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December 2021 Mechanism of folding a strip into isotetrahedra or rectangle dihedra
Kiyoko Matsunaga
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SUT J. Math. 57(2): 109-131 (December 2021). DOI: 10.55937/sut/1641859460


Any net of an isotetrahedron I (a tetrahedron with all congruent four faces) and a rectangle dihedra RD satisfies the Conway criterion. Does the converse proposition hold? If so, are there practical algorithms for it? The difficult part of the proof of it comes down to the following problem [5], [6]: Find the practical algorithm for folding a parallelogramic strip into I or RD. The process of how to cover a thin rectangular board by a long tape without making gaps or overlaps has been known as a folklore among natives in various places globally [1]. By generalizing the known folklore foldings, this paper gives the practical algorithms to obtain all isotetrahedra and rectangle dihedra into which a strip can be folded. As a result of it, it is also proved that there is no other way to fold a given strip into rectangle dihedra other than the known two types of folklore foldings.


The author would like to thank Jin Akiyama, David Eppstein, Stefan Langer- man and Joseph O'Rourke for their invaluable comments.


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Kiyoko Matsunaga. "Mechanism of folding a strip into isotetrahedra or rectangle dihedra." SUT J. Math. 57 (2) 109 - 131, December 2021.


Received: 28 December 2020; Published: December 2021
First available in Project Euclid: 21 April 2022

Digital Object Identifier: 10.55937/sut/1641859460

Primary: 52B45 , 52C20

Keywords: Conway criterion , Conway tile , foldability , isotetrahedron , rectangle dihedron , tile-maker

Rights: Copyright © 2021 Tokyo University of Science


Vol.57 • No. 2 • December 2021
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