Durer's Conjecture

Importance: High ✭✭✭
Subject: Geometry
» Polytopes
Keywords: folding
Recomm. for undergrads: yes
Posted by: dmoskovich
on: June 4th, 2012
Conjecture   Every convex polytope has a non-overlapping edge unfolding.

In 1525, Albrecht Dürer represented polytopes by cutting them open along edges and then flattening the surface onto the plane, without overlaps and without distorting the individual faces. Self-intersections are allowed during the unfolding process, but the final flattened surface must be free of overlaps. Whether a non-overlapping edge unfolding, as defined above, is possible for any convex polytopes was formulated by Shephard as a conjecture in 1975.


[D] A. Dürer, Unterweysung der Messung mit dem Zyrkel und Rychtscheyd. Nürnberg (1525). English translation with commentary by Walter L. Strauss The Painter's Manual, New York (1977).

[O] J. O'Rourke, How to fold it, Cambridge University Press, 2011, Book website

[P] K. Polthier Imagining maths- unfolding polyhedra

*[S] G.C. Shephard, Convex Polytopes with Convex Nets. Math. Proc. Camb. Phil. Soc., 78:389-403 (1975).

* indicates original appearance(s) of problem.