Durer's Conjecture

Importance: High ✭✭✭
Subject: Geometry
» Polytopes
Keywords: folding
polytope
Recomm. for undergrads: yes
Posted by: dmoskovich
on: June 4th, 2012

\begin{conjecture} Every convex polytope has a non-overlapping edge unfolding. \end{conjecture}

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.

Bibliography

[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, \href[Book website]{http://howtofoldit.org/}

[P] K. Polthier \href[Imagining maths- unfolding polyhedra] {http://plus.maths.org/content/os/issue27/features/mathart/index/}

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


* indicates original appearance(s) of problem.