Folding polyominoes with holes into a cube☆
Introduction
Given a piece of paper in the shape of a polyomino, i.e., a polygon in the plane formed by unit squares on the square lattice that are connected edge-to-edge, does it have a folded state in the shape of a unit cube? The standard rules of origami apply [3]; in particular, we allow each unit-square face to be covered by multiple layers of paper. Examples of this decision problem are given by the three puzzles by Nikolai Beluhov [2] shown in Fig. 1. We encourage the reader to print out the puzzles and try folding them.
Prior work [4] studied this decision problem extensively, introducing and analyzing several different models of folding. Beluhov [2] implicitly defined a grid model with the puzzles in Fig. 1: Fold only along grid lines of the polyomino; allow only orthogonal fold angles1 ( and ); and forbid folding material strictly interior to the cube. In this model, the prior work [4] characterizes which tree-shaped polyominoes (whose unit squares are connected edge-to-edge to form a tree dual graph) lying within a strip can fold into a unit cube, and exhaustively characterizes which tree-shaped polyominoes of ≤14 unit squares fold into a unit cube.
Notably, however, the polyominoes in Fig. 1 are not tree-shaped, and their interior is not even simply connected: The first puzzle has a hole, the second puzzle has two holes, and the third puzzle has a degenerate (zero-area) hole or slit. Arguably, these holes are what makes the puzzles fun and challenging. Therefore, in this paper, we embark on characterizing which polyominoes with hole(s) fold into a unit cube in the grid model. Although we do not obtain a complete characterization, we give many interesting conditions under which a polyomino does or does not fold into a unit cube.
The problem is sensitive to the choice of model. The other main model that has been studied in past work is the more flexible half-grid model, which allows orthogonal and diagonal folds between half-integral points, as in Fig. 2. The prior work [4] shows that all polyominoes of at least ten unit squares can fold into a unit cube in the half-grid model, leaving only a constant number of cases to explore, which were tackled recently [5]. Therefore, we focus on the grid model, which matches the puzzles of Beluhov [2].
If we generalize the target shape from a unit cube to polycube(s), there are polyominoes that fold in the grid model into all polycubes of at most a given surface area [6]. If we further forbid overlapping unit squares (polyhedron unfolding/nets instead of origami), this fold-all-polycubes problem has been studied for small polycubes [7], and there is extensive work on finding polyominoes that fold into multiple (two or three or more) different boxes [8], [9], [10], [11], [12].
Our results
- 1.
We show that any hole that is not one of five basic shapes of holes (see Fig. 3) always guarantee that a polyomino containing the hole folds into a cube; see Theorem 1 in Section 3.1. Polyominoes with exactly one of the five basic holes only sometimes allow folding into a cube.
- 2.
We identify combinations of two (of the five basic) holes that allow the polyomino to fold into a cube; see Section 3.2.
- 3.
We show that certain of the five basic holes or their combinations do not allow folding into a cube, that is, we show that subclasses of polyominoes with only specific basic hole(s) cannot be folded into a unit cube; see Section 4.
- 4.
We present an algorithm that checks a necessary local condition for folding into a cube; see Section 4.3.
- 5.
Whether this condition also constitutes a sufficient condition remains an open question; see Section 5.
- 6.
We conjecture that a slit of size 1 (see Fig. 3, second from left) never affects whether a polyomino can fold into a cube; see Section 4.2. However, we show that a slit of size 1 can be the deciding factor for foldability for larger polycubes.
Section snippets
Notation
A polyomino is a connected polygon P in the plane formed by joining together unit squares on the square lattice. We refer to the vertices of the n unit squares forming P as the grid points of P. We view P as an open region (excluding its boundary) which includes the n open unit squares of the form as well as some of the shared unit-length edges (and grid points) among these n unit squares. Notably, we do not require P to include the common edge between every adjacent
Polyominoes that do fold
In this section, we present polyominoes that fold. We start with polyominoes that contain a hole guaranteeing foldability.
Polyominoes that do not fold
In this section, we identify basic holes and combinations of basic holes that do not allow the polyomino to fold. First, we present some results that show how the paper is constrained around an interior grid point v. In particular, we consider situations when the induced polyomino of the four unit squares incident to v is connected; for an example consider Fig. 9.
Lemma 6 Four unit squares incident to a polyomino grid point v for which the induced polyomino is connected, cannot cover more than
Conclusion and open problems
We showed that, if a polyomino P does contain a non-basic hole, then P folds into . Moreover, we showed that a unit-square hole, size-2 slits (straight or L), and a size-3 U-slit sometimes allow for foldability.
Based on the presented results, we created a font of 26 polyominoes with slits that look like each letter of the alphabet, and each fold into . See Fig. 21, and http://erikdemaine.org/fonts/cubefolding/ for a web app.
We conclude with a list of interesting open problems:
- •
Does a
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was performed in part at the 33rd Bellairs Winter Workshop on Computational Geometry. We thank all other participants for a fruitful atmosphere. H. Akitaya was supported by NSF CCF-1422311 & 1423615. Z. Masárová was partially funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.
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