Abstract
For a given polygon P and a polyhedron Q, the folding problem asks if Q can be obtained from P by folding it. This simple problem is quite complicated, and there is no known efficient algorithm that solves this problem in general. In this paper, we focus on the case that Q is a box, and the size of Q is not given. That is, input of the box folding problem is a polygon P, and it asks if P can fold to boxes of certain sizes. We note that there exist an infinite number of polygons P that can fold into three boxes of different sizes. In this paper, we give a pseudo polynomial time algorithm that computes all possible ways of folding of P to boxes.
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Notes
- 1.
We do not give the formal definition of curvature. Intuitively, it indicates the quantity of paper around the point measured by its angle. See [6] for further details.
- 2.
For sake of simplicity, we do not define the labels of points in P on an edge shared by two rectangles of Q. We also do not define the label of a point p corresponding to the vertex \(v_i\) of Q.
- 3.
Some readers may consider the first phase is enough. However, we have not yet checked if some particles of polygons cause overlap on a face of Q. In other words, we have to check each face is made by particles of polygons by gluing without overlap or hole.
- 4.
The number of polyomino of area 30 is 2368347037571252.
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Acknowledgements
A part of this research is supported by JSPS KAKENHI Grant Number JP17H06287 and 18H04091.
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Mizunashi, K., Horiyama, T., Uehara, R. (2019). Efficient Algorithm for Box Folding. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_22
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