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Unfoldings of doubly covered polyhedra and applications to space-fillers

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Abstract

We study unfoldings (developments) of doubly covered polyhedra, which are space-fillers in the case of cuboids and some others. All five types of parallelohedra are examples of unfoldings of doubly covered cuboids (Proposition 1). We give geometric properties of convex unfoldings of doubly covered cuboids and determine all convex unfoldings (Theorem 1). We prove that every unfolding of doubly covered cuboids has a space-filling (consisting of its congruent copies) generated by three specified translates and three specified rotations, and that all such space-fillers are derived from unfoldings of doubly covered cuboids (Theorem 2). Finally, we extend these results from cuboids to polyhedra which are fundamental regions of the Coxeter groups generated by reflections in the 3-space and which have no obtuse dihedral angles (Theorem 3).

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Authors and Affiliations

  1. Faculty of Education, Kumamoto University, Kumamoto, 860-8555, Japan

    Jin-ichi Itoh

  2. Liberal Arts Education Center, Aso Campus, Tokai University Aso, Kumamoto, 869-1404, Japan

    Chie Nara

Authors
  1. Jin-ichi Itoh
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  2. Chie Nara
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Corresponding author

Correspondence to Jin-ichi Itoh.

Additional information

Communicated by Imre Bárány

Supported by Grant-in-Aid for Scientific Research No. 23540098, JSPS.

Supported by Grant-in-Aid for Scientific Research No. 23540160, JSPS.

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Itoh, Ji., Nara, C. Unfoldings of doubly covered polyhedra and applications to space-fillers. Period Math Hung 63, 47–64 (2011). https://doi.org/10.1007/s10998-011-7047-y

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  • DOI: https://doi.org/10.1007/s10998-011-7047-y

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