Covering Folded Shapes

TL;DR

This paper investigates whether folding a shape can require it to be scaled up to cover its folded version, analyzing various shapes and fold types with bounds and implications for computational origami.

Contribution

It introduces bounds on scale factors needed for covering shapes after folding, extending understanding of folding and covering problems in computational origami.

Findings

Bounds established for scale factors of convex shapes after folding

Analysis of single and arbitrary fold impacts on shape coverage

Connections made to universal cover and force closure problems

Abstract

Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects.