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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.)
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