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Continuous Flattening of the 2-Dimensional Skeleton of the Square Faces in a Hypercube

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Abstract

The surface of a 3-dimensional cube can be continuously flattened onto any of its faces, by moving creases to change the shapes of some faces successively, following Sabitov’s volume preserving theorem. Let \(C_n\) be an n-dimensional cube with \(n \ge 4\), and S be the set of its 2-dimensional faces, i.e., the 2-dimensional skeleton of the square faces in \(C_n\). We show that S can be continuously flattened onto any face F of S, such that the faces of S that are parallel to F, do not have any crease, that is, they are rigid during the motion.

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Authors and Affiliations

  1. Facalty of Education, Sugiyama Jyogakuen University, Nagoya, 464-8662, Japan

    Jin-ichi Itoh

  2. Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Nakano Campus, Nokano, Tokyo, 164-8525, Japan

    Chie Nara

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  1. Jin-ichi Itoh
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  2. Chie Nara
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Correspondence to Chie Nara.

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Jin-ichi Itoh and Chie Nara are supported by Grant-in Aid for Scientific Research (B) Japan Society for the Promotion of Science (15KT0020) and Research(C) (16K05258), respectively.

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Itoh, Ji., Nara, C. Continuous Flattening of the 2-Dimensional Skeleton of the Square Faces in a Hypercube. Graphs and Combinatorics 36, 331–338 (2020). https://doi.org/10.1007/s00373-019-02100-8

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  • DOI: https://doi.org/10.1007/s00373-019-02100-8

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