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Shephard's Conjecture

Shephard's conjecture states that every convex polyhedron admits a self-unoverlapping unfolding (Shephard 1975). This question is still unsettled (Malkevitch), though most mathematicians believe that the answer is yes.

It is known that Shephard's conjecture is false for non-convex 3-dimensional polyhedra (Bern et al. 1999, Malkevitch).

See also

Net, Unfolding

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References

Bern, M.; Demaine, E. D.; Eppstein, D.; and Kuo, E. "Ununfoldable Polyhedra." Proc. 11th Canadian Conference on Computational Geometry, pp. 13-16, 1999. Preprint dated 3 Aug 1999 available from http://arxiv.org/abs/cs.CG/9908003.Malkevitch, J. "Nets: A Tool for Representing Polyhedra in Two Dimensions." http://www.ams.org/featurecolumn/archive/nets.html.Malkevitch, J. "Unfolding Polyhedra." http://www.york.cuny.edu/~malk/unfolding.html.Malkevitch, J. "Le géométrie et la paire de ciseaux." La Recherche. No. 346, Oct. 2001. http://www.larecherche.fr/special/web/web346.html.Shephard, G. C. "Convex Polytopes with Convex Nets." Math. Proc. Camb. Phil. Soc. 78, 389-403, 1975.

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Shephard's Conjecture

Cite this as:

Weisstein, Eric W. "Shephard's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShephardsConjecture.html

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