Skip to main content Accessibility help
×
Hostname: page-component-76c49bb84f-h2dfl Total loading time: 0 Render date: 2025-07-10T07:22:05.556Z Has data issue: false hasContentIssue false

A Survey of Folding and Unfolding in Computational Geometry

Published online by Cambridge University Press:  27 June 2025

Jacob E. Goodman
Affiliation:
City College, City University of New York
Janos Pach
Affiliation:
City College, City University of New York and New York University
Emo Welzl
Affiliation:
Eidgenössische Technische Hochschule Zürich
Get access

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.

Information

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

Accessibility standard: Unknown

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×