11481

An Unfoldable Polyhedron

To unfold a polyhedron, cut it along some edges and flatten it out to form a net. (Allowing cuts on faces as well as edges is called a general unfolding.)
This Demonstration shows a "spiked tetrahedron" of Bern et al. [1]. It can only be unfolded with overlaps since one quarter of it (a "hat") can be unfolded only with overlaps.

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In [2] a polyhedron that can be edge-unfolded only with overlaps, is called an unfoldable polyhedron. The proof of the unfoldability of the spiked tetrahedron can be found in [2].
References
[1] M. Bern, E. D. Demaine, D. Eppstein and E. Kuo, "Ununfoldable Polyhedra," in Proceedings of the 11th Canadian Conference on Computational Geometry (CCCG'99), Vancouver, 1999 pp. 13-16. www.cccg.ca/proceedings/1999/fp38.pdf.
[2] E. D. Demaine and J. O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, New York: Cambridge University Press, 2007, pp. 318–320.
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