# A Comparison of Different Folding Models in Variations of the Map Folding Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Terminology

## 3. Outline

## 4. Simple Folding Model in 1D VOP

_{A}to x

_{G}, which are all integers. Then, we have x

_{B}− x

_{A}> x

_{C}− x

_{B}< x

_{D}− x

_{C}for the crimp and n − x

_{G}≤ x

_{G}− x

_{F}for the end-fold. The two states are respectively described by the following formulas, where o(x) indicates the squares folded to the coordinate x, s

_{i}(0 ≤ i ≤ n) indicates the squares whose lower-left vertex is located at point i before any fold, and tuples indicate the order of the squares from bottom to top.

**Step 1**. Compute the MV assignment of the map. Then, repeat the following three steps until either the map is shrunk to size 1 × 1 or the input order is determined to be unreachable. In the former case, the complete simple folding sequence could also be obtained.

**Step 2.**Find out the neighbor segments which are adjacent in the overlapping order. Decide the crease between them to be the first crease to fold. If no such neighbor segment exists, the input overlapping order is unreachable.

**Step 3.**According to the creases decided in Step 2, decide the crimps and end-folds on the 1D map by referring to Formulas (1) and (2). If no feasible crimp or end-fold exists, the input order is unreachable.

**Step 4**. Reduce the map to a new (smaller) map by applying the folding operations. Go back to Step 2 until the map is reduced to size 1 × 1.

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

_{2}and then unfold p

_{1}as the supposed unfolding of the original 1 × n map. (c) indicates the case that the k-th square and the (k + 1)-th square are not adjacent and the end of the (k + 1)-th square is invisible from outside. This time, first unfold p

_{3}, then unfold the outside (in this figure, the bottom) layer of p

_{2}as end-fold. After these unfolds, p

_{2}and p

_{3}becomes a whole part below the (k + 1)-th square. Unfold this whole part as a single layer, then unfold the (k + 1)-th square, and finally, unfold the remaining unfolded part of p

_{2}and p

_{1}according to the unfolding process of the original 1 × n map. In all the illustrations, p

_{1}, p

_{2}, and p

_{3}can be arbitrarily complicated; the other end of the map can also be extended to comprise all the possible cases. However, the unfolding process always follows the above manner. Correspondingly, we proved that for any 1 × (k + 1) map, its reachable folded states of the general folding are the same as its reachable folded states of the simple folding. Theorem 1 is proven. □

## 5. Simple Folding Model in Total VOP

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

_{1}to l

_{5}in order. Two input overlapping orders (l

_{3}, l

_{4}, l

_{5}, l

_{1}, l

_{2}) and (l

_{3}, l

_{4}, l

_{1}, l

_{5}, l

_{2}) correspond to the two states (a) and (b) in Figure 3, respectively. From the discussion before, their MV assignments are unique. For (a), {l

_{3}, l

_{4}}, {l

_{4}, l

_{5}} and {l

_{1}, l

_{2}} are the neighbor pairs adjacent in the overlapping order. Therefore, the corresponding creases c

_{1}, c

_{3}, and c

_{4}are firstly folded. The map is then reduced to the state illustrated in Figure 5a. On the opposite, for the states in Figure 3b, by the fact that l

_{3}, l

_{4}forms a neighbor pair adjacent in the overlapping order while l

_{5}and l

_{4}are not adjacent, c

_{3}is firstly folded while c

_{4}is not. Naturally, l

_{5}and l

_{2}should touch each other. However, the adjacent square pairs represent a state, as shown in Figure 5b, which is not valid.

## 6. Performances of Three Folding Models in Boundary VOP

#### 6.1. General Folding Model in Boundary VOP

_{i}

_{,j}(0 ≤ i < m, 0 ≤ j < n) refers to the square whose lower-left vertex is located at (i, j) before any fold. We further assume that the upfront side of s

_{0,0}is fixed. In the final folded state, on the bottom row, the creases between s

_{2i−1,0}and s

_{2i,0}would align on the y-axis, the creases between s

_{2i,0}and s

_{2i+1,0}would align on the line x = 1. On the top row, when m is odd (even), the creases between s

_{2i−1, m−1}and s

_{2i, m−1}would align on the y-axis (x = 1), the creases between s

_{2i, m−1}and s

_{2i+1, m−1}would align on the line x = 1 (y-axis).

_{i}

_{−1, p}, s

_{i}

_{, p}} and {s

_{j}

_{−1, q}, s

_{j}

_{, q}} whose creases align on the same line segment when completely folded, whether they penetrate each other or not, should be checked. Any penetrating state can be described as a permutation of (s

_{i}

_{−1, p}, s

_{j}

_{−1, q}, s

_{i}

_{, p}, s

_{j}

_{, q}). A similar property holds for the two vertical boundary sides. Checking the existence of such orders on four boundary sides totally costs O(m + n) time applying the stack structure introduced in [15].

_{c}

_{,d}< s

_{a}

_{,b}if s

_{c}

_{, d}is below s

_{a}

_{,b}in the final state. It is clear that < is a strict partial order because it satisfies:

- (
**irreflexivity**) there is no s_{a}_{,b}< s_{a}_{,b}; - (
**asymmetry**) if s_{a}_{,b}< s_{c}_{,d}, then there is no s_{c}_{,d}< s_{a}_{,b}; - (
**transitivity**) if s_{a}_{,b}< s_{c}_{,d}and s_{c}_{,d}< s_{e}_{,f}, then s_{a}_{,b}< s_{e}_{,f}.

_{c}

_{,d}to s

_{a}

_{,b}when s

_{c}

_{,d}< s

_{a}

_{,b}. Whether the input order follows a directed acyclic graph or not can be checked by a traverse. To conclude, we have Theorem 3. The proof is omitted because it follows the above analysis. The entire check takes O(m + n) time.

**Theorem**

**3.**

#### 6.2. Strict Inclusion Relations among Three Folding Models

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

## 7. Discussion

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A map is folded into m × n square layers. The overlapping order of the boundary squares in the folded state is given. (

**b**) An example of simple fold and unfold.

**Figure 2.**(

**a**) A 2 × 5 map which cannot be flat-folded; (

**b**) An example of the crimp and the end-fold.

**Figure 3.**A 1 × 20 map comprising five segments from l

_{1}to l

_{5}aligning in order. Two input overlapping orders correspond to the states (

**a**,

**b**). (

**a**) is reachable by the simple folding model while (

**b**) is not.

**Figure 4.**Three different cases of when gluing the (k + 1)-th square to a 1 × k map. The dashed segments illustrate the (k + 1)-th square. (

**a**) indicates the case where the (k + 1)-th square is adjacent to the k-th square; (

**b**) indicates the case that the k-th square and the (k + 1)-th square are not adjacent and the end of the (k + 1)-th square is visible from outside; (

**c**) indicates the case that the k-th square and the (k + 1)-th square are not adjacent and the end of the (k + 1)-th square is invisible from outside.

**Figure 6.**(

**a**) Three possible overlap states of a pair of horizontal neighbor squares and a pair of vertical neighbor squares. (

**b**) The notations and the coordinate system of the map.

**Figure 7.**(

**a**) A boundary overlapping order reachable for the general folding model while unreachable for the simple folding–unfolding model and the simple folding model. (

**b**) The left one indicates a boundary overlapping order corresponding to the smallest map with crease lines entirely assigned valleys that is only reachable by the general folding model; The right one indicates another boundary overlapping order corresponding to the map with crease lines entirely assigned valleys that is only reachable by the general folding model.

**Figure 8.**(

**a**) A boundary overlapping order reachable for the simple folding–unfolding model while unreachable for the simple folding model. (

**b**) The simplest instance reachable for the simple folding–unfolding model while unreachable for the simple folding model.

Problems | Simple Folding | Simple Folding–Unfolding | General Folding |
---|---|---|---|

1D VOP | A | = A = | A |

Total VOP | B_{1} | ⊊ B_{2} ⊊ | B_{3} |

Boundary VOP | C_{1} | ⊊ C_{2} ⊊ | C_{3} |

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Jia, Y.; Mitani, J.
A Comparison of Different Folding Models in Variations of the Map Folding Problem. *Appl. Sci.* **2021**, *11*, 11856.
https://doi.org/10.3390/app112411856

**AMA Style**

Jia Y, Mitani J.
A Comparison of Different Folding Models in Variations of the Map Folding Problem. *Applied Sciences*. 2021; 11(24):11856.
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**Chicago/Turabian Style**

Jia, Yiyang, and Jun Mitani.
2021. "A Comparison of Different Folding Models in Variations of the Map Folding Problem" *Applied Sciences* 11, no. 24: 11856.
https://doi.org/10.3390/app112411856