Efficient Folding Algorithms for Regular Polyhedra

TL;DR

This paper presents new efficient algorithms for folding polygons into regular polyhedra, extending previous work on boxes, with different complexities based on geometric properties, enabling pseudo-polynomial time solutions.

Contribution

The paper introduces four novel algorithms for folding polygons into various regular polyhedra, improving efficiency and extending prior folding algorithms to Platonic solids.

Findings

Algorithms for tetrahedron and cube are highly efficient.

Folding problem solvable in pseudo-polynomial time for regular polyhedra.

Extension to general deltahedra broadens applicability.

Abstract

We investigate the folding problem that asks if a polygon P can be folded to a polyhedron Q for given P and Q. Recently, an efficient algorithm for this problem has been developed when Q is a box. We extend this idea to regular polyhedra, also known as Platonic solids. The basic idea of our algorithms is common, which is called stamping. However, the computational complexities of them are different depending on their geometric properties. We developed four algorithms for the problem as follows. (1) An algorithm for a regular tetrahedron, which can be extended to a tetramonohedron. (2) An algorithm for a regular hexahedron (or a cube), which is much efficient than the previously known one. (3) An algorithm for a general deltahedron, which contains the cases that Q is a regular octahedron or a regular icosahedron. (4) An algorithm for a regular dodecahedron. Combining these algorithms, we can conclude that the folding problem can be solved pseudo-polynomial time when Q is a regular polyhedron and other related solid.