Edge-Unfolding Convex Polyhedra

Given an arbitrary convex polyhedron, determine whether it can always be cut along its edges and unfolded flat into a single, nonoverlapping, simple polygon.

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Abstract

The open problem asks whether every convex polyhedron can be cut along its edges and unfolded into a single simple polygon in the plane without overlap. The answer is conjectured to be affirmative, but no general result is known.

Background and progress

The problem was first stated by Shephard [Shephard, 1975] and has remained open for convex polyhedra [O'Rourke, 2000]. Convexity is essential to the unresolved question: nonconvex polyhedra with triangular faces are known that cannot be unfolded in the required way [Bern et al., 2003]. For convex polyhedra, the source reports a widespread expectation that the answer is yes. The original problem appears in the Open Problems Project at https://topp.openproblem.net/p9.

Reference papers

Ununfoldable polyhedra with convex facesMarshall Bern, Erik D. Demaine, David Eppstein, Eric Kuo, Andrea Mantler, Jack SnoeyinkComput. Geom. Theory Appl. · 2003
Folding and unfolding in computational geometryJoseph O'RourkeProc. 1998 Japan Conf. Discrete Comput. Geom. · 2000
Convex polytopes with convex netsGeoffrey C. ShephardMath. Proc. Camb. Phil. Soc. · 1975