Unfoldings of an envelope

An envelope is equivalent to a rectangle dihedron or a doubly-covered rectangle. It is cut along a tree graph that spans the four corners of the envelope to get a planar region. We show in Theorem 1 that every region satisfies the Conway criterion and so copies of the region tile the plane using onl...

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Bibliographic Details
Published in:European journal of combinatorics 2019-08, Vol.80, p.3-16
Main Authors: Akiyama, Jin, Matsunaga, Kiyoko
Format: Article
Language:English
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Summary:An envelope is equivalent to a rectangle dihedron or a doubly-covered rectangle. It is cut along a tree graph that spans the four corners of the envelope to get a planar region. We show in Theorem 1 that every region satisfies the Conway criterion and so copies of the region tile the plane using only translations and 180° rotations. Let P1 and P2 be two regions obtained by unfolding the same envelope along two non-crossing trees, respectively. Then we show in Theorem 2 that P1 is equi-rotational into P2, which means that P1 can be dissected into pieces that are hinged at corners, so that the pieces can be rigidly transformed into P2 by monotonous rotations at the hinges. In Theorems 3 and 4, we give the sufficient conditions for Conway tiles to be foldable into envelopes.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2018.02.023