Folding Polyominoes into (Poly)Cubes

TL;DR

This paper explores various models of folding polyominoes into polycubes, characterizes foldability for certain shapes, and provides algorithms for folding into specific polycubes, including trees and tetrahedra.

Contribution

It introduces a comprehensive framework of folding models, characterizes foldability for polyominoes into a cube, and develops efficient algorithms for specific folding problems.

Findings

Characterization of polyominoes that fold into a unit cube.

Linear-time algorithm for folding tree-shaped polyominoes into small polycubes.

Extension of folding concepts to triangular shapes and tetrahedra.

Abstract

We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^\circ$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.