upper bound on the number of times a piece of paper can be folded in half, depending on its thickness and length. In this paper, we generalize and refine Gallivan’s model. We first derive the corresponding lost-length formulas for alternate-side folding and apply them to both square and rectangular paper. We then develop improved models of folding that address limitations in Gallivan’s core assumption—that the fold behaves as a perfect semicircle with inner radius equal to the paper’s thickness—and derive new bounds accordingly. Finally, we introduce a constraint involving only the outermost layer, leading to a sharper folding limit. Potential applications and limitations of these theoretical results are discussed.