Efficient Folding Algorithms for Convex Polyhedra

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 f...

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Bibliographic Details
Published in:Discrete & computational geometry Vol. 70; no. 4; pp. 1499 - 1522
Main Authors: Kamata, Tonan, Kadoguchi, Akira, Horiyama, Takashi, Uehara, Ryuhei
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2023
Springer Nature B.V
Subjects:
Algorithms
Apexes
Combinatorics
Computational Mathematics and Numerical Analysis
Folding
Mathematics
Mathematics and Statistics
Polyhedra
Polynomials
Tetrahedra
Regular polyhedron (Platonic
Stamping
52C99
68W99
Pseudo-polynomial time algorithm
52B55
Computational origami
Folding problem
ISSN:0179-5376, 1432-0444
Online Access:Get full text
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Summary: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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ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-022-00415-7