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

2. The authors thank an anonymous referee of , 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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## Author information

Authors

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Correspondence to Ryuhei Uehara.