Abstract
In Chap. 3, we consider box, which is one of the most natural shapes in our daily life. From the viewpoint of mathematics, we have some other natural shapes. For example, we have five regular polyhedra which have been investigated since Archimedean’s period. However, surprisingly, they are not yet well investigated from the viewpoint of the unfolding. For example, it is not known that two regular polygons have a common net. That is, we do not know whether there is a polygon that can fold to two or more regular polyhedra. We give positive and negative results related to this interesting open problem.
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Notes
- 1.
Precisely, the “continued fraction expansion” used here is slightly different from the standard continued fraction representation. In the standard continued fraction expansion, \(a_i \ge 1\) for each term \(a_i\), all signs are “\(+\)”. On the other hand, our continued fraction expansion here also uses the sign of “−”, so that \(a_i>1\) holds when \(i>1\).
Reference
E.D. Demaine, J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Cambridge University Press, Cambridge, 2007)
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Uehara, R. (2020). Common Nets of (Regular) Polyhedra. In: Introduction to Computational Origami. Springer, Singapore. https://doi.org/10.1007/978-981-15-4470-5_4
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DOI: https://doi.org/10.1007/978-981-15-4470-5_4
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