Mechanism of folding a strip into isotetrahedra or rectangle dihedra

Abstract

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.