Edge-Unfolding Convex Polyhedra

Importance: Medium ✭✭
Author(s): Shephard, G.C.
Subject: Geometry
Keywords: folding
Recomm. for undergrads: no
Posted by: Erik Demaine
on: June 18th, 2019
Conjecture   Every convex polyhedron has a (nonoverlapping) edge unfolding.

An edge unfolding of a convex polyhedron consists of cutting along a spanning tree of the polyhedron's edges, then unfolding the remaining edges to bring all faces into a plane, without faces overlapping each other (thus resulting in a simple planar polygon).

The first explicit posing of this problem seems to be Shephard's 1975 paper [S], though the idea of edge unfolding goes back to Albrecht Dürer in 1525 [D]. See [DO] (Part III) for a survey.


[DO] Erik D. Demaine, Joseph O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007.

[D] Albrecht Dürer, The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Dürer for the Use of all Lovers of Art with Appropriate Illustrations Arranged to be Printed in the Year MDXXV, Abaris Books, New York, 1977. English translation by Walter L. Strauss of “Unterweysung der Messung mit dem Zirkel un Richtscheyt in Linien Ebnen uhnd Gantzen Corporen”, 1525.

*[S] Geoffrey C. Shephard, Convex polytopes with convex nets, Math. Proc. Camb. Phil. Soc. 78 (1975), 389–403.

* indicates original appearance(s) of problem.