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Overlapping Edge Unfoldings for Archimedean Solids and (Anti)prisms

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WALCOM: Algorithms and Computation (WALCOM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13973))

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Abstract

Herein, we discuss the existence of overlapping edge unfoldings for Archimedean solids and (anti)prisms. Horiyama and Shoji showed that there are no overlapping edge unfoldings for all platonic solids and five shapes of Archimedean solids. The remaining five Archimedean solids were also found to have edge unfoldings that overlap. In this study, we propose a method called rotational unfolding to find an overlapping edge unfolding of a polyhedron. We show that all the edge unfoldings of an icosidodecahedron, a rhombitruncated cuboctahedron, an n-gonal Archimedean prism, and an m-gonal Archimedean antiprism do not overlap when \(3\le n\le 23\) and \(3\le m\le 11\). Our algorithm finds three types of overlapping edge unfoldings for a snub cube, consisting of two vertices in contact. We show that an overlapping edge unfolding exists in an n-gonal Archimedean prism and an m-gonal Archimedean antiprism for \(n \ge 24\) and \(m \ge 12\). Our results prove the existence of overlapping edge unfoldings for Archimedean solids and Archimedean (anti)prisms.

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Acknowledgments

This work was supported in part by JSPS KAKENHI Grant Numbers JP18H04091, JP19K12098, and 21H05857.

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Authors and Affiliations

  1. School of Computer Science and Systems Engineering, Kyushu Institute of Technology, Fukuoka, Japan

    Takumi Shiota & Toshiki Saitoh

Authors
  1. Takumi Shiota
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  2. Toshiki Saitoh
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Correspondence to Takumi Shiota .

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Editors and Affiliations

  1. National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Chun-Cheng Lin

  2. National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Bertrand M. T. Lin

  3. University of Perugia, Perugia, Italy

    Giuseppe Liotta

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Shiota, T., Saitoh, T. (2023). Overlapping Edge Unfoldings for Archimedean Solids and (Anti)prisms. In: Lin, CC., Lin, B.M.T., Liotta, G. (eds) WALCOM: Algorithms and Computation. WALCOM 2023. Lecture Notes in Computer Science, vol 13973. Springer, Cham. https://doi.org/10.1007/978-3-031-27051-2_4

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  • DOI: https://doi.org/10.1007/978-3-031-27051-2_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-27050-5

  • Online ISBN: 978-3-031-27051-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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