## Abstract

The traditional waterbomb origami, produced from a pattern consisting of a series of vertices where six creases meet, is one of the most widely used origami patterns. From a rigid origami viewpoint, it generally has multiple degrees of freedom, but when the pattern is folded symmetrically, the mobility reduces to one. This paper presents a thorough kinematic investigation on symmetric folding of the waterbomb pattern. It has been found that the pattern can have two folding paths under certain circumstance. Moreover, the pattern can be used to fold thick panels. Not only do the additional constraints imposed to fold the thick panels lead to single degree of freedom folding, but the folding process is also kinematically equivalent to the origami of zero-thickness sheets. The findings pave the way for the pattern being readily used to fold deployable structures ranging from flat roofs to large solar panels.

## 1. Introduction

The waterbomb is a traditional origami (http://www.britishorigami.info/academic/lister/waterbomb.php). Commonly, two terms are related to it: waterbomb bases and waterbomb tessellations. There are two types of waterbomb bases: the eight-crease base and the six-crease base. The former is made from a square sheet of paper consisting of eight alternating mountain and valley creases around a central vertex (figure 1*a*). One of its typical tessellations is produced by four such bases tiling around a smaller square forming the square Resch pattern (figure 1*b*,*c*). The latter, consisting of two mountain and four valley creases shown in figure 1*d*, is more commonly known, and its tessellations range from a flat-foldable surface to a deformable tube known as the magic origami ball (figure 1*e*,*f*).

Both waterbomb origami structures were extensively investigated in the past. For instance, Hanna *et al.* and Bowen *et al.* established the bistable and dynamic model of the eight-crease waterbomb base [1,2]. Tachi *et al.* [3] worked on the rigidity of a six-crease origami tessellation with multiple degrees of freedom to achieve an adaptive freeform surface. On the application side, the first origami stent was made from the waterbomb tube aimed to achieve a large deployable ratio [4]. A worm robot [5] and a deformable wheel robot [6] were also proposed based on the magic origami ball.

In this paper, the focus is drawn on the six-crease waterbomb tessellation. Owing to its large deployable ratio between expanded and packaged states, it can be potentially used to fold large flat roofs and space solar panels. Although the waterbomb pattern is of multiple degrees of freedom, the symmetric folding is often preferred in most of research or artwork, which is done by constraining it with symmetric conditions and then controlling the motion to reach an ideal flat-foldable state. This is not easy in practice due to the fact that in rigid origami, the six-crease waterbomb base itself is a spherical 6*R* linkage with three degrees of freedom [7], thus the number of degrees of freedom for the pattern could increase significantly if the pattern consists of a large number of such bases.

The waterbomb pattern is primarily created for zero-thickness sheets just like all of the origami patterns. Yet, in most of the practical engineering applications, the thickness of the material cannot simply be ignored. Various methods have been proposed to fold thick panel. In one instance, tapered surfaces are used to fold a thick panel using the Miura-ori of zero-thickness sheet [8], whereas in the other, offsets at the edge of the panels were introduced to implement folding of thick panels using the square-twist origami pattern [9]. More recent research suggested to replace folds with two parallel ones to accommodate the thickness of materials [10]. In all of the above methods, the fundamental kinematic model in which origami is treated as a series of interconnected spherical linkages remained. Different from the above methods, the authors of this paper have also proposed an approach in which the fold lines were only allowed to be placed on the top or the bottom of flat thick panels. As a result, the spherical linkage assembly for the origami of zero-thickness sheet is replaced by an assembly of spatial linkages. We have proved that not only are the assemblies of such panels foldable, but also they can be folded compactly under certain conditions [11].

In this paper, we provide a comprehensive kinematic analysis on foldability of the waterbomb tessellation made from the six-crease waterbomb bases of both a zero-thickness sheet and a panel of finite thickness. Kinematically, the folding of a zero-thickness sheet is modelled as spherical 6*R* linkages, whereas that of a thick panel is treated as an assembly of the Bricard linkages. The analysis has revealed a number of very interesting features associated with waterbomb origami, including the existence of two folding paths for general waterbomb origami of zero-thickness sheets when it is folded symmetrically. Moreover, because the Bricard linkages are overconstrained [12,13], the increase in the number of degrees of freedom occurring for the origami of zero-thickness sheet does not materialize for thick panels.

The paper is structured as follows. Section 2 presents a detailed analysis on rigid foldability of the waterbomb tessellation for zero-thickness sheets. This is followed by the design and kinematic behaviour of its corresponding thick-panel origami in §3. Comparisons are made in §4 with further discussion for potential applications. The nomeclature is given in table 1.

## 2. Symmetric rigid folding of the waterbomb pattern of zero-thickness sheet

Consider a pattern made by tessellating six-crease waterbomb bases (figure 2*a*). The pattern consists of only two types of vertices, **D** and **W**, enlarged in figure 2*b*,*c*. The rigid origami folding around each vertex can be modelled kinematically as a spherical 6*R* linkage in which the creases act as revolute joints and the sheets between creases are rigid links. In general, a spherical 6*R* linkage is of three degrees of freedom, but this number is reduced to one if only the symmetric folding is allowed. In such a way, vertex **D** is regarded as a spherical 6*R* linkage with the geometrical parameters *α*_{12}=*α*_{34}=*α*_{45}=*α*_{61}=*α*,*α*_{23}=*α*_{56}=*π*−2*α*, where 0<*α*≤*π*/2. Imposing the line and plane symmetry conditions, i.e. *δ*_{1}=*δ*_{4} and *δ*_{2}=*δ*_{3}=*δ*_{5}=*δ*_{6}, to the closure condition of the linkage (see appendix A), we can then write the closure equations as
**W**, it becomes a plane-symmetric spherical 6*R* linkage with the geometric parameters *α*_{12}=*α*_{61}=*π*−*α*−*β*,*α*_{23}=*α*_{56}=*β*,*α*_{34}=*α*_{45}=*α*, where 0<*β*≤*π*/2, and
*ω*_{1} and *ω*_{3} of vertex **W** must be identical to that between *δ*_{1} and *δ*_{2} of vertex **D**. Replacing *δ*_{1} and *δ*_{2} in equation (2.1) with *ω*_{1} and *ω*_{3}, respectively, yields
**W**, we obtain two sets of equations. The first set is

The kinematic variables, or rotations about each crease, can be replaced by the dihedral angles between adjacent sheets connected by the crease. The relationship between the kinematic variables and dihedral angels is *δ*_{1}=*π*−*φ*_{1}, *δ*_{2}=*π*+*φ*_{2}, *δ*_{3}=*π*+*φ*_{3}, *δ*_{4}=*π*−*φ*_{4}, *δ*_{5}=*π*+*φ*_{5}, *δ*_{6}=*π*+*φ*_{6} for vertex **D** and *ω*_{1}=*π*−*ϕ*_{1}, *ω*_{2}=*π*−*ϕ*_{2}, *ω*_{3}=*π*+*ϕ*_{3}, *ω*_{4}=*π*−*ϕ*_{4}, *ω*_{5}=*π*+*ϕ*_{5}, *ω*_{6}=*π*−*ϕ*_{6}, for vertex **W**. Thus, the two sets of kinematic relationships of the waterbomb pattern presented by the dihedral angels become

Considering a pattern with *α*=2*π*/9, *β*=2*π*/9, and taking *ϕ*_{1} as an input, the variations of other dihedral angles at vertex **W** with respect to *ϕ*_{1} are plotted in figure 3*a*. There are two paths with the same starting point (*π*,*π*) and ending point (0,0): *path I* based on equations (2.5a–*e*) and *path II* on equations (2.6a–*f*). It indicates that vertex **W** can be folded compactly along two different paths. Yet for vertex **D**, with *φ*_{1}=*φ*_{4}=*ϕ*_{1}, there is only one path (figure 3*b*). Therefore, in general, the patterns with a large number of vertices **D** and **W** will fold in two different ways, from i, ii, iii, iv to v, or from i, viii, vii, vi to v, as demonstrated in figure 3*c*.

There are a few special cases of the waterbomb pattern which are most interesting. First, when *α*+*β*=*π*/2, creases along *z*_{2} and *z*_{6} at vertex **W** shown in figure 2*c* become collinear. As a result, they fold together like a single crease. *Path I*, given by equation (2.5), breaks down into two straight lines. A particular case with *α*=*β*=*π*/4 is shown in figure 4. At the first folding stage, *ϕ*_{2} (and *ϕ*_{6}) starts from *π* and finishes at 0 from i, xi, x to ix, whereas *ϕ*_{1}, *ϕ*_{3}, *ϕ*_{4} and *ϕ*_{5} remain to be *π*, then *ϕ*_{2} (and *ϕ*_{6}) is kept at constant 0 and *ϕ*_{1}, *ϕ*_{3}, *ϕ*_{4} and *ϕ*_{5} change from *π* to 0 along ix, viii, vii, vi and v. Both reach the compactly folded configuration. At the latter stage, vertex **W** behaves like a spherical 4*R* linkage, because *ϕ*_{2} and *ϕ*_{6} are frozen. The movement around vertex **W** will drive vertex **D** to move accordingly.

Second, equations (2.5) or (2.6) could give negative dihedral angles, which indicates a blockage occurring during folding, because physically the dihedral angles cannot be less than zero. By analysing equation (2.5b), it can be found that for *path I* when *α*+*β*>*π*/2, *ϕ*_{2} is always negative except at points (0,0) and (*π*,*π*). So a blockage is always there. From equation (2.6c), it can be found that on *path II* when *α*≠*β*, a blockage will occur when

For example, when *α*=7*π*/36, *β*=*π*/4, the kinematic curve between *ϕ*_{4} and *ϕ*_{1} is shown in figure 5*a*, and the folding sequences are demonstrated in figure 5*b*. Along *path I*, the pattern can be folded from a sheet at i to fully folded configuration at vii, whereas along *path II*, the folding process terminates at iii. The framed configurations are physically impossible owing to blockage, because these configurations correspond to cases where *ϕ*_{4} becomes negative. Even if the penetrations were allowed, folding along *path II* would end up in a fully folded configuration at vi that differs from that at vii along *path I*.

The physical blockage can also occur when *α*+*β*=*π*/2 but *α*≠*β*. Figure 5*c* shows a two-stage motion on *path I* and a blockage on *path II* for a pattern with *α*=*π*/6 and *β*=*π*/3. Based on the above analysis, the behaviour of the waterbomb tessellation can be summarized as follows.

(a) When

*α*+*β*<*π*/2 and*α*=*β*, there are two smooth folding paths with neither two-stage motion nor blockage.(b) When

*α*+*β*<*π*/2 and*α*≠*β*,*path II*is blocked and*path I*is smooth.(c) When

*α*+*β*=*π*/2 and*α*=*β*,*path I*is in two-stage motion, whereas*path II*is smooth.(d) When

*α*+*β*=*π*/2 and*α*≠*β*, both two-stage motion on*path I*and blockage on*path II*happen.(e) When

*α*+*β*>*π*/2 and*α*=*β*, only*path II*for vertex**W**is smooth, but vertex**D**is blocked. Thus, the whole pattern is blocked from compact folding.(f) When

*α*+*β*>*π*/2 but*α*≠*β*, both paths are blocked.

Among them, only cases (a)–(c) can have one or two smooth folding paths.

## 3. Folding thick panels with the waterbomb pattern

The waterbomb tessellation can also be used to fold panels with non-zero thickness. This is done by mapping the same pattern onto a thick panel while placing the fold lines either on top or bottom surfaces of the panel. Now, at **D** and **W**, there will still be six fold lines in places of creases, but these fold lines no longer converge to a vertex. In other words, dissimilar to the zero-thickness sheet, the distances between the adjacent fold lines are no longer zeros. In terms of the kinematic model, the spherical 6*R* linkage is now replaced by spatial 6*R* linkages. Among all possible spatial 6*R* linkages, the plane-symmetric Bricard linkage [13,15] is the most suitable one [11]. Let us select two Bricard linkages for **D** and **W**, respectively, figure 6*a*,*b*, with their link lengths being the panel thicknesses. As the linkages are overconstrained, the geometrical conditions of the linkage at **D** are
**W** are
*α* and *β* are the same as the sector angles of the origami pattern in figure 2*b*,*c* and *a* is the thickness of link 23 and *μ* is the proportion between the thickness of link 34 and link 23 in the vertex **W** of the thick-panel waterbomb pattern where *a*≠0 and *μ*≠0.

Applying the closure condition of the linkages leads to the following closure equations (see appendix A). For **D**, two sets of closure equations can be obtained, which are
**D**is *δ*′_{1}=2*π*−*φ*′_{1},*δ*′_{2}=*φ*′_{2},*δ*′_{3}=*π*+*φ*′_{3},*δ*′_{4}=2*π*−*φ*′_{4},*δ*′_{5}=*π*+*φ*′_{5},*δ*′_{6}=*φ*′_{6}. By conversion of the kinematic variables to the dihedral angels, the two sets of closure equations can be respectively rewritten as
**W**, which are
**W** is *ω*′_{1}=2*π*−*ϕ*′_{1}, *ω*′_{2}=*π*−*ϕ*′_{2},*ω*′_{3}=*ϕ*′_{3}, *ω*′_{4}=2*π*−*ϕ*′_{4},*ω*′_{5}=*ϕ*′_{5}, *ω*′_{6}=*π*−*ϕ*′_{6}, the two sets of closure equations now become
**D** and **W** need to be added, which are

— The first set of closure equations, equation (3.5), at

**D**and the first set of closure equations, equation (3.9) at**W**.

Because equations (3.5a) and (3.9a) are identical, the compatibility between **D** and **W**, equation (3.11), is satisfied automatically. Therefore, there is always a smooth folding path for the thick-panel origami for any *μ*≠0, figure 7*a*–*c*, in which *μ* is randomly selected as 0.5. By comparing equations (3.5) and (3.9) for the thick panel with equations (2.5) for the zero-thickness sheet, we can conclude that the thick-panel origami and the *path I* of the original waterbomb origami pattern are kinematically identical, as demonstrated by the folding sequence of the physical models in figure 7*d*. The motions of both structures are line and plane symmetric. Moreover, when *α*+*β*=*π*/2, *path I* becomes a two-stage motion, where *ϕ*′_{2} and *ϕ*′_{6} change from *π* to 0, whereas *ϕ*′_{1},*ϕ*′_{3},*ϕ*′_{4}, and *ϕ*′_{5} are kept to *π*, followed by the process that *ϕ*′_{1},*ϕ*′_{3},*ϕ*′_{4} and *ϕ*′_{5} move as a spatial 4*R* linkage. This linkage is actually a Bennett linkage. It eventually reaches the compact folding position. However, blockage could be occurred during the motion owing to the panel thickness, which could prevent the structure from being fully folded, see figure 8, in which *μ* is randomly selected as 0.7.

— The first set of closure equations, equation (3.5), at

**D**and the second set of closure equations, equation (3.10) at**W**.

Consider equations (3.5a) and (3.10a). Under the compatibility condition given by equation (3.11), there must be
*α*=*β*, another solution exists, which is
**W**. Only one folding path exists as shown in figure 9 for the case where *α*=7*π*/36, *β*=*π*/4 and *μ*=0.14. Note that this path matches that shown in figure 7*c* despite that in the latter, *μ* is randomly selected as 0.5. The motion behaviour of the thick-panel waterbomb remains the same as the zero-thickness origami in *path I*, and thus we name it *path I* for thick-panel origami. Moreover, when *α*+*β*=*π*/2, *μ*=0 from equation (3.12a). So it will not be considered.

Under the second solution, *μ*=1, given by (3.12b), equations (3.9) and (3.10) are different. In other words, together with equation (3.5), there are two sets of closure equations for the thick-panel origami with *μ*=1 that result in two folding paths. The first, based on equations (3.5) and (3.9), has been discussed earlier. The second, based on equations (3.5) and (3.10), is actually identical to equation (2.6) of the zero-thickness sheet. This shows that the corresponding folding path is kinematically identical to the *path II* of the waterbomb origami pattern of the zero-thickness sheet, and thus it is named as *path II* of the thick-panel origami. One of such example is shown in figure 10.

In thick-panel origami, there is also blockage because of collision of panels during the folding process. Generally, along *path I* of **W**, the blockage would appear when one of the dihedral angles becomes negative. The condition without blockage is *ϕ*′_{2}>0. Considering equation (3.9b) leads to *α*+*β*<*π*/2, which is the same conclusion as the zero-thickness origami pattern summarized in last section. To avoid the interference at **D** during the folding, 0<*α*≤*π*/4 must be satisfied.

—The second set of closure equations, equation (3.6), at **D**.

The other set of closure equations given by equation (3.6) at **D** signify that in the thick-panel case, there exists a folding path that violates the line symmetry. However, this path is practically always blocked, because *φ*′_{3} and *φ*′_{2}, *φ*′_{4}, and *φ*′_{1} always have opposite signs as indicated by equation (3.6b).

Therefore, the behaviour of the general thick-panel waterbomb can be summarized as follows.

(a) For any

*μ*≠0, when*α*+*β*<*π*/2, there is only one smooth folding path:*path I*.(b) For any

*μ*≠0, when*α*+*β*=*π*/2, there is one two-stage folding path,*path I*, with blockage.(c) For any

*μ*≠0, when*α*+*β*>*π*/2, there is one blocked folding path.In particular,

(d) For

*μ*=1, when*α*+*β*<*π*/2,*α*=*β*, there are two smooth folding paths, kinematically equivalent to*paths I*and*II*in the zero-thickness origami.(e) For

*μ*=1, when*α*=*β*=*π*/4,*path I*is in two-stage motion and blocked, but*path II*can achieve smooth folding.

Here, *paths I* and *II* cannot be switched from one to another once the motions are underway. The choice of folding paths has to be made at the start and end configurations. The detailed comparison on the kinematic behaviour of the general waterbomb tessellation of zero-thickness sheets and thick panels for different design parameters is given in table 2 of appendix B.

## 4. Conclusions and discussion

In this paper, we have analysed the rigid origami of the waterbomb tessellation of both zero-thickness sheets and thick panels under the symmetric motion condition. By introducing the plane-symmetric Bricard linkages to replace the spherical 6*R* linkages in the origami pattern, the thick-panel waterbomb structure has been successfully formed. The rigorous enforcement of compatibility conditions ensures the mobility and flat-foldability of the thick-panel origami. We have also proven that the thick-panel origami and that of the zero-thickness sheet are kinematically equivalent.

Despite the fact that the thick-panel origami is born from an existing origami of zero-thickness sheet, it has a number of advantages over its parent. First, kinematically the thick-panel origami structure is a mobile assembly of overconstrained Bricard linkages with only one degree of freedom, and thus no additional constraints are required to keep its motion symmetrical. This could be a great benefit for real engineering applications as its control system could become much more simple and reliable. Second, in general, the origami of waterbomb tessellation for zero-thickness sheets has kinematic singularity when it is flat and fully compact. However, for thick-panel origami, the singularity only appears when a very specific thickness is chosen. A suitable selection of the thickness of the panels make the latter possible to achieve compact folding without bifurcations. The unique motion path is certainly much desirable for most practical applications.

The waterbomb tessellation for the thick panels enables the structure to be folded compactly. The compactness of the package depends on the thickness coefficient and the number of vertices within the pattern. The pattern can be divided into strips formed by vertices **D** in the horizontal direction. Consider a pattern consisting of *m* strips, each with *n* vertices **D**. In the completely packaged configuration, the dimension in the vertical direction will be (*m*+1)/2 of the height of the larger triangles in vertex **D** and the cross-section dimensions are the width of the larger triangles in vertex **D** and the overall thickness as 2*n*(2+2*μ*)*a*, where *n* is the number of vertices **D** in the strip and *μ*≤1. *μ*>1 is not recommended because it results in panels with considerable thickness and, in turn, the overall thickness of the package when the panels are packaged. So the ratio between the area of a fully expanded shape and that of completely folded is about 4*n*. This indicates that the concept is very suitable to fold a structure in a long rectangular shape. On the other hand, to meet the geometrical conditions of the spatial linkages, each panel within the pattern could not be of the same thickness. As a result, the overall structure in the fully deployed configuration is flat but not absolutely even. However, for this waterbomb pattern, we have managed to make sure that one side of the expanded surface is completely flat, which enables the waterbomb origami pattern to be directly applicable to fold thick-panel structures such as solar panels and space mirrors.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

Y.C. and Z.Y. initiated the project, worked on this topic and wrote the paper. H.F. conducted all the equation derivation under the supervision of Y.C. and J.M. R.P. constructed all the models. J.M. did all the verification between the analytical and modelling results. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

This work was supported by the National Natural Science Foundation of China (projects no. 51275334, 51422506 and 51290293), the Ministry of Science and Technology of China (Project 2014DFA70710) and Air Force Office of Scientific Research of USA (R&D Project 134028).

## Acknowledgements

Y.C. acknowledges the support of the National Natural Science Foundation of China (projects no. 51275334, 51422506 and 51290293) and the Ministry of Science and Technology of China (Project 2014DFA70710). Z.Y. wishes to acknowledge the support of Air Force Office of Scientific Research (R&D Project 134028). Z.Y. was a visiting professor at Tianjin University while this research was carried out.

## Appendix A

According to the DH notation set-up in figure 11, the transformation matrix can be assembled as

which transforms the expression in the *i*+1th coordinate system to the *i*th coordinate system. The inverse transformation can be expressed as

As for spherical linkages, the axes intersect at one point, which means the lengths of each links are zeros and thus equation (A 3) reduces to

Equations (A 3) and (A 4) can be used to obtain the closure equations of the thick-panel waterbomb pattern and the original waterbomb origami pattern in the text, respectively.

## Appendix B

See table 2.

- Received December 11, 2015.
- Accepted May 5, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.