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Bounding the number of holes required for folding rectangular polyominoes into cubes

Florian Lehner, Benjamin Shirley

TL;DR

This work analyzes cube-foldability of rectangular polyominoes with holes under grid-edge folds, introducing the concept of hole cooperation and minimally cooperating hole sets. Using a formal folding framework with folding blueprints, facemappings, and layermappings, the authors derive structural bounds on hole interactions. They show that minimally cooperating sets for square, L-shaped, and U-shaped holes have size $2$, that $2$-tall slits exhibit similar bounds on large boards (with small-board exceptions), and that no finite bound exists when mixing hole types by constructing arbitrarily large minimally cooperating hole sets. These results illuminate how hole configurations govern foldability and have implications for algorithmic recognition and design of foldable polyomino patterns.

Abstract

We study the problem of whether rectangular polyominoes with holes are cube-foldable, that is, whether they can be folded into a cube, if creases are only allowed along grid lines. It is known that holes of sufficient size guarantee that this is the case. Smaller holes which by themselves do not make a rectangular polyomino cube-foldable can sometimes be combined to create cube-foldable polyominoes. We investigate minimal sets of holes which guarantee cube-foldability. We show that if all holes are of the same type, the these minimal sets have size at most 4, and if we allow different types of holes, then there is no upper bound on the size.

Bounding the number of holes required for folding rectangular polyominoes into cubes

TL;DR

This work analyzes cube-foldability of rectangular polyominoes with holes under grid-edge folds, introducing the concept of hole cooperation and minimally cooperating hole sets. Using a formal folding framework with folding blueprints, facemappings, and layermappings, the authors derive structural bounds on hole interactions. They show that minimally cooperating sets for square, L-shaped, and U-shaped holes have size , that -tall slits exhibit similar bounds on large boards (with small-board exceptions), and that no finite bound exists when mixing hole types by constructing arbitrarily large minimally cooperating hole sets. These results illuminate how hole configurations govern foldability and have implications for algorithmic recognition and design of foldable polyomino patterns.

Abstract

We study the problem of whether rectangular polyominoes with holes are cube-foldable, that is, whether they can be folded into a cube, if creases are only allowed along grid lines. It is known that holes of sufficient size guarantee that this is the case. Smaller holes which by themselves do not make a rectangular polyomino cube-foldable can sometimes be combined to create cube-foldable polyominoes. We investigate minimal sets of holes which guarantee cube-foldability. We show that if all holes are of the same type, the these minimal sets have size at most 4, and if we allow different types of holes, then there is no upper bound on the size.
Paper Structure (7 sections, 16 theorems, 15 figures)

This paper contains 7 sections, 16 theorems, 15 figures.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. The folding model
  4. Further Definitions and Basic Lemmas
  5. A bound for Square, L-shaped and U-shaped Holes
  6. A Bound for 2-Tall Slits
  7. No Bound for Unit Holes and 2-Tall Slits

Key Result

Lemma 2.2

Let $k,n\ge 2$ and let $P$ be a rectangular polyomino without holes. Then, in every folding of $P$ into $\mathcal{C}$, any pair of collinear creases are either both folded by $90^\circ$ or both folded by $180^\circ$. Moreover, either all horizontal or all vertical creases of $P$ are folded by $180^\

Figures (15)

  • Figure 1: The $4$ simple holes.
  • Figure 2: Standard Cube Layout
  • Figure 3: Facemapping of a non-trivially folded square hole
  • Figure 4: An L-shaped hole, which has overlapping closure with a square hole
  • Figure 5: An L-shaped hole, with a U-shaped hole that folds.
  • ...and 10 more figures

Theorems & Definitions (36)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 26 more