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Efficient Folding Algorithms for Convex Polyhedra

Abstract

We investigate a folding problem that inquires whether a polygon P can be folded, without overlap or gaps, onto a polyhedron Q for given P and Q. An efficient algorithm for this problem when Q is a box was recently developed. We extend this idea to a class of convex polyhedra, which includes the five regular polyhedra, known as Platonic solids. Our algorithms use a common technique, which we call stamping. When we apply this technique, we use two special vertices shared by both P and Q (that is, there exist two vertices of P that are also vertices of Q). All convex polyhedra and their developments have such vertices, except a special class of tetrahedra, the tetramonohedra. We develop two algorithms for the problem as follows. For a given Q, when Q is not a tetramonohedron, we use the first algorithm which solves the folding problem for a certain class of convex polyhedra. On the other hand, if Q is a tetramonohedron, we use the second algorithm to handle this special case. Combining these algorithms, we can conclude that the folding problem can be solved in pseudo-polynomial time when Q is a polyhedron in a certain class of convex polyhedra that includes regular polyhedra.

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Notes

  1. In some studies, a tetramonohedron is called an isotetrahedron or an isosceles tetrahedron.

  2. The authors thank an anonymous referee of [8], who mentioned this point.

  3. In this paper, we omit \(({\text {mod}}n)\) in \(\ell _i=(p_i,p_{i+1\,({\text {mod}}n)})\) for simplicity.

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Correspondence to Ryuhei Uehara.

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Part of this study was presented at CCCG 2020. Portions of this research was supported by JSPS KAKENHI Grant Numbers 17K00017, 18H04091, 20H05964, and 21K11757.

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Kamata, T., Kadoguchi, A., Horiyama, T. et al. Efficient Folding Algorithms for Convex Polyhedra. Discrete Comput Geom (2022). https://doi.org/10.1007/s00454-022-00415-7

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  • DOI: https://doi.org/10.1007/s00454-022-00415-7

Keywords

  • Computational origami
  • Folding problem
  • Pseudo-polynomial time algorithm
  • Regular polyhedron (Platonic
  • Stamping

Mathematics Subject Classification

  • 68W99
  • 52B55
  • 52C99