A Survey of Folding and Unfolding in Computational Geometry

Summary

We survey results in a recent branch of computational geometry: folding and unfolding of linkages, paper, and polyhedra.

1. Introduction

Folding and unfolding problems have been implicit since Albrecht Dürer [1525], but have not been studied extensively in the mathematical literature until recently. Over the past few years, there has been a surge of interest in these problems in discrete and computational geometry. This paper gives a brief survey of most of the work in this area. Related, shorter surveys are [Connelly and Demaine 2004; Demaine 2001; Demaine and Demaine 2002; O'Rourke 2000]. We are currently preparing a monograph on the topic [Demaine and O'Rourke ≥2005].

In general, we are interested in how objects (such as linkages, pieces of paper, and polyhedra) can be moved or reconfigured (folded) subject to certain constraints depending on the type of object and the problem of interest. Typically the process of unfolding approaches a more basic shape, whereas folding complicates the shape. We define the configuration space as the set of all configurations or states of the object permitted by the folding constraints, with paths in the space corresponding to motions (foldings) of the object.

This survey is divided into three sections corresponding to the type of object being folded: linkages, paper, or polyhedra. Unavoidably, areas with which we are more familiar or for which there is a more extensive literature are covered in more detail. For example, more problems have been explored in linkage and paper folding than in polyhedron folding, and our corresponding sections reflect this imbalance. On the other hand, this survey cannot do justice to the wealth of research on protein folding, so only a partial survey appears in Section 2.5.