Journal of Geometry

, Volume 107, Issue 1, pp 61–75 | Cite as

Continuous flattening of truncated tetrahedra

  • Jin-ichi Itoh
  • Chie NaraEmail author


Each Platonic polyhedron P can be folded using a continuous folding process into a face of P so that the resulting shape is flat and multilayered, while two of the faces are rigid during the motion. In previous works, explicit formulas of continuous functions for such motions were given and the same result as above was shown to hold for any tetrahedron. In this paper, we show that a truncated regular tetrahedron can be folded continuously via explicit continuous folding mappings into a flat (folded) state, such that two of the hexagonal faces are rigid. Furthermore, given any general tetrahedron P and any truncated tetrahedron Q of P, we show that if Q contains the largest inscribed sphere of P and satisfies some condition, then Q can be folded continuously into a flat folded state such that two of the hexagonal faces of Q are rigid during the motion.

Mathematics Subject Classification

Primary 52Bxx Secondary 52Cxx 


Tetrahedron continuous flattening truncated tetrahedron rhombus 


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© Springer Basel AG 2015

Authors and Affiliations

  1. 1.Faculty of EducationKumamoto UniversityKumamotoJapan
  2. 2.Organization for the Strategic Coordination of Research and Intellectual PropertiesMeiji UniversityNakanoJapan

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