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Continuous flattening of convex polyhedra. (English) Zbl 1349.51011
Márquez, Alberto (ed.) et al., Computational geometry. XIV Spanish meeting on computational geometry, EGC 2011, dedicated to Ferran Hurtado on the occasion of his 60th birthday, Alcalá de Henares, Spain, June 27–30, 2011. Revised selected papers. Berlin: Springer (ISBN 978-3-642-34190-8/pbk). Lecture Notes in Computer Science 7579, 85-97 (2012).
Summary: A flat folding of a polyhedron is a folding by creases into a multilayered planar shape. It is an open problem of E. D. Demaine et al. [Geometric folding algorithms. Linkages, origami, polyhedra. Cambridge: Cambridge University Press (2007; Zbl 1135.52009)] that every flat folded state of a polyhedron can be reached by a continuous folding process. Here we prove that every convex polyhedron possesses infinitely many continuous flat folding processes. Moreover, we give a sufficient condition under which every flat folded state of a convex polyhedron can be reached by a continuous folding process.
For the entire collection see [Zbl 1253.68016].

MSC:
51M15 Geometric constructions in real or complex geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
52B10 Three-dimensional polytopes
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[1] Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.A.: Star unfolding of a polytope with applications. SIAM J. Comput. 26, 1689–1713 (1997) · Zbl 0891.68117 · doi:10.1137/S0097539793253371
[2] Alexandrov, A.D.: Convex Polyhedra. Monographs in Mathematics. Springer, Berlin (2005); Translation of the 1950 Russian edition by Dairbekov, N.S., Kutateladze, S.S., Sossinsky, A.B.
[3] Chen, J., Han, Y.: Shortest paths on a polyhedron. In: Proc. 6th Ann. ACM Sympos. Comput. Geom., pp. 360–369 (1990) · doi:10.1145/98524.98601
[4] Demaine, E.D., Demaine, M.D., Lubiw, A.: Flattening polyhedra (2001) (unpublished manuscript)
[5] Demaine, E.D., O’Rourke, J.: Geometric folding algorithms, Lincages, Origami, Polyhedra. Cambridge University Press (2007) · doi:10.1017/CBO9780511735172
[6] Itoh, J.-I., Nara, C.: Continuous Flattening of Platonic Polyhedra. In: Akiyama, J., Bo, J., Kano, M., Tan, X. (eds.) CGGA 2010. LNCS, vol. 7033, pp. 108–121. Springer, Heidelberg (2011) · Zbl 1349.51010 · doi:10.1007/978-3-642-24983-9_11
[7] Kaneva, B., O’Rourke, J.: An implementation of Chen and Han’s segment algorithm. In: Proc. 12th Canadian Conf. Comput. Geom., pp. 139–146 (2000)
[8] Pak, I.: Inflating polyhedral surfaces (2006), http://www.math.ucla.edu/ pak/papers/pillow4.pdf
[9] Sabitov, I.: The volume of polyhedron as a function of its metric. Fundam. Prikl. Mat. 2(4), 1235–1246 (1996) · Zbl 0904.52002
[10] Sabitov, I.: The volume as a metric invariant of polyhedra. Discrete Comput. Geom. 20, 405–425 (1998) · Zbl 0922.52006 · doi:10.1007/PL00009393
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