Rigid folding equations of degree-6 origami vertices.

Rigid origami, with applications ranging from nano-robots to unfolding solar sails in space, describes when a material is folded along straight crease line segments while keeping the regions between the creases planar. Prior work has found explicit equations for the folding angles of a flat-foldable degree-4 origami vertex and some cases of degree-6 vertices. We extend this work to generalized symmetries of the degree-6 vertex where all sector angles equal 60 ∘ . We enumerate the different viable rigid folding modes of these degree-6 crease patterns and then use second-order Taylor expansions and prior rigid folding techniques to find algebraic folding angle relationships between the creases. This allows us to explicitly compute the configuration space of these degree-6 vertices, and in the process we uncover new explanations for the effectiveness of Weierstrass substitutions in modelling rigid origami. These results expand the toolbox of rigid origami mechanisms that engineers and materials scientists may use in origami-inspired designs.

Introduction

Over the past 10 years, there has been a surge of interest in applications of origami, the art of paper folding, in engineering [1], physics [2] and architecture [3]. Of particular use to these fields is rigid origami, where we attempt to flex a flat sheet made of stiff polygons joined edge-to-edge by hinges, called the crease pattern of the origami [4]. Such flexible crease patterns have the advantage of being flat and thus easy to manufacture, and the mechanics present during the folding and unfolding process operates independent of scale, whether used for deploying large solar panels in space [5] or microscopic polymer gel membranes [6].

An essential ingredient for the design of rigid origami structures is a thorough understanding of the configuration space of the rigidly folding crease pattern, a union of manifolds in , with the number of creases, where each coordinate tracks the folding angle (the signed deviation from the unfolded state) of a crease. Knowing explicit relationships between the folding angles during the rigid folding and unfolding process allows designers to calculate relative folding speeds between adjacent panels and plan rotational spring torque to actuate each crease in order to have the structure self-fold to a desired state [7,8]. Normally engineers prefer rigid origami crease patterns that possess a single degree of freedom (d.f.), such as crease patterns that consist only of degree-4 vertices, because their configuration spaces are completely understood [9].

In this paper, we show how configuration spaces can be similarly well-understood for rigid origami vertices with higher d.f. if we impose symmetry on the folding angles. This approach is of use in applications because actuators for folding creases can often be programmed to impose symmetry on the rigid folding, e.g. by using springs of prescribed strengths. Additionally, adding symmetry constraints will reduce the d.f. of the rigid origami vertex, making the folding mechanics more controllable for design.

The idea of using crease pattern symmetry to reduce d.f. and obtain folding angle relationships is not new (see for instance [7,10-13]). However, in this study we undertake the first complete examination of all symmetry possibilities of the degree-6 origami vertex whose sector angles between creases are all . After establishing background and prior work in §2, we will compute the number of symmetrically different patterns we can place on the sequence of folding angles that are rigidly foldable in §3, and find explicit equations for the folding angle relationships in each case in §4. Our methods will also allow us to generalize away from the ‘all angles’ crease pattern, so long as the folding angle symmetry can be preserved.

We will also see that, as in the degree-4 case, the Weierstrass substitution leads to especially nice folding angle relations in many of our degree-6 cases. We offer an explanation as to why this is the case in §5 and offer a list of open questions for future study.

Background on rigid foldings

We follow the notation from [14]. A crease pattern is a straight-line embedding of a planar graph on a compact region . We define a continuous function to be a rigid folded state of the crease pattern on if for every face of (including the faces that border the boundary of ) we have that restricted to is an isometry and our folded image has no self-intersections.[1] Each crease line will border two faces, and the signed supplement of the dihedral angle between these faces in a rigid folded state is called the folding angle of under . We denote the set of folding angles for all the creases in a rigid folded state by the folding angle function . If then the crease is unfolded and if (resp. ) then is said to be a valley (resp. mountain) crease. If any crease has then that crease has been folded flat and represents the largest magnitude a folding angle can achieve.

If has creases then the parameter space of the crease pattern is , where each point has coordinates equal to a possible folding angle of crease in a rigid folded state . The configuration space of the crease pattern is the subset of points such that there exists a rigid folded state whose folding angle function is . We then say that a crease pattern has a rigid folding if there exists a path and a parameterized family of rigid folded states on such that the folding angle function for is given by .

One of the main problems in rigid origami theory is to describe the configuration space of a given crease pattern, which can be done by finding expressions for the folding angle functions. Certainly a first step is to do this for single-vertex crease patterns , where only one vertex of is in the interior of . The case where this vertex is degree-4 and flat-foldable (meaning each crease can be folded flat simultaneously) is completely understood and characterized as follows:

Theorem 2.1.

Let be a degree-4 flat-foldable vertex whose plane angles around the vertex are (in order) , and , where , arranged among the creases as in figure 1. Let be the folding angle function for a rigid folded state of . Then there are only two possibilities for an explicit representation of , which we call mode 1 and 2: and where

Figure 1

(a) Modes 1 and 2 for a flat-foldable, degree-4 vertex. (b) The configuration space is the union of two curves, one for each mode. Bold creases are mountains, others are valleys. (Online version in colour.)

See [7,14] for a proof. Also, the fact that the opposite plane angles around the vertex are supplementary follows from Kawasaki’s Theorem, and the fact that we must have one mountain and three valley creases (or vice-versa) as in figure 1a follows from Maekawa’s Theorem. These are two basic theorems on flat-foldable vertices and can be found in [14].

The fact that the folding angle expressions in theorem 2.1 are linear when parameterized by the Weierstrass substitution is surprising. We refer to the constants and as folding angle multipliers of their respective rigid foldings.

Also note that the case where gives us a crease pattern, called a bird’s foot, with a line of reflective symmetry. This results in , implying that there is only one rigid folding mode.

A tool that can be used to prove theorem 2.1, and that we will employ in this paper, involves rotations about the crease lines to simulate the rigid folding. Let be the orthogonal matrix that rotates by about the line containing crease in , where we place the vertex of at the origin and let lie in the -plane. Define Then implies that , the identity matrix [15]. It has been proven [16,17] that a vertex is rigidly-foldable if and only if it is 2nd-order rigidly-foldable, meaning that the second-order Taylor expansion (about the origin) of , say where the folding angles are parametrized by time , equals . Writing the folding angles as , the Taylor expansion is Therefore, the linear and quadratic terms must be zero. The linear term equalling zero corresponds to the origami vertex being infinitesimally rigidly foldable at the origin (i.e. the unfolded state). The second-order terms being zero is what guarantees a finite-length rigid folding motion away from the origin. Following [17], we can compute these second-order terms from (2.3) and arrive at the equation where is the zero matrix, , , and we consider our single-vertex creases to be vectors (where at the unfolded state, when ). As argued in [17], we may assume the folding angle acceleration terms are zero at the origin (i.e. we may assume is constant near the origin because rigid folding paths through the origin will always be mountain-valley symmetric). Thus, for a rigid folding to exist in a neighbourhood of the origin, we need to find velocities that make the first matrix sum in equation (2.4) the zero matrix. Surprisingly, often this local approximation around the origin leads to folding angle functions that hold globally for the whole configuration space.

Example 2.2 (Trifold).

As an example, we will use the second-order matrix identity (2.4) to find the folding angle function for , the degree-6 vertex whose plane angles are all (figure 2a) and that rigidly folds with folding angle symmetry (figure 2b) for the creases . (This was first derived in [7,18].) We call this rigid folding symmetry the trifold way to fold this vertex.

Figure 2

The trifold example. (a) The degree-6, all--angle single-vertex crease pattern . (b) A symmetric rigid folding of . (c) The configuration space curves, which is a subset of the full space, with sample points illustrated by their respective rigid foldings. (Online version in colour.)

We set for . Then the first matrix sum in (2.4) becomes Setting this equal to zero and solving for yields two possible folding modes: and . There are many relationships that could satisfy these linear differential equations at the origin, such as anything of the form for the first mode. However, taking a page from the degree-4 case, one should consider the Weierstrass substitution. It turns out that a modified Weierstrass substitution does the trick, and the global folding angle relationships for the trifold are as proved in [7,18]. These curves are shown in figure 2c.

In general, a rigid folding vertex of degree will have d.f. and so will have 3 d.f. as it rigidly folds. Unlike the degree-4 case, there are no known elegant folding angle equations for degree-6 rigid foldings; while such equations can be computed using standard kinematics methods, the resulting equations are quite unwieldy (see §4f). The symmetry imposed in example 2.2, however, reduces the d.f. to one, and very nice folding angle equations result. We can think of the equation (2.5) as carving out a curve for this particular symmetric case from the larger configuration space .

Different symmetries of the degree-6 vertex

It therefore makes sense to explore all the different symmetry types that the folding angles of can achieve and see if their folding angle equations can be found. A rigid folding of with no symmetry would have all different folding angles, . If we equate the different with colours, then classifying the differently symmetric rigid foldings of with different (for ) is almost like finding symmetrically different 6-bead bracelets where the beads are coloured with colours. The latter is a classic combinatorics problem and can be solved via Burnside’s Lemma or Polya enumeration. But these two problems are not exactly the same; using numbers for colours, Polya enumeration would distinguish and as differently coloured bracelets, but these would equate to the same kind of folding angle symmetry for , one with three consecutive, equal folding angles, then two equal folding angles, then a third folding angle. Therefore, we need to enumerate and classify not just the different -coloured bracelets with six beads, but the -coloured bracelet patterns on six beads.

Polya enumeration is not enough to solve this problem; while it can be a helpful tool, one needs to examine the individual terms in the cycle index generating function to see if folding symmetries are being counted multiple times. This can be done by hand, but we double-checked by writing Python code to conduct a thorough enumeration. Then for each type of bracelet pattern, we used equation (2.4) to check if a rigid folding existed in a neighbourhood of the origin; if solving the system of equations that (2.4) generated led to a null or complex solution, then a rigid folding would not exist for that pattern.

Our results are shown table 1 and illustrated in figure 3 (aside from the trifold, which was previously shown in figure 2). It may seem surprising that so few rigid folding patterns exist from the numerous -colour bracelet patterns, but many of them reduce to the main folding patterns we discovered: trifold, bowtie, opposites, igloo and two pair (to be described in §4). For example, the 3-colour bracelet pattern (123232), which converts to folding pattern , generates equations in (2.4) that simplify to and the trifold equations (2.5).

Table 1

Bracelet colourings and rigid folding symmetries for .

no. of colours k	k-colour bracelet patterns	rigid foldings	(symmetric pattern, folding name, d.f.)
1	1	0
2	7	2	(121212, trifold, 1), (122122, bow tie, 1)
3	14	1	(123123, opposites, 2)
4	10	2	(123432, igloo, 2), (112234, two pair, 1)
5	3	1	(112345, almost general, 2)
6	1	1	(123456, fully general, 3)

Figure 3

The symmetric rigid folding cases from table 1 (excluding the trifold). The number sequence with each case indicates the respective bracelet pattern (equivalent to the indices of the folding angles, in order). (Online version in colour.)

Equations for symmetric modes

We now describe the equations for symmetrically-different rigid foldings of degree-6 vertices obtained from equation (2.4), generalizing them beyond the all--angle case when possible while retaining the folding angle symmetry. Note that the local approximation of equation (2.4) does not guarantee its extension, globally, to the whole configuration space, and therefore when such a local-to-global extension is possible we will aim to provide a proof that is independent of equation (2.4).

Generalized trifold, (121212)

We can generalize the trifold rigid origami from example 2.2 by letting the sector angles of the crease pattern alternate and , as in figure 4a, while still having the folding angle pattern , meaning that the rigid folds will still have rotational symmetry about the vertex. The second-order approximation of the folding equations near the origin in this case extend globally and give us the following:

Figure 4

(a) The generalized trifold crease pattern. (b) A sequence of mode 1 rigid foldings. (c) Configuration space of the generalized trifold for a few values. (Online version in colour.)

Theorem 4.1.

The generalized trifold crease pattern in , has two rigid folding modes. Mode 1 has while the mode 2 equation is the same but with and reversed.

Note that if then equation (4.1) becomes equation (2.5) of the trifold in example 2.2.

Proof.

We follow a similar approach to the proof of equation (2.5) given in [7]. By the rotational symmetry of the crease pattern and folded state, the matrix equation (2.3) simplifies to where we assume that the paper lies in the -plane, is along the positive -axis, and are the matrices that rotate about the - and -axes by , respectively. This is equivalent to saying that is a rotation of about some axis. A common fact (derived from Rodrigues’ rotation matrix formula) is that the trace of such a rotation matrix will equal . On the other hand, where and . The factorization of in equation (4.2) is obtained from a sequence of trigonometric identity manipulations. Since , we can see that the second factor of (4.2) does not represent our desired configuration space for the generalized trifold. For example, if we set , then this factor simplifies to the constant , which is never zero. Therefore, the first factor must equal zero, and the resulting equation has solution curves for as shown in figure 4c for various values of . As seen in [7], this type of equation can be re-written in terms of and . Doing this results in the equivalent expression (2.5) and its mode 2 counterpart.

Generalized bow tie, (112112)

The bow tie symmetry has folding angles in the pattern . When folded on there is only one folding mode up to rotation about the vertex. If we generalize so that the crease pattern still has horizontal and vertical symmetry, as shown in figure 5a, we have two folding modes that still produce a bow tie shape (figure 5b). The mountain-valley (MV) assignment labeled mode 1 of the generalized bow tie appears in the well-studied waterbomb tessellation; see vertices and in the crease pattern of figure 7b. Thus the folding angle equations of this case are already known and were first reported in [10]. The mode 2 MV assignment does not appear in the waterbomb tessellation and seems to be new. For completeness, we include the explicit folding angle equations for both modes in the following theorem.

Figure 5

(a) The generalized bow tie crease patterns, modes 1 and 2. (b) Rigid foldings of the bow ties. (c) Configuration space of the generalized bow tie for a few values, where the modes converge when . (Online version in colour.)

Figure 7

(a) The plane symmetry of a folded . (b) The waterbomb tessellation crease pattern and rigid folding simulation. All the vertices are degree-6 and have either bow tie ( and ) or igloo symmetry ( and ). (Online version in colour.)

Theorem 4.2.

The generalized bow tie crease pattern in figure , has two rigid folding modes, parametrized by

Note that when both of the equations in (4.3) become the same.

The result may be proved using the vertex-splitting technique from [19], and indeed this is exactly what Zhang and Chen do in [12] to derive the mode 1 equation. That is, in the mode 1 case we split the degree-6 bow tie vertex in the crease pattern into two flat-foldable degree-4 vertices and in the crease pattern as in figure 5a. This new 2-vertex crease pattern must be kinematically equivalent to the mode 1 generalized bow tie, since if we shorten the segment in the folding angles will not change. Thus in the limit the folding angles in must be the same as those in . Furthermore, the two vertices in are bird’s feet and force the creases to fold with the desired bow tie symmetry. Specifically, theorem 2.1 gives us that .

In the mode 2 case, we split the vertex in along a different line to produce two degree-4 flat-foldable vertices and in with sector angles , as shown in figure 5a. In the creases bounding the sector angle must have different MV parity, and only one choice for these creases matches the desired bow tie symmetry (the one shown in figure 5a). This will be kinematically equivalent to the mode 2 rigid folding of , as in the mode 1 case. In this MV assignment, the degree-4 vertices in are folding in mode 2 of theorem 2.1, and therefore we have that , as desired.

Generalized opposites, (123123)

The pattern is the only 3-colour bracelet pattern whose corresponding rigid folding symmetry for does not reduce to either the trifold, the bow tie, or simply does not rigidly fold. We generalize this pattern in a way that matches the symmetry , as shown in figure 6a, denoting the crease pattern by . Note that if then this crease pattern becomes that of the generalized bow tie, and if then we get mode 1 of the generalized bow tie and if or then we get mode 2. Therefore, we would expect that the configuration space of the generalized opposites pattern should contain the configuration spaces of the different modes of the generalized bow tie as subsets. This immediately suggests that if we consider the generalized opposites configuration space to be points , then cannot be a one-dimensional curve, but must be a two-dimensional surface. In other words, the kinematics of the generalized opposites must have two d.f.

Figure 6

(a) The generalized opposites crease pattern. (b) Rigid foldings of the opposites pattern. (c) Configuration space of the opposites pattern with curves (through the corners and the origin) showing the three ways that the bow tie symmetry is a special case of the opposites symmetry. (Online version in colour.)

We verify this by considering the second-order approximation of the rigid folding around the origin, as in §2. We set where , , , , , and . Then the first matrix sum in (2.4) becomes

Setting this equal to zero and reparametrizing with the Weierstrass substitution gives the folding angle approximation near the origin Surprisingly, equation (4.4) extends to the whole configuration space . That is, we can express any of the in terms of the other two and generate rigid folding simulations like those in figure 6b. Note that a proof of this that does not rely on the second-order approximation is not possible with the vertex-splitting technique of the §4b (or the parallel pleat transform of the §4d) because reducing to a crease pattern with only degree-4 vertices will necessarily result in a 1-d.f. system, and we know that is 2-d.f.

In figure 6c, we see the configuration space given by equation (4.4). On this surface we have drawn the three different bow tie special cases of , as mentioned previously. Indeed, if we set in (4.4) and either , , or we obtain the folding angle equations for the generalized bow tie in theorem 4.2.

Generalized igloo, (123432)

The igloo from figure 3 and table 1 has folding angles of the form , or bracelet pattern. To generalize this, we maintain the line of reflection symmetry of this pattern and allow to be the sector angles between creases and and to be the angle between and . We call this crease pattern (figure 7a).

Like the bow tie, the igloo symmetry pattern appears in the waterbomb tessellation pattern, as seen at vertices and in figure 7b. Explicit folding angle equations for this waterbomb tessellation instance of the igloo pattern, even when generalized to , can be found in [10]. However, this does not tell the whole story of . As stated in [10], when the waterbomb tessellation is folded symmetrically, so that the bow tie and igloo vertices rigidly fold symmetrically, then its rigid folding is only 1-d.f. But the generalized igloo on its own is a 2-d.f. rigid origami. We capture this, as well as equations for the whole configuration space of in the following theorem.

Theorem 4.3.

The symmetric degree-6 crease pattern is a 2-d.f. rigid origami, and the configuration space is given by and

To see that the generalized igloo is a 2-d.f. rigid origami, imagine that we place the vertex of at the centre of a unit radius sphere so that the folded origami intersects at a closed spherical linkage. Fix the sector of between creases and choose any . This will determine the position of the creases and , and if we draw two circles of radii at these creases’ endpoints on , then the intersection of these circles will determine either 0, 1, or 2 possible locations for the crease . Thus, any pair will determine a rigid folding of , and any rigid folding of will give us such a pair , making this a 2-d.f. system.

As noted in [10], all rigid foldings of are reflection-symmetric about the plane that contains the creases and . We let this plane be the -plane and position the crease to be on the positive -axis, as shown in figure 7a. Then, keeping fixed, we imagine folding the sectors of paper between the creases , , and . To replicate the folding angle at , the sector would have to lift from the -plane by an angle of , while the folding angles at and will simply be and , respectively. We let . Then the image of after this folding will be In order for the folding of to be symmetric about the -plane, we must have that the -coordinate of (4.7) is zero. This simplifies to Isolating the terms in (4.8) to create and simplifying a bit further gives (4.5). Performing the same computation on except flipped so that the crease is fixed to the positive -axis gives an equation on , which is (4.6).

Equations (4.5) and (4.6) can be used to create 2-d.f. rigid folding simulations of . The configuration space for that we obtain from these equations in the parameter space is shown in figure 8 (we actually used the singularity-free equation (4.8) to compute this image).

Figure 8

The configuration space in parameter space. The mode 1 and mode 2 curves from theorem 4.4 are indicated, while the third indiccated curve is the rigid folding of in a symmetric folding of the waterbomb tessellation derived by Chen et al., (eqns (2.4a) and (2.4b) in [10]). (Online version in colour.)

We see in theorem 4.3 a glimmer of the Weierstrass substitution in the 2-d.f. system , although it is not as apparent or elegant as in the generalized opposites pattern . However, several 1-d.f. slices of the configuration space can be found that show instances where the Weierstrass substitution, like the degree-4 and trifold cases, reveals linear relationships between some folding angles.

One way to find such 1-d.f. slices is to use the parallel pleat transform from [18] (also known as the double line method). This method takes as input a rigidly-foldable crease pattern and creates a new crease pattern where each crease of is replaced in by a pair of parallel creases , (a pleat) and the vertices of degree in are replaced by an -sided polygon in . This is done in such a way that the vertices in are all degree-4 and flat-foldable, so that the folding angle equations of theorem 2.1 can be used to rigidly fold . Thus, the folding angle of in will equal the sum of the folding angles of and in .

This parallel pleat transform has additional constraints; see [18] for more details. But in the case of the generalized igloo it works very well and is shown in figure 9b, where the folding angle multipliers of the creases in are shown and lead to two different modes for this crease pattern (one where the ‘igloo door’ pops in and one where it pops out). These multipliers along with theorem 2.1 give us folding angle relationships for as shown in the following theorem.

Figure 9

(a) The generalized igloo crease pattern. (b) The parallel pleat transform applied to the igloo crease pattern, with creases labelled with their folding angle multipliers for modes 1 and 2 (red multipliers are mountain and blue are valley creases). (c) Illustrations of rigid folding modes 1 and 2 of the generalized igloo. (Online version in colour.)

Theorem 4.4.

The generalized igloo crease pattern in figure , has two rigid folding modes parameterized by and for mode 1, and for mode 2 and

In the parallel pleat transform, we choose our pleats in to be perpendicular to the central polygon. This means that the folding angle multipliers of the degree-4 flat-foldable vertices in will all be of the form for . See figure 9b for reference.

In mode 1, we choose the parallel pleats made from to have folding multiplier 1, so if is the folding angle of these two creases in , we have that . Next, the parallel pleats made from crease in have, according to theorem 2.1, folding angles . Therefore, as desired. For crease we obtain , and for crease we have . When simplified, these produce the mode 1 equations as stated in the theorem. The equations for mode 2 follow similarly.

A few notes:

As noted in [18], the parallel pleat transform offers an explanation for why sometimes the modified Weierstrass substitution appears in rigid folding angle equations, as it arises naturally when adding the two folding angles of a pleat that are parametrized with for some constant and where .

The folding angles , and are all expressed in terms of in theorem 4.4, and thus the configuration space of this case, which is four-dimensional, can be visualized by slices, as shown in figure 10.

Figure 10

Generalized igloo example configuration spaces in mode 1 with (a) , and (a) , . (Online version in colour.)

While parametrizations with Weierstrass substitutions and are successful in theorem 4.4 for and , it does not seem to be a successful way to completely express the relationship (although the equations given look tantalizingly close).

When the first equation in modes 1 and 2 reduces to , meaning that crease is never folded and so is actually a rigid folding vertex of degree 5, as shown in figure 10b. Also in this case the second equation in theorem 4.4 becomes in mode 1 and in mode 2, giving a direct linear relationship between these folding angles. (This also happens in the case where .)

The curves that modes 1 and 2 in theorem 4.4 make in the configuration space are shown in figure 8. Also in this figure we show a different 1-d.f. curve discovered by Chen et al. in [10] that captures the kinematics of in a symmetric folding of the waterbomb tessellation (specifically, eqns (2.4a) and (2.4b) in [10], which also demonstrate the prevalence of the Weierstrass substitution).

Example 4.5 (Resch triangle twist tessellation).

As an example of how the equations we have seen thus far can be applied to a larger crease pattern, we consider the triangle twist tessellation shown in figure 11. This was designed by Ron Resch in the 1960s [20,21] and has been studied extensively for rigid folding and metamaterial purposes [22-24].

Figure 11

A rigid folding simulation of the Resch triangle twist tessellation using a combination of trifold and igloo equations. (Online version in colour.)

As a rigid folding, the Resch pattern has many degrees of freedom [24]. But if we insist on folding it with rotational symmetry about the centre vertex , then it becomes 1-d.f. The vertex is a trifold rigid folding with , and thus its rigid folding follows (say, in mode 1 of) equation (2.5). The folding angles for vertices - can then be given by the 1-d.f. generalized igloo mode 2 equations in theorem 4.4. This provides the folding angle inputs needed to determine the folding angles for vertices - using the 2-d.f. generalized igloo equation (4.5) and (4.6) in theorem 4.3.

Two pair, (112234)

The symmetry for , which we call the two pair case, has folding angle relationships that are more difficult to characterize than the previous cases considered. The vertex-splitting and parallel pleat techniques do not work in this case because the pattern lacks rotational and reflection symmetry. That is, splitting the vertex into two flat-foldable degree-4 vertices will not work because there is no way to do this while separating the two creases with folding angle and the two creases with folding angle ; both need to be separated because flat-foldable degree-4 vertices cannot have consecutive creases with equal folding angles while rigidly folding, by theorem 2.1. The parallel pleat transform requires the vertex to either have a line of reflection symmetry or certain types of rotation symmetry in order for the folding angle multipliers of theorem 2.1 to be consistent across the vertices of the parallel pleat crease pattern. The symmetry has neither, and thus this technique will not work for symmetry.

However, parts of the matrix product in (2.3) can be used to describe the configuration space for the case, as we will see in what follows.

Theorem 4.6.

The two pair symmetric rigid folding for is a 1-d.f. system whose configuration space is described by the following equations:

Let us orient the crease pattern as shown in figure 12a. We fix the face between creases and and imagine cutting along creases and to remove one face of the crease pattern. We pick a value for and fold creases and accordingly. This determines the placement of crease , and there will be finitely many values of that, when used to fold creases and , will place so as to form a angle with the placement of . Thus the two pair rigid folding symmetry is a 1-d.f. system.

Figure 12

(a) The two pair crease pattern symmetry. (b) The slice of the two pair configuration space, the figure eight in blue, with the rigid folding of two points illustrated. The red curves on the periphery are solutions to equation (4.11) that cause the paper to self-intersect, with one example shown. (Online version in colour.)

Specifically, folding along and then by folding angle moves the crease into position On the other hand, folding along and then by folding angle moves the crease into position Our crease lines are of unit length, and we want the vectors resulting from (4.12) and (4.13) to form an angle of so that we may insert the face of that we cut away. Thus their dot product should equal . Simplifying this gives us the relationship between the and folding angles shown in equation (4.11).

Taking a different approach, we can rewrite the matrix product (2.3) equalling the identity as follows: Simplifying the entries of the two matrix products in (4.14) gives us equation (4.10), while simplifying the entries gives us (4.9).

Note that equation (4.9) illustrates the symmetry of the two pair case; if is a point in the configuration space, then so is .

The configuration space curve of the valid values that satisfy equation (4.11) forms a figure eight and is shown in figure 12b, along with images of two different rigid folding examples with the same value of radians. The other curves shown also represent values satisfying equation (4.11) but are not valid because they force the paper to self-intersect.

The equations in theorem 4.6 defy manipulation attempts to get more simple relations between the folding angles using Weierstrass substitutions. One reason for this is precisely because, as previously mentioned, we were unable to reduce the symmetry case for to a kinematically equivalent 1-d.f. crease pattern with only degree-4 flat-foldable vertices.

Fully and almost general cases

As previously stated, the fully general rigid folding of has no known way to elegantly express its configuration space. Yet it can be computed using standard kinematic analysis. For completeness, and as a way to contrast with our previous results for the symmetric cases, we summarize one such method here, based on the approach described by Balkcom [25,26].

We cut along the crease to split it into a left side and a right side , as shown in figure 13a. The fully general degree-6 vertex rigid folding has 3 d.f., and so we let , , and be our independent variables and we seek to find folding angle functions for and . Fix the face between creases and . Then folding along by followed by folding along by will move into position In the other direction, we fold along by , then along by , and finally along by to move to position The vectors (4.15) and (4.16) must be equal, and the -coordinate of (4.15) is only dependent on (since the matrix does not affect the -axis). In fact, the -coordinate of (4.15) is , and equating this to the -coordinate of (4.16) gives us Since cosine is an even function, equation (4.17) gives us two solutions for , one positive and one negative. For we could equate the other coordinates of (4.15) and (4.16), but a more robust formula can be obtained by removing the term from (4.15), as the result will be a vector that forms an angle of with (4.16). Letting and be (4.16), we have (The full expression for computed from (4.16) is quite large and uninspiring; it is omitted to save space.)

Figure 13

(a) The set-up for analysing the fully general rigid folding of , with one rigid folding shown. (b) Three slices of the configuration space of . (c) More slices stacked on the axis. (Online version in colour.)

Equations (4.17) and (4.18) can be used to simulate general rigid foldings of , and an example is shown in figure 13a. Note that in this 3-d.f. system, not all values of produce solutions for and ; e.g. some combinations will force creases and to be too far away from each other. Figure 13b shows slices of the configuration space, for , , and , with the valid points shaded in. Figure 13c shows more such slices stacked on the axis to give a sense of the inadmissible region inside this configuration space.

The rigid folding symmetry of , which is the bracelet pattern and is referred to as the ‘almost general’ case in table 1, is the least amount of symmetry we could try to impose on the fully general case. As such, we cannot expect this 2-d.f. rigid folding to have elegant folding angle expressions. However, one may set in equations (4.17) and (4.18) to obtain expressions for use in rigid folding simulations of this case.

Conclusion

We have seen that while capturing the kinematic folding angle relationships for rigid foldings of the degree-6 vertex crease pattern leads to unwieldy and complicated equations, imposing symmetry on the folding angles yields, in most cases, elegant folding angle expressions, sometimes even linear equations when parameterized with a Weierstrass substitution. Our proofs have also revealed an explanation for why Weierstrass substitutions work so well in such equations: in each 1-d.f. symmetric rigid folding of where Weierstrass substitutions were effective, we were able to reduce the degree-6 crease pattern to a degree-4, flat-foldable (multiple-vertex) crease pattern where the Weierstrass substitution definitely applies by theorem 2.1. That some higher-degree crease patterns are kinematically-equivalent to degree-4 flat-foldable crease patterns seems a logical reason why Weierstrass substitutions are so effective in rigid origami.

There are many avenues for future work from the results of this paper. What follows is a list of suggestions and open questions.

Can the folding angle expressions presented in this paper be further simplified? In the engineering and scientific literature one can find several different (yet equivalent) ways to express the degree-4 flat-foldable folding angle relationships (e.g. [8,9]), but those shown in theorem 2.1 are favourable since they illustrate the tangent half-angle linearity. The degree-6 vertex symmetries presented here that use the Weierstrass substitution are likewise favourable. But, for example, could the 2-d.f. equations for in the generalized igloo (123432) pattern (in theorem 4.3) be improved?

How can we explain the effectiveness of Weierstrass substitutions in 2- or higher-d.f. rigid foldings, such as in the opposites (123123) case? Such cases cannot be reduced to 1-d.f. degree-4 crease patterns, so a different reasoning, perhaps one purely algebraic, might lurk within such equations.

In rigid vertex folding symmetries without rotational or reflection symmetry, reduction to degree-4 flat-foldable crease patterns does not seem possible, as seen in the two pair pattern. Since the two pair pattern is a 1-d.f. rigid folding, it might have simple folding angle relationships yet to be discovered. Can other techniques be found to handle such cases?

We hope that the degree-6 folding angle relationships presented here will offer new tools to mechanical engineers and materials scientists looking for more origami methods to incorporate into their designs. As Tachi et al. show in [7], using the folding angle expressions in equation (2.5) to program spring actuators on the creases of a rigidly folding trifold mechanism results in global convergence of the rigid folding to a desired state. The new folding angle relationships we have presented should work equally well and provide new rigid folding mechanisms to expand designers’ rigid origami vertex repertoire from degree-4 to include degree-6 vertices as well.