# zbMATH — the first resource for mathematics

On reversibility among parallelohedra. (English) Zbl 1374.52018
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, 14-28 (2012).
Summary: Given two convex polyhedra $$\alpha$$ and $$\beta$$, we say that $$\alpha$$ and $$\beta$$ are a reversible pair if $$\alpha$$ has a dissection into a finite number of pieces which can be rearranged to form $$\beta$$ in such a way that no face of the dissection of $$\alpha$$ includes any part of an edge of $$\alpha$$, no face of the dissection of $$\beta$$ includes any part of an edge of $$\beta$$, the pieces are hinged on some of their edges so that the pieces of the dissection are connected as in a tree-structure, all of the exterior surface of $$\alpha$$ is in the interior of $$\beta$$, and all of the exterior surface of $$\beta$$ comes from the interior of $$\alpha$$. Let $$\mathfrak{P}_{i}$$ denote one of the five families of parallelohedra (see Section 2 for the corresponding definitions). In this paper, it is shown that given an arbitrary canonical parallelohedron $$P$$, there exists a canonical parallelohedron $$Q \in \mathfrak{P}_{i}$$ such that the pair $$P$$ and $$Q$$ is reversible for each $$\mathfrak{P}_{i}$$.
For the entire collection see [Zbl 1253.68016].
##### MSC:
 52B45 Dissections and valuations (Hilbert’s third problem, etc.)
Full Text:
##### References:
 [1] Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups, 3rd edn. A Series of Comprehensive Studies in Mathematics, vol. 290. Springer (1999) · Zbl 0915.52003 · doi:10.1007/978-1-4757-6568-7 [2] Boltyanskii, V.G.: Equivalent and Equidecomposable Figures. D. C. Health and Co. (1963), Translated and adapted from the first Russian edition (1956) by Henn, A.K., Watts, C.E. [3] Boltyanskii, V.G.: Hilbert’s third problem. V. H. Winton & Sons (1978), Translated by Silverman, R.A. · Zbl 0388.51001 [4] Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge University Press, New York (1997) · Zbl 0939.52008 · doi:10.1017/CBO9780511574917 [5] Frederickson, G.N.: Hinged Dissections: Swinging and Twisting. Cambridge University Press, New York (2002) · Zbl 1130.00003 [6] Frederickson, G.N.: Piano-Hinged Dissections: Time to Fold. Cambridge University Press, New York (2006) · Zbl 1126.52014 [7] Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, New York (2007) · Zbl 1135.52009 · doi:10.1017/CBO9780511735172 [8] Akiyama, J., Nakamura, G.: Dudeney Dissection of Polygons. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 14–29. Springer, Heidelberg (2000) · Zbl 0965.52003 · doi:10.1007/978-3-540-46515-7_2 [9] Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of triangles and convex quadrilaterals – all hinge points interior to the sides of the polygons. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry. Algorithms and Combinatorics, vol. 25, pp. 43–63 (2003) · Zbl 1077.52501 · doi:10.1007/978-3-642-55566-4_3 [10] Fedorov, E.S.: An introduction to the theory of figures. In: Notices of the Imperial Mineralogical Society (St. Petersburg) Ser. 2, vol. 21, pp. 1–279 (1885); Republished with comments by Akad. Nauk. SSSR, Moscow (1953) (in Russian) [11] Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics (2005) · Zbl 1133.52301 [12] Dolbilin, N., Itoh, J., Nara, C.: Geometric realization on affine equivalent 3-parallelohedra (to be published) · Zbl 1349.52013 [13] Akiyama, J., Seong, H.: On the reversibilities among quasi-parallelogons (to be published)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.