## Abstract

Rigid origami is concerned with the folding of rigid thin-walled structures. The materials from which the structures are made are not allowed to deform or bend, but can rotate freely about pre-arranged creases. One of the challenges in rigid origami is the flat folding of a shopping bag with a rectangular base if the bag is made of rigid materials. The problem is not only mathematically interesting but also has practical implications as many consumer goods are packaged in box-shaped cartons or cardboard boxes. In this paper, a new crease pattern has been proposed that allows a tall box-shaped bag with a rectangular base to be rigidly folded flat. Rigid folding conditions are established, and solutions that meet these conditions are found numerically. Simulations and experiments carried out demonstrate that the solution works. The new pattern represents the first practical solution for tall bags and can lead to direct applications in the packaging industry. Moreover, the folding analysis can be used to design an automated packaging process for folding box-shaped stiff cartons.

## 1. Introduction

Rigid origami is concerned with the folding of flat sheets or three-dimensional thin-walled structures about a set of pre-arranged creases. The materials are rigid and therefore are not allowed to bend or deform. Only rotations about the creases are permitted. One of the challenges in rigid origami is rigid and flat folding of a shopping bag with a rectangular base, should the bag be made of rigid materials. Conventional paper shopping bags are usually collapsed flat for storage and opened up before use. They commonly carry the crease pattern shown in figure 1*a*. It has been proved that the bags cannot be rigidly folded flat following such a pattern (Balkcom *et al*. 2004) unless they are made of relatively flexible materials such as paper.

Folding rigid shopping bags is not only mathematically interesting, but it also has practical implications. Many consumer products are packed in cartons made of sheet materials less flexible than paper. Typically these cartons are box-shaped, made with both top and bottom open to allow flat packing (Cannella & Dai 2006; Dai & Cannella 2008). Special robots have been developed to automate the packaging process (Dubey & Dai 2006). There have been no solutions that enable box-shaped tall cartons with their bottom base fixed to be practically folded flat.

Several strategies were proposed to address the problem in the past. It is commonly known that the short bag, whose height is no greater than half of its depth, is rigidly collapsible using the crease pattern shown in figure 1*b*. Based on it, Balkcom *et al*. (2004) suggested a folding method, that is, at first rolling the upper parts inward continuously until a tall bag becomes a short bag and then adopting the folding procedure for the short bag. This is practical only if the thickness of the bag is negligible, which is often untrue. They also conjectured another folding pattern, as shown in figure 1*c*, yet the proof of its effectiveness was absent. Details of their approach can be found in Balkcom (2004). More recently, two more rigid folding schemes were proposed (Balkcom *et al*. 2009), but they are only for specific bags, i.e. partially taped bags or cubical bags with equal width, depth and height. No practical solution has been found for rigid and flat folding of tall bags.

Here, we report a solution for folding a rigid tall bag. The solution is practical without the thickness issue associated with the rolling method and is applicable to bags of various dimension ratios. In the derivation, we adopt a simple vector analysis method to describe the folding of the bag despite more sophisticated tools for modelling rigid origami existing (Belcastro & Hull 2002; Streinu & Whiteley 2005; Wu & You 2010). This is because the model that we are dealing with is relatively simple and the vector-based method gives good physical intuition on what happens in reality.

The layout of the paper is as follows. First, we introduce the new crease pattern, followed by establishment of the conditions for its rigid foldability. Then we discuss the influences and selection of the pattern parameters, as well as restrictions and possible variations in the proposed pattern. Using the same mathematical framework, we have also demonstrated that the conjectured pattern given by Balkcom *et al*. (2004) actually does not work.

## 2. New crease pattern

A new crease pattern, shown in figure 1*d*, is obtained by adding several new creases to the traditional pattern. Selection of an angle *ϕ* is to be discussed later. Carrying the new pattern, the tall bag now consists of a lower short bag and the upper portion. Since the short bag can be rigidly folded, foldability of the tall bag depends on how the upper part folds along with the short bag. Hence, the focus here is to link the rigid folding of the upper portion of a tall bag with that of the lower short bag.

Based on the new pattern, the bag is now composed of four identical quarters, each containing one corner of the bag. Assuming that symmetry is maintained during folding, only one quarter, shown in figure 2, needs to be analysed for simplicity. Let *ρ* and *ω* represent the angles between panel ACIJ and panel CGHI against the base plane, respectively. There must be 0≤*ρ*≤90^{°}. Around valley crease CI, the dihedral angle between panels ACIJ and CGHI is *ρ*+*ω*, which must satisfy,
2.1

To analyse folding of the upper portion, let us first imagine panels CEF and CEG were removed. Panels CIHG and CFD then became two flaps that fold rigidly with the short bag. Rigidity of a folded configuration then depends on whether these two flaps can be reconnected by adding CEF and CEG back. To ensure their connection, vertices E, F and G must form a triangle throughout the entire folding process, thus there must be 2.2 and 2.3

In addition, due to symmetry, edge EF should never extrude beyond the plane of symmetry on which lines KD and DF lie. Mathematically, that is, 2.4

The above four conditions govern the rigid folding of a tall bag. They can be expressed in terms of the dihedral angles and bag dimensions. To do so, a Cartesian coordinate system, with the origin at vertex A, *x* and *y* axes along baselines AK and AJ, is set up, as shown in figure 2. Denote by *h*, *w* and *d* the height, width and depth of a tall bag, respectively. Conditions (2.2)–(2.4) can be written in terms of *ϕ*, *h*′, *d*, *ρ* and *ω*, where *h*′=*h*−*d*/2 is the height of the upper portion of the bag, see inequalities (A1)–(A3) in appendix A.

Conditions (2.1)–(2.4) restrict the attainable ranges of changing *ρ* and *ω* during folding. When a tall bag is open, *ρ*=90^{°} and *ω*=90^{°}, whereas *ρ*=0 and *ω*=90^{°} when it is flatly collapsed. Rigid folding therefore only becomes possible if there is at least one continuous path within the attainable ranges of *ρ* and *ω* linking those two states.

## 3. Examples

Consider a bag with *h*=1.5*d* (*h*′=*d*) and *w*=2*d*. Conditions (2.1)–(2.4) are plotted in the *ω* versus *ρ* diagram shown in figure 3 with *ϕ* ranging from 60^{°} to 85^{°}. For instance, the folding conditions for *ϕ*=75^{°} is given in figure 3*f*, where *P*_{1} (*ρ*=90^{°} and *ω*=90^{°}) corresponds to the state when the bag is open, and *P*_{2} (*ρ*=0 and *ω*=90^{°}) to the fully collapsed state. The shaded area is the attainable region where all of the four conditions are satisfied. It can be seen that when *ϕ*=75^{°} there is more than one continuous path within the shaded area available linking *P*_{1} to *P*_{2}, as shown in figure 3*f*, one of which is path *P*_{1}−*P*_{3}−*P*_{2}. Figure 4*a*–*f* shows an animation of the folding process along this particular path. When the bag reaches the state shown in figure 4*f*, the vertical part can be easily turned flat by rotation about the mid creases. A rigid bag was made to validate this solution. It was made of durable paper with thin steel sheet bonded to the panels between creases to create rigidity. The folding sequence of the bag is given in figure 5, which demonstrates that our solution works.

The selection of *ϕ* is important for rigid and flat folding of the bag. For the same bag, conditions (2.1)–(2.4) are plotted in figure 3*b* for *ϕ*=82^{°}. Note that now the attainable region has been separated into two, thus no continuous path between *P*_{1} to *P*_{2} exists, indicating that the bag will get stuck during folding. Computer simulation shown in figure 6 demonstrates that this is indeed the case.

The exact effect of *ϕ* on the foldability is given next.

## 4. Discussion

### (a) Pattern angle *ϕ*

The pattern angle *ϕ* is the only design parameter of the proposed new pattern. Its selection is vital to determine whether the new pattern leads to rigid and flat folding of the bag. Its influence can be shown by a series of plots of conditions (2.1)–(2.4) in figure 3 for the bag with *h*=1.5*d* and *w*=2*d*. It is obvious that increasing *ϕ* leads to a gradual shrinkage of the waist of the attainable region. When *ϕ*=81.72^{°}, the waist of the attainable region, shown in figure 3*d*, becomes a point at *ρ*=48.70^{°} and *ω*=99.75^{°}. The attainable region splits into two areas for *ϕ*>81.72^{°}. At *ϕ*=81.74^{°}, figure 3*c*, condition (2.4) splits into two curves. The same occurs to condition (2.2) when *ϕ* is increased further as shown in figure 3*a*,*b*. The maximum value for *ϕ* is 81.72^{°} in order to achieve rigid and flat folding of the tall bag.

For bags of other dimensions, the maximum *ϕ* is found to be related to *h*/*d*, see table 1. For relatively large *h*/*d*, it approaches 80^{°}; while for *h*/*d* close to 0.5, which is the short bag case, it comes to near 90^{°}.

On the other hand, there is also a minimum *ϕ* due to the restriction that two inclined creases on the front or back panels of the bag should not intersect each other, thus,
4.1

It has also been noticed that condition (2.4) has branches and these branches interfere with the attainable region when *ϕ* approaches its minimum given in (4.1). Figure 3*h* shows such an example for a bag with *h*=1.5*d*, *w*=2*d* and *ϕ*=60^{°}. Around *P*_{2} (*ρ*=0 and *ω*=90^{°}), a branch shown in dot grey line emerges in addition to that shown in solid grey line. So, *P*_{2} would be outside the attainable region if the additional branch is activated. Folding simulation indicates that activation of this branch requires the corner fold CE in figure 2 to be a valley during folding, i.e. it has to fold inwards, as shown by the two configurations in figure 7. This is a motion bifurcation. Thus, the branch of condition (2.4) has to be kept inactive for the bag to be fully flat foldable.

### (b) The minimum width *w*

For tall bags usually *w* is a given dimension. If *w* is too small, vertex F shown in figure 2 will collide with its counterpart from the opposite side during folding. There are two ways to avoid this problem. One is to impose a minimum width condition, that is, half the bag width must be no less than the largest inward distance of F,
4.2
which can also be written in terms of *h*, *d* and *ρ* as
4.3
The minimum width can be easily obtained from (4.3). For example, for a bag with *h*=1.5*d*, the minimum *w* is found to be 1.53*d*. Table 2 lists the minimum *w* for various given ratios of *h*/*d*.

The other possibility is to add additional creases around vertex F and its counterpart to avoid collision. Because it involves further complications in the crease patterns, we do not discuss it further here.

## 5. Conclusion and final remark

We have proposed a new crease pattern that allows a tall bag with a rectangular base to be rigidly flat-foldable. The rigid folding conditions of the pattern are established. Because of the highly non-linear nature of these conditions, solutions have been found numerically, leading to a set of valid pattern angles and width of the bag. Despite these restrictions, the folding scheme works for most of the tall bags. Simulations and experiments that have been carried out have shown that this is the case.

The new crease pattern is the first practical solution to the challenging problem in rigid origami and computational geometry. It can lead to direct applications in the packaging industry. The folding analysis could be used to design an automated packaging process for hard box-shaped cartons.

It needs to be pointed out that Balkcom *et al*. (2004) also conjectured a pattern, shown in figure 1*c*, which was thought to be rigidly foldable. As a final remark, we are to demonstrate that this pattern does not lead to rigid and flat folding.

First, we remove some secondary panels from the conjectured pattern shown in figure 1*c*, leading to a simplified base bag which should have at least a single degree of freedom during folding if it were foldable, see figure 8. The base bag must first be rigidly flat-foldable in order for the entire bag to be so. For the crease pattern on the base bag shown in figure 8*b*, we choose angle *σ* as the design parameter, which is the angle between the inclined crease DQ and horizontal crease CD.

Now analyse the rigid foldability of the base bag. During folding, relative positions of all creases and edges have to change, see figure 8*c*. Among them, particular attention is drawn on edges DQ and CM. The distance between them can be written as
5.1
The minus sign is added to let *d*_{1}>0 when the base bag is open.

To achieve rigid flat folding, *d*_{1} must remain positive (or negative) during the entire folding process. Should *d*_{1} switch sign during folding, DQ would have to penetrate into a panel ACMN opposite, an impossible proposition in reality. Hence, the rigid flat-folding cannot be done since the folding process terminates when *d*_{1} reaches 0.

Now set up the same Cartesian coordinate system as previously, shown in figure 8*c*. The same dihedral angles *ρ* and *τ* are used to describe the folding of the base bag though only one of them is independent, see appendix B. Distance *d*_{1} can subsequently be obtained. Details of derivation can be found in appendix B. Since 0<*σ*<90^{°}, by assigning *σ* with arbitrary values within its range the function of *d*_{1}/*d* with regard to *τ* can be derived, with *τ* changing between 90^{°} (open configuration) and 0 (fully folded configuration). Again since the relationship is highly non-linear, we use *d*_{1}/*d* versus *τ* curves in figure 9 to examine the sign of *d*_{1}. The trend is rather consistent: for any non-zero *σ*, *d*_{1}/*d* always switches signs. Hence, the base bag is not rigid and flat foldable, and neither is the tall bag.

## Acknowledgements

W.W. wishes to acknowledge the financial support from the University of Oxford in the form of a Clarendon Scholarship. Z.Y. is grateful to Erik D. Demaine, Martin L. Demaine and John Ochsendorf of MIT who first exposed him to the exciting problem of the rigid origami. The authors also thank Mrs A. W. May for proof-reading the manuscript.

## Appendix A. Mathematical derivation for rigid foldability of the proposed pattern

To obtain conditions (2.2)–(2.4) in terms of *ρ* and *ω*, we first express all the vectors along the creases and edges using the Cartesian coordinate system shown in figure 2. Denote by *τ* and *η* the inclined angles of DK and FD with respect to the bottom plane, respectively. For a bag with height, width and depth *h*, *w* and *d*, respectively, and the vertical height of the upper portion being *h*′=*h*−*d*/2, there are
Moreover,
where can be derived from the following nonlinear equation system,
Furthermore, as
it can be found that
indicating that the short bag has only one degree of freedom during folding, and thus only one of the angles, *ρ* or *τ*, is independent.

Also for panel CDF, there must be resulting in Now substituting the above expressions into conditions (2.2)–(2.4) yields A1 A2 and A3 where and .

## Appendix B. Mathematical derivation for the base bag of the conjectured pattern

In order to calculate *d*_{1} given by (5.1), we first find the edge vectors at an arbitrary folded shape of the base bag shown in figure 6*c*, where *ρ* and *τ* are variables though only one of them is independent as (see appendix A). They are
can be obtained by solving the following equation system,

Substituting the above expressions into equation (5.1), we obtained *d*_{1} (or *d*_{1}/*d*) in terms of *τ* for a given *σ*.

- Received February 18, 2011.
- Accepted March 1, 2011.

- This journal is © 2011 The Royal Society