Self-Foldability of Rigid Origami

In mechanical designs based on rigid origami, the folding motion is often achieved by rotational actuators that apply bending moments at each foldline between facets. This self-folding principle is very powerful in different applications; it can be used to design reprogrammable matter [1], self-folding machines [2], or to obtain 3D microstructures based on printing patterns [3]. When using self-folding techniques, we always encounter a common question: Is a given configuration possible to be realized by a set of actuators? If so, how? Indeed, the rigid foldability condition [4,5] is a necessary condition for a rigid origami to be actuated as a mechanism; however, this is not at all sufficient. For example, consider a vertex composed of three mountains and three valleys. This can be actuated into two clearly distinct configurations even though they follow the same MV assignment (Fig. 1). Once the vertex starts to pop up or pop down, it cannot flip to the other side without unfolding everything. This bifurcating nature and multistability of origami vertices is an interesting phenomena attracting the attention of scientists [6–8]. However, from the viewpoint of mechanical design, we often want to avoid such bifurcations. The objective of this paper is to predict the way the model folds from the flat unfolded state and to enable a design of motion via the computation an appropriate set of applied moments. The example of Fig. 1 tells us that we need extra care in order to achieve such a goal. This is not a straightforward problem, as we can observe that even with a set of proper mountain valley folding moments, we may easily encounter bifurcations at the flat state. This flat state singularity is the very reason that the folding of rigid foldable origami such as the Miura-ori or the Resch pattern is difficult at the beginning but is easy when correctly folded for a finite amount.

In this paper, we will pose the mathematical problem of self-folding: To know if there exists a set of moments applied on each hinge, which we later call a driving force, to enable a desired folding motion of a given crease pattern. We give basic theoretical tools for self-folding with which we can analyze and design rigid origami mechanisms based on self-folding.

We also consider a subproblem of self-foldability: Self-foldability with rotational spring driving forces that we define rigorously later in this paper. In this situation, we have a physical implementation of actuators where each edge has a prescribed target fold angle and independently tries to get closer to the target fold angle when actuated, as if torsion springs are attached to creases. We define a mathematical model of such a system and call it a rotational spring driving force. Rotational spring driving forces can model actual methods used in self-folding contexts, e.g., multilayered shape memory materials [1,2] or polymer gel [8]. If a rigid origami is self-foldable with a rotational spring driving force, this can be a very robust system, programmable without active sensing or instantaneous feed-back control.

In this paper, we concentrate on the problem of finding a rotational spring driving force that uniquely self-folds from and to desired rigid origami states. Is this always possible? If not, can we characterize uniquely self-foldable patterns? We believe that the general problem leads to interesting open problems in mathematics and theoretical computer science. As evidence, we will investigate the self-foldability of some interesting examples that can be helpful to grasp the essence of the self-folding problem.

Self-foldability is the problem of asking if a rigid folding path from a flat unfolded state to a 3D-folded state can be actuated using a set of driving force (rotational moment) functions without causing a bifurcation at any state. Here, we ignore the inertial effects; in such a system, the springs and dampers dominate, and thus, we can assume that we instantaneously obtain a critical (angular) velocity proportional to the (moment of) force. Self-foldability is an especially important problem at the flat unfolded state where rigid origami constraints degenerate.

Definition 1 (Configuration Space). For a rigid origami with n creases (edges), a configuration is the set of fold angles ρ₁,…, ρ_n of creases 1,…, n of the crease pattern; this can be represented as a point$ρ=[ρ1,…,ρn]T$in n-dimensional parameter space, where n is the number of creases. A rigid folding, i.e., valid configuration, is a configuration which satisfies a set of constraints derived from isometries (Eq.(1)below) and nonintersection of the facets. The set of rigid foldings from the same rigid origami is a subset of the parameter space and is called the configuration space.

Definition 2 (Continuous Folding). A well-behaved continuous rigid folding$ρ(s)$from a rigid folding$ρ(0)$to$ρ(starget)$is an arc length-parameterized piecewise C¹ curve in a valid configuration space in n-dimensional parameter space. Because it is piecewise C¹, there are at most two tangent vectors$v+(s0):=lims→s0+(dρ(s)/ds)$and$v−(s0):=lims→s0−(dρ(s)/ds)$at a valid configuration.

Consider a set of well-behaved continuous foldings passing through a configuration. The set of tangent vectors of such foldings at the configuration form a region when projected on the unit sphere, which we call valid tangents.

We believe that “well-behaved folding” defined as a piecewise smooth curve captures any process of an actual folding motion that has only a finite number of singular positions. At singular positions, especially at the origin (the flat, unfolded state), the two velocity vectors defined capture the irreversible nature of self-folding and self-unfolding, i.e., self-unfolding is usually easier than self-folding. Figure 2 illustrates the above definitions. Valid tangents may form a nonconnected region on the unit sphere, and the self-foldability problem is deeply related to the geometry of valid tangents.

Definition 3 (Driving Force). A driving force is a continuous vector field in the parameter space:$f(ρ)=[f1(ρ),…,fn(ρ)]T$. A driving force is conservative if it is the negative gradient of some C¹ scalar field$U(ρ)$, i.e.,$f(ρ):=−∇U(ρ)$. We call$U(ρ)$the potential energy.

Also, we call a conservative driving force and potential energy a rotational spring driving force and rotational spring potential energy, respectively, if the potential energy is an additively separable function, i.e., the function can be represented as the sum of functions of each fold angle:$U(ρ)=U1(ρ1)+⋯+Un(ρn)$⁠.

In physical sense, a driving force represents a set of moments applied to the hinges. We call it a “force” in the generalized sense, as an energy gradient in some coordinate system. As already mentioned, rotational spring driving forces yield more robust and easy-to-implement systems of self-folding than general potential energy or nonconservative forces that require instantaneous feedback control.

Definition 4 (Constrained Force). We define the constrained forces along a well-behaved continuous rigid folding $ρ(s)$ to be $f+(s):=v+(s)·f(ρ(s))$ and $f−(s):=v−(s)·f(ρ(s))$⁠. We call the former the forward force and the latter the backward force. If a driving force is conservative, then, $f+(s0)=−lims→s0+(∂U/∂ρ)·(dρ/ds)=−lims→s0+(dU(ρ(s))/ds)$⁠.

Definition 5 (Self-Foldable). A well-behaved continuous folding $ρ(s)$ from a rigid folding $ρ(0)$ to $ρ(starget)$ is self-foldable by driving force $f(ρ)$ if at any point $ρ(s)$ for s ∈ [0, s_target), the forward force f₊(s) is positive and takes a local maximum among the valid tangents at s. Here, we can observe the continuity of vector directions by calculating an intersection of the configuration space with a sphere of radius ϵ around the point when ε → 0. We call a well-behaved continuous folding $ρ(s)$ uniquely self-foldable if $ρ(s)$ is the only well-behaved continuous folding, that is, self-foldable by f.

and $R(ℓi,ρi)$ is an orthogonal matrix representing the rotation by angle ρ_i about an axis along $ℓi$ passing through the origin. As Eq. (1) is a set of polynomial equations of cosines and sines of the fold angles on a compact domain (−π ≤ ρ_i ≤ π for all i), the valid set of variables is a closed set. We also define the nonintersection condition to allow the paper to touch (but not penetrate) itself. This implies that the configuration space of an origami disk is a closed set.

This gives three equations for each interior vertex of a crease pattern. The configuration space is tangent to at least a e_in – 3v_in-dimensional linear space for an origami model with e_in creases and v_in interior vertices. However, in the flat unfolded state of $L=ℓ$⁠, the third row of Eq. (3) degenerates, giving us a higher-dimensional tangent space with at least e_in − 2v_in dimensions. The flat state forms a connecting point of otherwise separated configuration space components.

Consider a degree-4, flat-foldable single vertex with sector angles α,β,π − α,π − β (Fig. 3). Its fold angles ρ₀, ρ₁, ρ₂, ρ₃ form a four-dimensional parameter space, and the relationships between these angles give us a valid configuration space that is the union of the following two modes [12] (see the Appendix for details, Fig. 3)

Notice that $0<|p|<1$ and $0<|q|<1$⁠. The folding modes 1 and 2 are each 1-manifolds embedded in four-dimensional parameter space that intersect only at t = 0 (Fig. 4). We, respectively, call them configuration curves 1 and 2. Note that at t = 0, the valid tangents lie within the two-dimensional space defined by Eq. (3).

Theorem 1. For any rigidly foldable, flat-foldable degree-4 vertex with an arbitrary starting and target configurations, there exists a rotational spring driving force that makes the vertex uniquely self-foldable.

We call the subsets of curve 1 separated by $ρ=0$ manifolds 1+ and 1−, such that curve 1+ includes $ρT$⁠.

We now claim that if there exists a potential energy function $U(ρ)$ with the conditions that

1. (1)

   U monotonically decreases along mode 1 toward the target state.

2. (2)

   U monotonically decreases along mode 2 toward the flat state.

then, $U(ρ)$ uniquely self-folds from any state to the target state along the shortest path from the initial and targets shapes.

The proof is as follows. Assume that above conditions are satisfied. If we start from a point on curve 1+, then a continuous rigid folding along the shortest path to target position $ρT$ always has positive forward force f₊(s) > 0 because of condition 1. Therefore, this path is uniquely self-foldable by U.

Consider that the initial point is on curve 2. Then, a continuous rigid folding along the shortest path from any point $ρ≠0$ on curve 2 to $ρ≠0$ is uniquely self-foldable by U because of condition 2. Similarly, if we start from a point on curve 1−, the shortest path to 0 is uniquely self-foldable by U because of condition 1.

Once we arrive at $ρ=0$ there are four possible paths on which to travel. Here, the one along curve 1+ is uniquely chosen as the tangent vector v₊ because

1. (1)

   The tangent direction toward 1+ is strictly energy decreasing as f₊(s) > 0 because of condition 1.

2. (2)

   The tangent direction toward 1− is strictly energy increasing as f₊(s) < 0 because of condition 1.

3. (3)

   The tangent direction along curve 2 in either direction is energy increasing since f₊(s) < 0 once we move away from $ρ=0$⁠.

Therefore, if we start from a point on curve 2 or 1−, the path from the point to 0 and through curve 1+ to $ρT$ is uniquely self-foldable by U.

Since ρ₀(t) − τ₀ and ρ₃(t) − τ₃ has the same sign, this takes the value 0 only at the target state.

is zero only at t = 0.

Thus, the potential energy given by Eq. (8) uniquely self-folds along a valid rigid folding path from an arbitrary configuration to $ρ=ρT$⁠. □

Equation (8) can be simply realized by using rotational springs with rest angles set to target angles; for edge i, we attach a spring with moment f_i = k(τ_i − ρ_i) (proportional to the angle difference between current and target states). Here, we use the same stiffness k = 2 for all of four creases. Also, this is just an example. We may construct a different set of driving forces with different nonlinear springs such that it also satisfies conditions 1 and 2.

As seen in Theorem 1, the driving force must be carefully designed in order for the self folding to pass through $ρ=0$ without getting caught on any bifurcations of the configuration space manifold. We can generalize the necessary conditions for unique self-foldability at the flat, unfolded state.

Lemma 2 (Perpendicular Constraints). A well-behaved continuous rigid folding$ρ(s)$from the unfolded state$ρ(0)=0$with tangent vectorv₊ is uniquely self-foldable only if the driving forcefat the unfolded state is perpendicular to every tangent vector in the valid tangents not connected to the projection ofv₊ or −v₊ on the unit sphere.

Proof. Consider the spherical projection of valid tangents at the unfolded state, which is a closed set because the configuration space is a closed set. These projected tangents are arc-wise connected to the projection of the vector v₊ or −v₊. Consider a valid tangent vector v_a that is not connected to v₊ or −v₊ and assume, for the sake of contradiction, that v_a is not perpendicular to f(0), i.e., $va·f(0)≠0$⁠. Because the configuration is in a flat state, flipping all mountains and valleys of the valid folding is also a valid folding by symmetry. Therefore, there are two folding paths with tangent ±v_a, one of which must make a positive dot product with f, i.e., f₊ > 0. Since v_a is in a closed domain, there is a vector v_max that is arc-wise connected to v_a that locally maximizes $vmax·f(0)$⁠. Since v_max is not v₊ or −v₊, there is another self-folding motion, which contradicts the uniqueness of our self folding. □

As a consequence of Lemma 2, we are able to get the following important necessary condition.

Lemma 3 (Infinitesimal Dimension Constraints). Consider an origami model at the flat, unfolded state and the tangentvof a desired well-behaved continuous rigid folding. The desired folding is uniquely self-foldable at the flat state only if the dimension m of the solution space of first-order constraints given by Eq.(3)is strictly larger than the number of dimensions n of the linear space spanned by every tangent vectorv_a not connected tovor −vvia valid tangents.

Proof. Since the configuration space is tangent to the linear space defined by Eq. (3), we have that m ≥ n. Assume that m = n, then the first-order solution space is exactly the linear space composed of the vectors v_a. Let $v1,…vn$ be linearly independent vectors spanning this space. Then, v can be written as a linear combination of these vectors. However, if the model is uniquely self-foldable in the direction of v, then, the force f must satisfy $vi·f=0$ for i = 1,…, n, which results in v·f = 0. Therefore, it is not possible to design a driving force f that self-folds in the direction of v such that $v·f>0$⁠. □

Lemma 4 (All Positive Constraints). A well-behaved rigid folding$ρ(s)$from the unfolded state$ρ(0)=0$with tangent vectorv₊, whenv₊ and −v₊ are in separate components of valid tangents, is uniquely self-foldable only if the force at the unfolded state forms non-negative dot product, f₊ ≥ 0, for every tangent vector connected tov₊ via valid tangents.

Proof. Assume that there is a vector v_a arc-wise connected to v₊ through valid tangent but forms negative dot product: v_a·f < 0. Then, its opposite vector −v_a, which is connected to −v+ though valid tangents, satisfies −v_a·f > 0. There is a vector v_max that is arc-wise connected to −v_a that locally maximizes v_max·f(0). Since v_max is not v₊, there is another self-folding motion. □

The combination of these necessary conditions are useful to the design of driving forces that make the model uniquely self-fold in a desired way. However, this is not sufficient even locally at the flat state; if the subset of valid tangents that is connected to v₊ forms a wiggly boundary, it will have another local maximum of f₊. In Sec. 6, we use these conditions to prove or disprove unique self-foldability of a rigid origami based on degree-4 vertices.

In this section, we consider the family of 1DOF rigidly foldable origami generated by connecting vertices of Theorem 1. An obvious example of this is a Miura-ori, but it is not restricted enough for our purposes. A large variety of over-constrained mechanisms can be generated [12–14] by using the linear relationship between opposite fold angles described in tangent of half angle formulas. Such structures also inherit the same nature of the single vertex having separate modes. This makes the structures reconfigurable and reprogrammable into different shapes, but at the same time this makes it difficult for them to self-fold.

Lemma 5. If the interior vertices of a well-behaved continuous rigid folding form an a × b quadrangular grid, then it has exactly a + b linearly independent vectors within the first-order constraints in the flat state.

because $ℓ2$ and $ℓ3$ are not parallel. Consider the global model of a quadrangular grid. If we arbitrarily choose $ρ˙$ for the b edges on the top and the a edges on the left, then this will sequentially determine $ρ˙$ for connecting edges. This leaves an (a − 1) × (b − 1) grid with top and left edges assigned with $ρ˙$⁠. This process sequentially determines the infinitesimal folding angle of each edge. □

As the most simple case, consider connecting two flat-foldable vertices as in Fig. 5. In this case, each vertex can choose its mode from 1 to 2 as in Theorem 1, whose combination yields four possible modes.

Theorem 6. An origami made of two flat-foldable degree-4 vertices is not uniquely self-foldable from a flat state.

where each row represents a seven-dimensional infinitesimal mode. The rank of this matrix is 3. Also, the 3 × 7 subset matrix composed of an arbitrary selection of three rows has rank of 3. Now, assume that one of the modes v is uniquely self-foldable. Then, the left-over tuple of modes form linearly independent vectors of a three-dimensional space. Because of Lemma 5, we have a 2 + 1 = three-dimensional tangent space, and by Lemma 3, v cannot be a unique self-folding mode. □

This means that if there is a self-folding, then, there is always another valid self-folding mode. Comparing the numbers of folding modes and infinitesimal modes provides good insight toward this type of problem.

As another example, consider a regular square twist in Fig. 6 with free mountain-valley assignment. In this structure, all vertices share the same angle $α=(π/2)$ and β, and thus, the coefficients are

Theorem 7. A regular square twist model is not uniquely self-foldable from a flat state.

whose rank is 4. Removing row 1 or 5 does not change the rank of the matrix. Because of symmetry, any submatrix composed of five arbitrary rows is also rank 4. According to Lemma 3, this cannot uniquely self-fold from the flat state. □

This will give only two solutions (σ₁₂, σ₂₃, σ₃₄, and σ₄₁) = (+++−) or (− − −+), visualized in Fig. 7.

Theorem 8. The irregular square twist in Fig. 7 is uniquely self-foldable from and to arbitrary states by a rotational spring force.

In mode 2, the energy function is the summation of functions monotonically decreasing toward the flat-state, which is again monotonically decreasing toward the flat-state. In mode 1, the energy function is the summation of functions monotonically decreasing toward the target state, which is again monotonically decreasing toward the target state. Therefore, the rotational spring energy in Eq. (10) uniquely self-folds to $ρT$ from any configuration. □

Equation (10) can be realized by using rotational springs with rest angles set to target angles: for edge i, we attach a spring with moment f_i = k_i(τ_i − ρ_i) (proportional to the angle difference between current and target states). Here, the stiffness for the creases shared by two interior vertices is k_i = 4 (i = 1…4), and the stiffness for the creases shared only by one vertex is k_i = 2 (i = 5…8).

Finally, we come back to the degree-6 vertex shown in Fig. 1. As already mentioned, the driving force toward the correct mountain and valley assignment, $f=(f,−g,f,−g,f,−g)T$ (f, g > 0), at the flat state can self-fold into either of pop-up or pop-down states since both of them share the same mountain-valley assignment, and thus, this driving force will not uniquely self-fold. We will show an insight toward designing the same vertex to uniquely self-fold into one of the states in Fig. 1.

This vertex with six creases has three degrees-of-freedom in a generic state. For simplicity, we assume threefold symmetry and treat it as a 1DOF mechanism. As this is a more restrictive configuration space, unique self-folding in this set-up does not imply actual self-folding; we will discuss the generalization later.

This forms a configuration space composed of two curves intersecting at the flat state (Fig. 8). The folding path can be simplified as follows:

at the flat state. Thus, a set of weak valley and stronger (approximately 3.73 times stronger) valley force assignments is necessary, instead of a native assignment following that of the target shape (alternating mountains and valleys). In fact, in the symmetric case a proper driving force uniquely self-folds the vertex.

Theorem 9. The regular degree-6 vertex with threefold symmetry constraints is uniquely self-foldable from and to arbitrary states by a rotational spring force.

Thus, if we use the tangent of quarter angle as the axes for the parameter space, the configuration space is represented by straight lines (Fig. 9). As the derivative of $tan(ρ/4)$ with respect to ρ is positive in −180 deg < ρ < 180 deg, the signs of the forward force along a folding path are preserved in this remapped parameter space.

Therefore, our objective is to construct a potential energy that strictly minimizes toward $ρT$ along mode 1 and strictly minimizes to $ρ=0$⁠. We may construct such a rotational spring energy by considering an off-configuration target $ρT′=(τA′,τB′)$⁠, which is, in the remapped parameter space, the intersection point of the ray passing through origin toward the direction of $[2+3,1]T$ and the line passing through $ρT$ perpendicular to the tangent v₁ at $ρT$⁠. Such a point always exists because the folding path exists on a straight line.

This energy is quadratic in this remapped parameter space, and strictly minimizes to $ρ=0$ along mode 2 and strictly minimizes to $ρT$ along mode 1, and thus, energy (17) uniquely self-folds under a threefold symmetry constraint. □

We conjecture that this symmetric path is actually a valid self-folding path even without the symmetry constraint. This can be verified by considering the orthogonal projection f_proj of the driving force f to the solution space of first-order constraints. Figure 10 illustrates the crease lines in the folded position. Applying the spherical laws of sines, we obtain that the vector position of the creases L_i (i = 0,…,5) in mode 1 can be represented as

for some c > 0, i.e., the projected driving force is parallel to the symmetric folding mode 1 for any 0 ≤ t ≤ 0.5 and 0 ≤ t_T ≤ 0.5. This means that the constrained force v·f₁ is maximized in the direction of v = v₁, so there is a symmetric self-folding motion along mode 1. Although we did not check if constrained force along v₁ is the only local maximum, we believe this is so. If it is, the rotational spring force can uniquely self-fold a degree-6 vertex.

In this paper, we proposed a mathematical model of self-folding, specifically for unique-self folding by rotational spring driving forces. We provided necessary conditions for self-foldability that serve as tools to analyze and design self-foldable crease patterns. Using these tools, we demonstrated several results: a degree-4 flat-foldable vertex is uniquely self-foldable; two-vertex and regular twist models based on these vertices are not uniquely self-foldable, but we can design a nonregular twist, that is, uniquely self-foldable. Also, we demonstrated the self-foldability of a degree-6 vertex using the driving force with alternating strong and weak valleys. We believe that these tools can be a basis for future design methods of mechanisms and robotics based on origami. In particular, each of the examples with self-foldability forms a configuration space branching out to different 1DOF mechanisms, while we can assign driving forces that correctly make it choose one of the target modes. This controllability leads to the design of reprogrammable origami systems that can fold into different mechanisms. However, the characterization of self-foldability is still an open problem, and we would like to explore further in this direction.

The first author was supported by the JST PRESTO program and JSPS KAKENHI 16H06106. The second author was supported by NSF grant EFRI ODISSEI 1240441.

Parameters in Crease Patterns and Folded States

e_in =

number of creases in a crease pattern

$ℓi$ =

normalized direction vector of edge i from the vertex in a flat unfolded state

L_i =

normalized direction vector of edge i from the vertex in a folded state

p, q =

coefficients of a degree-4 flat-foldable origami vertex

ρ_i =

fold angle, i.e., the supplementary angle of dihedral angle at the fold line i

v_in =

number of interior vertices in a crease pattern

τ_i =

target fold angle of fold line i

Variables in the Parameter Space

$f(ρ)$ =

driving force (a set of applied moments). Vector

f(s) =

constrained force along v, defined by $f(ρ(s))·v(s)$

s =

an arclength parameter of a folding path

t =

a parameter of a folding path

t =

a vector representing the current configuration by tangents of half fold angles. i-th element is $tan 12ρi$

$U(ρ)$ =

potential energy of the applied moments. Scalar.

v(s) =

normalized tangent vector of a folding path in parameter space

$ρ$ =

a vector representing the current configuration. i-th element is ρ_i

$ρ(s)$ =

a folding path

$ρT$ =

configuration of the target position. i-th element is τ_i, the target folding angle of fold line i

Theorem 10. The configuration space of a degree-4 flat-foldable vertex is the union of Eq.(4).

Proof. For necessity, consider that there exists a folded state, and consider a spherical intersection of the vertex with a unit sphere. This is a four-bar spherical linkage with edge lengths of sector angles.

Now, we can check the symmetry of Eq. (4) by replacing α by π − α or α by β. This means that we can safely assume that α is (one of) the smallest angle(s) of four sector angles.

Also, because the configuration space is symmetric with respect to $ρ=0$⁠, assume that this vertex is popped down as the pop-up state can always expressed by negating all fold angles of pop-down state. From single-vertex rigid foldability [4], the vertex must have three valley creases c_a, c_b, c_c forming sector angles strictly less than π, and 1 mountain crease c_d.

Now, the pop-down state is possible if and only if

1. (1)

   α < π − β and the assignment of c₀, c₁, c₂, c₃ is V, M, V, V.

2. (2)

   α = π − β and the assignment of c₀, c₁, c₂, c₃ is V, 0, V, 0.

3. (3)

   α < β and the assignment of c₀, c₁, c₂, c₃ is M, V, V, V.

4. (4)

   α = β and the assignment of c₀, c₁, c₂, c₃ is 0, V, 0, V.

where M, V, 0 refer to mountain, valley, and uncreased, respectively. The folding of case 2 satisfies mode 1 with p_a = 0. Similarly, the folding of case 4 satisfies mode 2 with p_b = 0.

In case 3, the vertex is expressed as a quadrangle on the sphere whose interior is the front side of the vertex. The quadrangle is convex at c₁, c₂, c₃, and concave at crease c₀, so the geodesic segment between c₀ and c₂ lies inside this quadrangle (Fig. 11). Therefore, segment c₀c₂ divides the quadrangle into two triangle composed of convex angles. Let γ < π denote the length of c₀c₂. Now, we consider the opposite point $c′3$ of c₃. Then, the angle $∠c0c′3c2$ equals π − ρ₃ because $c3c′3=π$⁠. Now, triangle c₀c₁c₂ and $c0c′3c2$ are congruent to each other both of them share segment lengths α, β, γ < π. Because these triangle share the same base angles, $ϕB=π−ϕD$ and $ϕC=π−ϕA$⁠. Then, we get that $ρ0=ϕB+ϕC−π=π−ϕA−ϕD=ρ2$⁠.

Such a folding can be represented by the mode 2 of Eq. (4). Similarly, the folding of case 1 satisfies mode 1 of Eq. (4).

For sufficiency, Eq. (4) satisfies Eq. (1). Therefore, Eq. (4) represents the entire configuration space of the vertex.