On Rigid Origami II: Quadrilateral Creased Papers
OOn Rigid Origami II: Quadrilateral CreasedPapers
Zeyuan He, Simon D. Guest ∗ December 31, 2018
Abstract
This paper describes several new variations of large rigid-foldable quadri-lateral creased papers, which are generated by “stitching” together rigid-foldable Kokotsakis quadrilaterals. These creased papers are constructedwith the following additional requirements: (a) There is at least one rigidfolding motion for which no folding angle remains constant. (b) Thequadrilateral creased paper is infinitely extendable in both longitudinaland transverse directions. (c) The sector angles, which define the creasedirections, can be solved quadrilateral by quadrilateral. This work is basedon a nearly complete classification of rigid-foldable Kokotsakis quadrilat-erals from Ivan Izmestiev. All quadrilateral creased papers described inthis paper have one degree of freedom in each branch of their rigid foldingmotion.
Keywords : rigid-foldable, folding, degree-4, Kokotsakis quadrilateral
The most common forms of origami pattern are based on a quadrilateral mesh.In this article we will discuss the rigid-foldability of a quadrilateral creasedpaper, which is more precisely defined as
Definition 1. A quadrilateral creased paper satisfies: (see Figure 1(a))(1) The Euler characteristic is 1.(2) The crease pattern is a quadrilateral mesh.(3) No row or column of inner creases forms a cycle.A vertex or crease is inner if it is not on the boundary of the creased paper, apanel is inner if none of its vertex is on the boundary of the creased paper.We note that the rigid-foldability defined in [1] briefly means a continuumof the rotation of rigid panels along the inner creases without self-intersection,which does not require all the inner creases to be folded. However, such innercreases seem redundant in most cases. Thus in this article we only consider thecase where all the inner creases are folded. ∗ [email protected], [email protected]. Department of Engineering, University of Cam-bridge, Cambridge CB2 1PZ, United Kingdom. a r X i v : . [ m a t h . M G ] D ec a) (b) Figure 1: (a) shows a rigid-foldable quadrilateral creased paper, which is themain object of study in this article. The polyhedral surface bounded by a dashedcycle is a Kokotsakis quadrilateral. (b) is an example of a “ring” structuredisallowed by Remark 2; such systems will be discussed in a future article.
Remark 1.
We do not require a creased paper to be either developable orflat-foldable. A developable creased paper has a rigidly folded state where allthe folding angles are zero (at this planar state the Gaussian curvature is 0everywhere). A flat-foldable creased paper has a rigidly folded state where allthe folding angles are ± π . Remark 2.
Here we require the Euler Characteristic to be 1, which makes thequadrilateral creased paper 1-connected and topologically a disk (or a “paper”).We also require that no row or column of inner creases form a cycle, otherwisethe rigid-foldability condition will be different from Theorem 1. An example ofsuch a disallowed system is the “ring” structure shown in Figure 1(b). In thiscase the rigid-foldability condition will heavily rely on the length (number ofedges) of the cycle. Actually, when dealing with stacking [2] or cylindrical [3]structures, the analysis of such “ring” structure is indeed necessary, and will bediscussed in a future article of this series.There have been many results concerning a rigid-foldable quadrilateral creasedpaper. The majority of them focus on a developable and flat-foldable quadri-lateral creased paper, which is an important subset of the quadrilateral creasedpapers considered here. Tachi [4] first described the sufficient and necessarycondition for a developable and flat-foldable quadrilateral creased paper to berigid-foldable, and other papers followed with further examples and design meth-ods [5, 6]. If we go out of this subset, [7] introduced some non-developable andflat-foldable quadrilateral creased papers, while [8, 9, 10, 11] introduced somedevelopable and non-flat-foldable quadrilateral creased papers.We want to provide a full description of rigid-foldable quadrilateral creasedpapers, without any restriction on the developability or flat-foldability. In thisarticle we apply general tools developed in the first paper in this series to thispractical problem [1]. In that paper, where we put forward a theoretical frame-work to discuss rigid origami, and review important previous results, we provideall the underlying definitions used in the present paper.2efore coming to the main text we want to discuss why quadrilateral creasedpapers are a common object of study. Experience shows that, a more genericcondition can be less useful, so it is reasonable to start from a paper with simplecrease pattern. First we note that we don’t consider a paper with holes, since wehave shown that the constraints around holes are more complex than vertices[1]. Then there are two aspects that need to be justified: why we choose allthe inner vertex to be degree-4, and why we choose all the inner panels to bequadrilateral.Concerning degree-4 vertices, for a paper without holes, we have demon-strated that the configuration space of the whole creased paper is essentiallyconstrained by a system of polynomial equations, and is the intersection of theconfiguration space of all its single-vertex creased papers [1] (a degree-n single-vertex creased paper is the restriction of a creased paper on panels incident toa degree- n inner vertex). Specifically, there are three polynomial constraints onfolding angles in a single-vertex creased paper. We have shown that a single-vertex creased paper with no more than three inner creases is not rigid-foldable,except when two of them are co-linear. Based on the analysis above, it is rea-sonable to start with a creased paper that only contains degree-4 inner vertices.Concerning quadrilateral inner panels, suppose there are V vertices, E creasesand F panels. from the Euler’s formula V − E + F = χ (1)where χ is the Euler characteristic of the creased paper. When V is large, E ∼ V , so F ∼ V . That means if we want a creased paper to be freelyextended or refined when every inner vertex is degree-4, every inner panel shouldon average be quadrilateral. While there exist hybrid degree-4 creased paperswhose inner panels are, for instance, a tiling of triangles and hexagons (such asin the kagome pattern), it is reasonable to start with a creased paper whose innerpanels are all quadrilaterals. Note that the vertices, creases and panels incidentto the boundaries do not place additional constraints on the configuration spaceof a creased paper if ingoring self-intersection.From another perspective, any rigid origami is essentially a real algebraicsystem on all the folding angles, which means if the number of variables is muchmore greater than the number of constraints, with possibility 1 this creasedpaper is rigid-foldable. Note that the way of counting the constraints is providedin Section 4.1 of [1]. Therefore the rigid-foldability of an under-constrained rigidorigami, such as a triangular creased paper, is not so interesting. On the otherhand, analyzing the rigid-foldability of an over-constrained rigid origami is tofind the collection of singular points in the associated algebraic system, whichis more intriguing and difficult.From the next section we will start to discuss the collection of rigid-foldablequadrilateral creased papers. In order to present further results we need somepreliminaries. 3 a) c α (b) α α α c c c ρ ρ ρ ρ α α α α ρ ρ ρ ρ - ρ ξ Figure 2: (a) A degree-4 single-vertex creased paper. We label these sectorangles and corresponding inner creases counter-clockwise. (b) Two non-trivialrigidly folded states with the outside edge of the single-vertex creased paperdrawn on a sphere as arcs of great circles, assuming the panel corresponding to α is fixed when we change the magnitude of ρ . Generically, there are two setsof folding angles ρ , ρ , ρ , which are symmetric to ξ and colored by red andblue. As shown in Figure 2, a degree-4 single-vertex creased paper has four sectorangles α , α , α , α and four folding angles ρ , ρ , ρ , ρ whose correspondinginner creases are c , c , c , c . To study the rigid-foldability of a quadrilateralcreased paper, the first step is to clarify the relation among the folding anglesaround its each degree-4 vertex, which is the main part of the configuration space of a degree-4 single-vertex creased paper. The configuration space is describedas the collection of all possible folding angles under a given set of sector angles,as well as the order of stacking of panels at some special rigidly folded states[1]. Here the information on order (we call it the order function ) is not asimportant as the relation between folding angles, because we can always checkpossible orders of stacking when the sector and folding angles are given.In Section 2 of [12], a complete analysis on the configuration space of adegree-4 single-vertex creased paper is presented. In this paper the relations areexpressed in terms of tan ρ i in a compact form, which involves complex analysisand elliptic function. Note that we assume the sector angles < α i < π . Thisis reasonable because if a sector angle equals to π (without loss of generality,suppose α = π ), then α + α + α ≥ π ;(1) if α + α + α = π , ρ = ρ ; ρ = ρ = 0 .(2) otherwise, ρ = ρ ; ρ , ρ = { a, b } or {− a, − b } , where a, b are non-zeroconstants. 4 a) (b) This foldingangle is 0
Figure 3: (a) The initial rigidly folded state of a quadrilateral creased paperdescribed in Proposition 1, where a folding angle is . (b) shows the finalrigidly folded state of (a), where some panels clash and the rigid folding motionhalts. The mountain and valley creases are colored red and blue. (a) and (b)are plotted by Freeform Origami [4].which is essentially folding along a crease. Additionally, if a sector angle isgreater than π (without loss of generality, suppose α > π ), there is no essentialdifference if substituting α with π − α , which means for a set of sector angles < α i < π , changing α i to π − α i will only result in greater possibility ofself-intersection of panels. Here we will introduce the sufficient and necessary condition (also called theloop condition [4]) for a quadrilateral creased paper to be rigid-foldable. Westart with a special case, where each inner vertex is degree-4, but there is noinner panel.
Proposition 1. (From Theorem 5 in [1]) If a creased paper satisfies:(1) It has Euler Characteristic 1 and each inner vertex is degree-4.(2) The interior of crease pattern is a tree.(3) All the sector angles are in (0 , π ) , and for each inner vertex, the four innercreases does not form a cross.then this creased paper is generically rigid-foldable. Especially, if the creasedpaper described above is also developable, at the planar folded state (where allthe folding angles are ) it is rigid-foldable.Proposition 1 gives a way to design a degree-4 creased paper with no cycle inthe interior of the crease pattern, as shown in Figure 3(a) and 3(b). Note thatthis excludes some non-generic cases [1], such as the example shown in Figure4(a) and 4(b). 5 a) (b))) ρ Figure 4: (a) and (b) are the front and rear perspective views of a creased paper,where two single-vertex creased papers share two panels. The sector angles areshown in degrees. This rigidly folded state is not rigid-foldable because thecommon folding angle ρ cannot exceed π/ in the top single-vertex creasedpaper, and cannot be below π/ in the bottom single-vertex creased paper,although both of the single-vertex creased papers are rigid-foldable. This figureis replicated from [1]. Remark 3.
Some creased papers have neither inner panels nor inner vertices,and can be regarded as piecewise 1-dimensional origami. Some results have beengiven in sections 11.2 and 11.3 of [13], but we will not consider such creasedpapers further in this article.The creased paper described in Proposition 1 is like a “tree” structure, andin the next theorem we will show that this tree structure is closely related tothe rigid-foldability of a quadrilateral creased paper.
Theorem 1.
Consider a quadrilateral creased paper. If and only if(1) the corresponding tree structure (see Figure 5(b)) is rigid-foldable in aclosed interval of all the folding angles;(2) for all i, j , the following equation is satisfied simultaneously in this closedinterval θ i,j ≡ φ i,j (2)then the quadrilateral creased paper is rigid-foldable in this closed interval.Figure 5 defines the folding angles θ i,j and φ i,j . Proof.
As it is straightforward to move between the quadrilateral creased paperand the tree structure by cutting or gluing, the sufficiency and necessity arenatural.
Remark 4.
Any creased paper may have a number of branches in its rigidfolding motion [1]. For a developable quadrilateral creased paper, we can dis-tinct different branches of rigid folding motion from different mountain-valleyassignment. This is because for each branch of every degree-4 single-vertexcreased paper, the sign of every folding angle remains the same when the rigidfolding motion moves away from . Each inner vertex should be incident to6 a) (b) column 1row 1row 2 column 2column 3 θ ϕ θ θ θ θ θ ϕ ϕ ϕ ϕ ϕ ρ ρ ρ column 4 Figure 5: (a) shows a rigidly folded state of a quadrilateral creased paper, and(b) is its corresponding tree structure, where we “cut” inner creases connectingadjacent columns to make the interior of crease pattern have no cycle. ρ j arethe folding angles of the top row. θ i,j , φ i,j and their corresponding inner creasesare colored red and blue, respectively.three mountain creases and one valley crease, or three valley creases and onemountain crease. However, mountain-valley assignment is not applicable forthe distinction of the branches of rigid folding motions of a non-developablequadrilateral creased paper. Remark 5.
The quadrilateral creased paper described in Theorem 1 has thesame rigid folding motion as its corresponding tree structure, therefore generi-cally it has one degree of freedom (1-DOF) in each branch of the rigid foldingmotion. Nevertheless, the degree of freedom might be greater than 1 at somespecial points in the configuration space, such as the planar and flat (all thefolding angles are ± π ) rigidly folded states. This can be examined from calcu-lating the rank of Jacobian of constraints on the configuration space providedin [14]. Definition 2. A Kokotsakis quadrilateral is a polyhedral surface in R , whichconsists of one center quadrilateral (the base); four side quadrilaterals, oneattached to each side of this center quadrilateral; and four corner quadrilateralsplaced between each two outer consecutive side quadrilaterals (see Figure 1(a)).From Theorem 1, the next corollary is natural.7 orollary 1.1. A quadrilateral creased paper is rigid-foldable if and only ifthe restriction of a rigidly folded state on each Kokotsakis quadrilateral is rigid-foldable.Now we explain the process of “stitching”.
Definition 3.
For a Kokotsakis quadrilateral Q , we say that another Kokot-sakis quadrilateral Q can stitch with Q in the longitudinal or transverse direc-tion when Q can share two vertices with Q in the longitudinal or transversedirection.The rigorous work in [12], gives a nearly-complete classification of all rigid-foldable Kokotsakis quadrilaterals. Each type of rigid-foldable Kokotsakis quadri-lateral is described by a system of equations (most of them are trigonometric,some are exponential or elliptic) on its sector angles, although a solution is notguaranteed. Therefore given any quadrilateral creased paper, we can interpretit by stitching together Kokotsakis quadrilaterals, either from the same or dif-ferent types. In other words, if we can find all possible stitching of rigid-foldableKokotsakis quadrilaterals, we will get the complete description of a rigid-foldablequadrilateral creased paper. To our interest, in the area of rigid origami, wehope to find as many methods as possible to construct a large quadrilateralcreased paper that satisfies the following three extra conditions.(1) It has at least one non-trivial rigid folding motion. Definition 4.
Following the terminology introduced in [12], A rigid fold-ing motion of a creased paper is called trivial if a folding angle remainsconstant in this rigid folding motion. A trivial rigid-foldable Kokotsakisquadrilateral only has trivial rigid folding motion.We think this requirement is reasonable because when designing a creasedpaper, a not folded crease seems redundant. If a large quadrilateral creasedpaper contains a trivial rigid-foldable Kokotsakis quadrilateral, any rigidfolding motion of this large quadrilateral creased paper will be trivial,therefore we exclude from consideration the 4 trivial rigid-foldable Kokot-sakis quadrilaterals (Type 3.8 in [12]).(2) It is infinitely extendable in both longitudinal and transverse directions.We require the technique of constructing this creased paper can be ap-plied to an m × n ( m, n → ∞ ) quadrilateral mesh, which also means thequadrilateral mesh can be arbitrarily refined. This requirement enlargesthe range of application of a quadrilateral creased paper, such as to ap-proximate a target surface by 1-DOF rigid origami [15].(3) The sector angles can be solved quadrilateral by quadrilateral, i.e. thedesign of the entire quadrilateral creased paper never requires the solu-tion of equations where variables are sector angles from more than oneKokotsakis quadrilateral. 8e raise this requirement mainly because even if a quadrilateral creasedpaper is designed from possible stitchings, generically the number of totalconstraints will be much more than the number of variables as the size in-creases. A compromise is making the design process to be as follows. Startfrom a Kokotsakis quadrilateral. Set its sector angles as known variablesand solve the sector angles of the neighboring Kokotsakis quadrilaterals,quadrilateral by quadrilateral. Continue doing this until all the sectorangles are solved. Two advantages of this requirement are:(a) each step completed by the above process will result in a rigid-foldablecreased paper.(b) when adding a new Kokotsakis quadrilateral, we don’t need to adjustthe solved part because there is enough design freedom.In the next section, we will demonstrate several methods to construct alarge rigid-foldable quadrilateral creased paper satisfying the three additionalrequirements above. We understand that this method will not be comprehensive,i.e. there may be large quadrilateral creased papers that can only be solved asa whole, but it is unlikely practicable to find such cases. First we provide a useful conclusion, that all types of large quadrilateral creasedpapers presented in this section are also rigid-foldable after “switching somestrips”, as described below.
Definition 5.
We define the operation of switching a strip as being to replaceall the sector angles on a column or a row of panels by their complements to π .Here we have extended the definition of switching a boundary strip in [12]. (seeFigure 6) Theorem 2.
Switching a strip preserves the rigid-foldability of a quadrilateralcreased paper.
Proof.
We first consider switching a transverse strip as shown in Figure 6.The sector angles on panels colored grey are replaced by their complementsto π . We use x, y, z, u, v, w, r, s to represent the tangent of half of the cor-responding folding angles ( tan ρ ) on these labeled inner creases. FollowingLemma 4.5 in [12], we know after switching a strip { x, y, z, u, v, w, r, s } →{ x, − y, z, u, − v, w, − r − , − s − } . The other related folding angles on this stripare changed in the same way. Therefore from Theorem 1, switching a strippreserves the rigid-foldability of a quadrilateral creased paper. The proof forswitching a longitudinal strip is similar.Now we list some possible types of rigid-foldable quadrilateral creased pa-pers. The justification for this categorization is provided in Section 5.9 a) (b) xyuw -r -1 -s -1 zv rs x -yuw z-v Figure 6: (a) is a quadrilateral creased paper. We use x, y, z, u, v, w, r, s to represent the tangent of half of the corresponding folding angles ( tan ρ )on these labeled inner creases. In (b), the sector angles on panels coloredgrey are replaced by their complements to π , and after switching a strip { x, y, z, u, v, w, r, s } → { x, − y, z, u, − v, w, − r − , − s − } . The other related fold-ing angles on this strip are changed in the same way. Generically, switching astrip will make a developable creased paper non-developable. (a) (b) α π - α α α π - α α β β β β β γ π - γ γ π - γ γ π - γ γ π - γ γ π - γ γ π - γ γ π - γ γ π - γ γ π - γ γ π - γ γ γ γ γ γ Figure 7: An example of the orthodiagonal type. (a) shows the relation amongthe sector angles. The geometry here is not representive. (b) is a rigidly foldedstate of the developable case described in Remark 6. Each column or row ofinner creases are co-planar, and each plane formed by a column of inner creasesis orthogonal to each plane formed by a row of inner creases. (b) is plotted byFreeform Origami [4], where the mountain and valley creases are colored redand blue. 10 .1 Orthodiagonal
For this case, as shown in Figure 7(a), the independent input sector angles are α j ( j ∈ [0 , n ] ) and β i ( i ∈ [1 , m − ). The other sector angles γ ij can be solvedone by one as follows.(1) In the first row, cos α cos γ = cos α cos β cos α j − cos γ ,j = cos α j cos γ ,j − , j ∈ [2 , n ] (3)(2) From the second row, tan β i − tan β i = tan γ i − ,j tan γ i,j , i ∈ [1 , m − , j ∈ [1 , n ]cos β i cos γ i +1 , = cos β i +1 cos γ i, , i ∈ [1 , m − γ i,j − cos γ i +1 ,j = cos γ i,j cos γ i +1 ,j − , i ∈ [1 , m − , j ∈ [2 , n ] (4)Switching some strips will give another orthodiagonal quadrilateral creasedpaper. Typical geometric features of this category are:(1) Each column or row of inner creases are co-planar.(2) Each plane formed by a column of inner creases is orthogonal to each planeformed by a row of inner creases. Hence the name “orthodiagonal”. Remark 6. If β = π − α and β i +1 = π − β i ( i ∈ [1 , m − ), this quadrilateralcreased paper will also be developable, as for the example shown in Figure 7(b).If β = α and β i +1 = β i ( i ∈ [1 , m − ), this quadrilateral creased paper willalso be flat-foldable. This category is the union of flat-foldable quadrilateral creased paper describedin [4] and its variation from switching some strips. Typical geometric featuresof this category are:(1) If at each inner vertex the sum of opposite sector angle equals π , thisisogonal type is flat-foldable. Otherwise it is not flat-foldable.(2) The absolute value of folding angles on a row or column of inner creasesare equal.This category is named isogonal because at every inner vertex opposite sectorangles can be equal after switching some strips. Note that the property of “arigidly folded state can guarantee a rigid folding motion” described in [4] isspecial for the isogonal type because here the existence of a rigidly folded stateis equivalent to equation (2). For other types the rigid-foldability condition onsector angles is more complex, which makes similar conclusion not useful.11 ρ ρ ρ ρ ρ ρ (a) (b) α β γ δ α β γ δ α β γ δ α β γ δ α β γ δ α β γ δ Figure 8: (a) shows how we label the sector angles of a Kokotsakis quadrilateral.(b) is a linear unit, we label the sector and folding angles to illustrate the linearrelationship among the tangent of half of the folding angles. In both (a) and(b) the geometries are not representative.
This category is formed by stitching linear units to generate a row of Kokotsakisquadrilaterals in the transverse direction, and repeating this row in the longitu-dinal direction. A linear unit is the union of two degree-4 single-vertex creasedpapers sharing two panels (see Figure 8(b)). The reason for this name is, foreach linear unit we have tan ρ c tan ρ and tan ρ c (cid:48) tan ρ , where c, c (cid:48) are constants depending on the sector angles. There are 5 linear units denotedby types 5.1–5.5, following the notation in Section 3.5 of [12], where the relationamong sector angles and the value of linear coefficient c are also provided (seeFigure 9). To allow repeating, κ and κ in the linear unit 5.1 should eitherboth be sin α − β / sin α + β or both be cos α − β / cos α + β . When stitching linearunits to form a row, they should be placed in their rotated position , which isorthogonal to the regular position shown in Figure 9. Definition 6.
Stitching a linear unit with another linear unit means these twolinear units form a Kokotsakis quadrilateral. They do not share vertices, butthe folding angles on the matching creases must be the same. Linear unit 1 canstitch with linear unit 2 if δ + δ + δ + δ = 2 π and corresponding constants c = c (cid:48) . We say two linear units are stitched in the linear direction .Here the linear direction is transverse, which is why we call this type trans-verse linear repeating . To allow repeating, apart from requiring δ + δ + δ + δ =2 π , we should also set γ + γ + γ + γ = 2 π (See Figure 8(a)). With the restric-tion above and given suitable input sector angles, the rest of the sector anglescan be solved numerically. An important notice is there might be “degenera-tion” (which is discussed in Section 6) in the stitching of linear units. Thesedegenerations can be found both analytically and numerically, and we providethis information in Table 1. Although it will not result in negative results, therelation among sector angles will be more special. In order not to make redun-dant classification, we will not split the this type into smaller classes. Someexamples are shown in Figure 10. 12 π - α α π - β α α π - α π - α γ α π - β β π - α δ π - δ δ π - δ α α δ δ α δ α δ β α α β α β β α β π - α α π - β α π - β β π - α α δ β γ α δ β γ α δ β γ α δ β β α α β α β β α Figure 9: We use 5.1 to 5.5 to represent different types of linear units. Specificrelations among the sector angles of each linear unit and the value of linearcoefficient c are listed in Section 3.5 of [12]. Types 5.1(b), 5.2(b), 5.3(b) and5.5(b) are developable, while other types are non-developable.2(a) 2(b) 3(a) 3(b) 5(a) 5(b)2(a) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Table 1: This table shows when the stitching of linear units degenerates. (cid:13) means the Kokotsakis quadrilateral formed by two linear units will degenerateto a Kokotsakis quadrilateral in the parallel repeating type (Figure 10(a)). (cid:13) means it will degenerate to a Kokotsakis quadrilateral in the orthodiagonal type(Section 4.1). (cid:13) means both (cid:13) and (cid:13) are possible. The difference between transverse and longitudinal linear repeating is that thelinear direction is respectively transverse and longitudinal. Here the linear unitsare in the regular position . We label a linear unit in each case with a dashedred cycle in Figure 10(a) and 11(a). Longitudinal linear repeating is formed bystitching linear units 5.1 and 5.3 to generate a row of Kokotsakis quadrilateralsin the transverse direction, and repeating this row in the longitudinal direction.To allow repeating, apart from requiring δ + δ + δ + δ = 2 π , we should alsoset α + α + α + α = 2 π (See Figure 8(a)). Here κ and κ in the linearunit 5.1 should either both be sin α − β / sin α + β or both be cos α − β / cos α + β .An example is shown in Figure 11(a) and 11(b). This category is formed by stitching type 3.2 to form a row of Kokotsakisquadrilaterals, and repeating this row in the longitudinal direction. To allowrepeating, apart from requiring δ + δ + δ + δ = 2 π , we should also set α + α + α + α = 2 π (See Figure 8(a)). An example is shown in Figure 11(c)and 11(d). 13 a) (b)(c) (d)(e) (f) β α π - α π - δ γ δ π - β π - γ β α π - α π - δ γ δ π - β π - γ Figure 10: Examples of the transverse linear repeating, which is formed bystitching a row of linear units, and repeating this row in the longitudinal di-rection (orthogonal to the linear direction). We label all the sector angles indegrees. (a) is formed by stitching “parallel linear units”, which is bounded bya red dashed cycle. We name this type as “parallel repeating”. This is a typicalspecial rigid-foldable quadrilateral creased paper, and some stitchings in thetransverse linear repeating will degenerate to this case. (b) is a rigidly foldedstate of (a). (c) is formed by 5.1(b)–5.2(b)–5.2(b)–5.1(b). (d) is a rigidly foldedstate of (c). (e) is formed by 5.1(b)–5.3(b)–5.5(b)–5.1(b). (f) is a rigidly foldedstate of (e). The independent input sector angles are colored red in (a), (c) and(e). Other sector angles are solved numerically. (b), (d) and (f) are plotted byFreeform Origami [4], where the mountain and valley creases are colored redand blue. 14 c) (d)(e) (f)(a) (b)
Figure 11: (a) is an example of the longitudinal linear repeating, formed bystitching linear units 5.1 and 5.3 to generate a row of Kokotsakis quadrilaterals,and repeating this row in the longitudinal direction (“along” the linear direction).Here a linear unit 5.1(b) is bounded by a dashed red cycle to show the differenceof linear direction in the transverse and longitudinal linear repeating. (b) is arigidly folded state of (a). (c) is an example of conic repeating, formed bystitching type 3.2 to generate a row of Kokotsakis quadrilaterals, and repeatingthis row in the longitudinal direction. (d) is a rigidly folded state of (c). (e)is an example of the hybrid type. The first Kokotsakis quadrilateral is type6.1, its adjacent row and column are formed by stitching of type 1-2, the restpart is of isogonal type. (f) is a rigidly folded state of (e). The independentinput sector angles are colored red in (a), (c) and (e). Other sector angles aresolved numerically. We label all the sector angles in degrees. (b), (d) and (f)are plotted by Freeform Origami [4], where the mountain and valley creases arecolored red and blue. 15 a) (b) Figure 12: (a) is an example of the parallel mixed type, where the independentinput sector angles are colored red. Other sector angles are solved numerically.We label all the sector angles in degrees. The upper part is a linear repeatingtype shown in Figure 10(e), and the lower part is a parallel repeating typeshown in Figure 10(a). (b) is a rigidly folded state of (a), plotted by FreeformOrigami [4]. The mountain and valley creases are colored red and blue. Wealso show another mountain-valley assignment in the upper part compared toFigure 10(e).
This category is formed by the following steps. An example is shown in Figure11(e) and 11(f).(1) Choose the first Kokotsakis quadrilateral from types 1-1, 6.1, 6.2, 7.2, 7.3,7.7 or 7.8.(2) In both the row and column incident to the first Kokotsakis quadrilat-eral, choose types 1-1, 1-2, 1-3, or 1-5 to stitch with the first Kokotsakisquadrilateral.(3) Solve the rest of this quadrilateral creased paper as we do for the isogonaltype.
This type is provided in [9] and [11]. It is developable and has 3 independentinput sector angles α, β, γ . Setting δ = 2 π − α − β − γ , the sector angles ofthis creased paper are α, β, γ, δ, π − α, π − β, π − γ or π − δ . At each vertex,the sector angles are α, β, γ, δ or π − α, π − β, π − γ, π − δ . We can interpretthe tiling type as a combinatorial special solution of the stitching of types 3.2,which is not formed by repeating. 16 .8 Parallel Mixed Given a rigid-foldable quadrilateral creased paper, this type is formed by stitch-ing a parallel repeating type (see Figure 12) in the longitudinal direction.
In this section we will describe some of the underlying mathematical structurethat leads to the different types presented in Section 4. This section heavilyrelies on the classification of rigid-foldable Kokotsakis quadrilaterals from IvanIzmestiev. Throughout this section, we make extensive use of the labellingscreme presented in Section 3 of [12]. Specifically, 1.1, 1.2; 2; 3.1, 3.2; 4.1,4.2; 5; 6.1–6.3; 7.1–7.10. Each type 5 consists of two of the five linear units described in Section 4.3, we describe a type 5 as, for instance, 3(a)–5(b), or as,for instance, 2-2 when we do not need to denote (a) or (b).From Section 2.4 of [12], a degree-4 single-vertex creased paper is classifiedas follows, based on different types of the configuration space.
Definition 7.
A degree-4 single-vertex creased paper with the sector angleslabeled as α, β, γ, δ is said to be:(1) of elliptic type, if equation (5) has no solutions;(2) of conic type, if equation (5) has exactly one solution;(3) a deltoid , if it has two pairs of equal adjacent sector angles, and an an-tideltoid , if it has two pairs of adjacent sector angles complementing eachother to π ;(4) an isogram , if it has two pairs of equal opposite sector angles, and an antiisogram , if it has two pairs of opposite sector angles complementingeach other to π . α ± β ± γ ± δ = 0 (5)When stitching different types of rigid-foldable Kokotsakis quadrilateral, ver-tices that are shared between the quadrilaterals must of course be of the sametype. Note that if two Kokotsakis quadrilaterals cannot stitch with each other,switching a strip (see Definition 5) cannot make them stitchable since it doesnot change the type of a vertex, at most switching from deltoid to antideltoid,or isogram to antiisogram. We also need to check whether other relations amongthe existing sector angles are compatible with the new Kokotsakis quadrilateralwe plan to stitch.Type 1.1 can only stitch with itself, forming one category of large rigid-foldable quadrilateral creased paper, called orthodiagonal (Section 4.1). Thegeometric properties mentioned in Section 4.1 can be examined directly. Wehave considered possible rotation of Kokotsakis quadrilaterals, but it does notincrease the variation.For type 2, stitching with itself forms one category of large rigid-foldablequadrilateral creased paper, called isogonal (Section 4.2). Other stitching pos-sibilities of type 2 are included in the discussions on type 5.For other types we provide a Proposition to a further discussion.17 roposition 2. (1) For any type of rigid-foldable Kokotsakis quadrilateral, ifthe 8 sector angles around two adjacent vertices are given and compatiblewith the constraints on sector angles, there is enough design freedom tosolve for the other 8 sector angles.(2) For types 5, 6, 7.2, 7.4, 7.5, 7.7, 7.8 and 7.9, if the given 12 sector anglesaround three vertices are compatible with the constraints on sector angle,then if the unsolved vertex is an (anti)isogram, there is enough designfreedom for the other 4 sector angles.(3) For type 7.6, if the given 12 sector angles around three vertices are com-patible with the constraints on sector angles, then if the unsolved vertexis elliptic, there is enough design freedom for the other 4 sector angles.(4) For other types, even if the given 12 sector angles around three verticesare compatible with the constraints on sector angles. Generically there isno solution for the other 4 sector angles. Proof.
The equations of rigid-foldability condition on sector angles in each typeof 1.2 and 3–7 are independent. If the 8 sector angles around two adjacentvertices are given without contradiction to the constraints on sector angles, thenumber of remaining equations is no more than 8, and we say there is enoughdesign freedom for the other 8 sector angles. If we are given 12 sector anglesaround three vertices which are compatible with the constraints on sector angles,only for the cases mentioned in (2) and (3) the number of remaining equationsis 4, otherwise it is greater than 4. In this sense, we say generically there is nosolution for the other 4 sector angles.Proposition 2 only provides an estimate. However, we have verified thisestimate numerically. Apart from the cases given in Proposition 2(2) and 2(3),we have not found it to be possible to obtain a large quadrilateral creased paperjust by solving equations quadrilateral by quadrilateral. Therefore our nextstep is to consider special solutions. Note that from Theorem 2, many othervariations based on these special solutions can be obtained by switching somestrips.
For special solution 1, the method is to stitch compatible categories of rigid-foldable Kokotsakis quadrilaterals in one row by solving equations quadrilateralby quadrilateral, and repeating this row in the longitudinal direction to obtaina large quadrilateral creased paper, which makes the creased paper rowwise-periodical. From Proposition 2 we know solving a row is possible, and whenconstructing special solutions in the longitudinal direction, we should considerthat each time when we solve 8 sector angles based on 8 known sector angles,the number of additional constraints to allow such special solutions plus thenumber of original constraints should not exceed 8. We will explain why onlyusing repeating here in Section 5.3.Consider generating the row of Kokotsakis quadrilaterals for repeating, apartfrom the initial quadrilateral, each time we add a quadrilateral, solve 8 sectorangles based on 8 known sector angles. Therefore we should first analyze whichtypes of rigid-foldable Kokotsakis quadrilateral can generate a new rigid-foldable18okotsakis quadrilateral by repeating, if the 8 sector angles around two adjacentvertices are given. We note that the Kokotsakis quadrilaterals described in[12] might be able to repeat either “horizontally” or “vertically”. To give usmaximum freedom, and allow either type of repeating, here we describe theplacement as “regular” if the placement is aligned with the description in [12], or“rotated” if the placement is orthogonal. For regular position, an extra condition( α + α + α + α = 2 π ), must be added to allow repetition because the sum ofsector angles on each inner panel should be π . For rotated position, the extracondition becomes γ + γ + γ + γ = 2 π . Proposition 3.
The following statements describe, for each type of rigid-foldable Kokotsakis quadrilateral, whether it is possible to solve 8 sector anglesnumerically based on 8 known sector angles such that after repeating we willstill obtain a rigid-foldable Kokotsakis quadrilateral.(1) For types 3.1 and 3.2, by repeating we can obtain another rigid-foldableKokotsakis quadrilateral.(2) For type 1-1, at each of 4 vertices there is a coefficient κ , which can bechosen from sin α − β / sin α + β or cos α − β / cos α + β . If the number of eachchoice is 0, 2 or 4, for either the regular or rotated position by repeatingwe can obtain another rigid-foldable Kokotsakis quadrilateral, otherwisewe cannot obtain a rigid-foldable Kokotsakis quadrilateral.(3) For type 5 (excluding 1-1), for the rotated position (where the linear di-rection is transverse) by repeating we can obtain another rigid-foldableKokotsakis quadrilateral, if κ and κ in the linear unit 5.1 are both sin α − β / sin α + β or both cos α − β / cos α + β .(4) For types 1-3, for the regular position (where the linear direction is lon-gitudinal) by repeating we can obtain another rigid-foldable Kokotsakisquadrilateral, if κ and κ in 5.1 are both sin α − β / sin α + β or both cos α − β / cos α + β . However, for other units in type 5 in the regular position gener-ically we cannot obtain a rigid-foldable Kokotsakis quadrilateral.(5) For type 7.1, for either the regular or rotated position, by repeating wecan obtain another rigid-foldable Kokotsakis quadrilateral.(6) For types 7.4 and 7.5, for the rotated position by repeating we can ob-tain another rigid-foldable Kokotsakis quadrilateral, while for the regularposition generically we cannot obtain another rigid-foldable Kokotsakisquadrilateral.(7) For other types generically after repeating we cannot obtain a rigid-foldableKokotsakis quadrilateral. Proof.
Statements (1) and (5) can be verified directly.Statements (2), (3) and (4): For type 5, repeating in either the regularposition or the rotated position preserves the linear relations among the tan-gent of half of the folding angles. For linear unit 5.1, if κ and κ are both sin α − β / sin α + β or both cos α − β / cos α + β , the linear coefficient c remains thesame after repeating in either the regular position or the rotated position, oth-erwise only the sign of c is changed. For linear unit 5.3, c remains the same after19epeating in either the regular position or the rotated position. For linear units5.2–5.5, c remains the same after repeating in the rotated position, while in theregular position changes its value. Since when two linear units stitch with eachother the linear cofficient must be the same, we can draw these conclusions.Note that type 3-3 is not suitable for repeating, which is explained in statement(7).Statement (6): Repeating for the rotated position can be verified directly.Repeating for the regular position needs two extra constraints for rigid-foldability,and the total number of constraints will exceed 8.Statement (7): For all the other types, either the additional constraints toallow repeating contradicts with the original constraints on sector angles, or thenumber of additional constraints to allow repeating plus the number of originalconstraints on sector angles is greater than 8. In this sense we say genericallythese types are not suitable for repeating.From Proposition 3 we know the possible types that can be used to stitchwith other types, which also determines the first Kokotsakis quadrilateral in thisrow. However, our numerical studies have lead to the following restrictive ob-servations about Kokotsakis quadrilaterals. We will discuss these Observationsin Section 6. Observation 1.
We only find non-real numerical solution for Kokotsakis quadri-laterals of types 3.1, 6.3, and 7.9.
Observation 2.
The only numerical solution we have found for a Kokotsakisquadrilateral of type 7.1 degenerates to a Kokotsakis quadrilateral in the “par-allel repeating” type shown in Figure 10(a).
Observation 3.
Kokotsakis quadrilaterals of types 7.4 and 7.5 have numericalsolutions, but are rigid in Freeform Origami [4].Then considering the compatibility among all the types, the results for spe-cial solution 1 are listed as follows. The row of Kokotsakis quadrilaterals to berepeated can be constructed by the following ways.(1)
Transverse Linear Repeating : Stitch among the linear units of type 5 inthe rotated position (Section 4.3). The linear direction is transverse.Because the linear relation among the tangent of half of correspondingfolding angles, linear units can only stitch with linear units in the lineardirection. Note that only using 1-1 we will obtain the isogonal type, whichis presented in Section 4.2.(2)
Longitudinal Linear Repeating : Stitch among type 1-3 in the regular po-sition (Section 4.4). The linear direction is longitudinal.Just from the type of vertex in type 1-3, we can also choose types 1-3,6.1, 7.3 7.4 and 7.5 as the first Kokotsakis quadrilateral. However, fromObservation 3 we will not choose 7.4 and 7.5. More careful examinationon types 6.1 and 7.3 shows that the additional requirements to allow re-peating contradicts with the rigid-foldability condition on sector angles.20ote that choosing 1-1 we will obtain the isogonal type.(3)
Conic Repeating : Stitch among type 3.2. (Section 4.5)Apart from linear units, from Observations 1–3 the only remaining typeis 3.2. What we then need to consider is choosing the first Kokotsakisquadrilateral in this row, which can be types 3.2, 5-5, and 7.8. However,if type 5-5 is chosen, this will become a special case of 3.2. More carefulexamination on type 7.8 shows that the additional requirements to allowrepeating contradicts with the rigid-foldability condition on sector angles.
For special solution 2, the method is to construct a row and a column of rigid-foldable Kokotsakis quadrilaterals, then consider the stitching of solvable casesmentioned in Proposition 2(2) and 2(3) to obtain a large quadrilateral creasedpaper. Here we also need to make sure that this step of “solving one vertex basedon three vertices” can proceed in both longitudinal and transverse directionsindefinitely, which excludes type 7.6. Considering solving an (anti)isogram, onetype of large quadrilateral creased paper can be constructed by the steps listed inSection 4.6, called the hybrid type. Note that a type 5 Kokotsakis quadrilateralcannot have two linear directions except for 1-1. From Observations 1 and 3 wewill not consider types 6.3, 7.4, 7.5 and 7.9 when choosing the first Kokotsakisquadrilateral.
Apart from repeating in special solution 1, it is natural to consider whether otheroperations are possible after constructing a row of rigid-foldable Kokotsakisquadrilaterals. For instance, the new rows can be generated by substitutingsome sector angles by their complements to π , or switching the position of somesector angles. From our observation, generically such operations(1) do not result in another rigid-foldable Kokotsakis quadrilateral.(2) need two or more extra conditions for ensuring the sector angles on eachinner panel to be π , where the total number of constraints exceeds 8.Therefore we will not consider such operations in this article. We do, how-ever, admit that repeating is only an elementary special solution, and possiblythese operations could be applied in some non-generic case.Further, there are some special solutions which are more dependent on sym-metry. First, we have the tiling type provided in [9] and [11] (Section 4.7).However, it is unclear to us how to use “tiling” to generate a large quadrilateralcreased paper from a solved row of rigid-foldable Kokotsakis quadrilaterals. Sec-ond, the parallel mixed type (Section 4.8), which is based on the compatibilityof the parallel repeating quadrilateral creased paper. Some insights on findingmore types of rigid-foldable quadrilateral creased papers are provided in Section6. 21 Discussion
Here we point out some considerations for further exploration of large rigid-foldable quadrilateral creased papers.(1) Although for a large quadrilateral creased paper, generically the number ofconstraints for rigid-foldability is much greater than the number of sectorangles, it maybe still possible to solve this system as a whole and findsome special solutions.(2) Following the idea of solving the sector angles quadrilateral by quadri-lateral, in Section 5 we move forward only by comparing the number ofconstraints and variables. However, this is not strict for non-linear sys-tems. By studying the rigid-foldability conditions more carefully theremay be further more special solutions. Then we might be able to go be-yond the known types and find more general rigid-foldable quadrilateralcreased papers.
In this article, the orthodiagonal, isogonal, parallel repeating, tiling and parallelmixed types do not require the solution of equations, while other types aredesigned by specifing some input sector angles and solving the rest of the sectorangles numerically. We will comment on these numerical solutions here.(1) When making examples we used fsolve in MATLAB with random inputsector angles and initial values. The numerical solutions can be sensitiveto the input sector angles and initial values. Given a set of input sectorangles there might be no solution, or only complex solutions, as mentionedin Observation 1. Sometimes an exact numerical solution can be obtainedonly after trying many different input sector angles and initial values.Changing the position of input sector angles will also affect the numericalresult.(2) Most of equations in the rigid-foldability condition are trigonometric, butsome are exponential or elliptic. Trigonometric equations can be trans-formed into polynomial equations with the following relations: cos α = 1 − tan α
21 + tan α , sin α = 2 tan α
21 + tan α (6)which has the advantage that the numerical solution of a system of poly-nomial equations has been well studied. It would be possible to applymore advanced numerical methods here for better results.(3) In Observation 3, we mentioned that types 7.4 and 7.5 can be solved, butdo not pass the test for rigid-foldability in Freeform Origami. This mightbe because the rigid-foldability constraints on sector angles are derived inthe complexified configuration space, where the real folding angles maybeisolated. 22nother phenomenon in the numerical solution is called degeneration , whichcan be divided into two types.(a) In the numerical result, an (anti)deltoid or (anti)isogram may degenerateto a miura-ori vertex or a cross. A conic vertex may degenerate to an(anti)deltoid or (anti)isogram. An elliptic vertex may degenerate to aconic vertex. Further degeneration is also possible.(b) There might be unexpected relations among the sector angles, such asin Figure 10(e) and 10(f), some inner creases in different linear units areparallel; and in Figure 11(a) and 11(b) when solving new sector angles ina row some values recur.Apart from a vertex becoming a cross, such degenerations are acceptable inthe numerical results. However, sometimes negative numerical results appear inthe form of degeneration, as mentioned in Observation 2. After finding all the sector angles of a large quadrilateral creased paper, thelength of inner creases in a row and a column is adjustable, which can makea quadrilateral creased paper have various configuration. When plotting thecreased paper based on a set of known sector angles, the length of inner creasesshould be adjusted to avoid intersection of inner creases at some points otherthan vertices.
As we know more about rigid-foldable quadrilateral creased papers, a natu-ral question is how to count the number of branches of rigid folding motion.Especially, for a developable quadrilateral creased paper, the mountain-valleyassignment can be used to distinct different branches. The problem is a gen-eral theory for counting the number of branches is still not clear, althoughgiven a specific example we have some techniques (at least, enumeration) todeal with it. Another approach is linking this problem to graph coloring, [16]counts the number of mountain-valley assignments for local flat-foldability of aMiura-ori, while how to identify those guarantee global flat-foldability (whichare the exact number of branches) requires some unknown techniques. Generi-cally, a rigid-foldable quadrilateral creased paper with more symmetry will havea greater number of branches. One observation is, with no degeneration, manyof the large rigid-foldable quadrilateral creased papers presented here only have1 branch of rigid folding motion, whereas for some degenerated cases, this num-ber increases exponentially with respect to the number of vertices. This topicwill be discussed in a future article of the series.
Based on a nearly complete classification of rigid-foldable Kokotsakis quadrilat-erals from Ivan Izmestiev, this paper describes several new variations of large23igid-foldable quadrilateral creased papers, without any restriction on the de-velopability or flat-foldability. These new variations still have one degree offreedom, but the rigid folding motions are more irregular.
We thank Ivan Izmestiev and Grigory Ivanov for helpful discussions at the work-shop organized by the Erwin Schrodinger Institute on "Rigidity and Flexibilityof Geometric Structures". We also thank Martin Van Hecke for helpful discus-sions at the 7th International Meeting on Origami, Science, Mathematics, andEducation.
References [1] Zeyuan He and Simon D. Guest. On Rigid Origami I: Piecewise-planar Paperwith Straight-line Creases. arXiv preprint arXiv:1803.01430 , 2018.[2] Mark Schenk and Simon D. Guest. Geometry of Miura-folded metamaterials.
Proceedings of the National Academy of Sciences , 110(9):3276–3281, 2013.doi: 10.1073/pnas.1217998110.[3] Tomohiro Tachi and Koryo Miura. Rigid-foldable cylinders and cells.
J. Int.Assoc. Shell Spat. Struct , 53(4):217–226, 2012.[4] Tomohiro Tachi. Generalization of rigid-foldable quadrilateral-mesh origami.
Journal of the International Association for Shell and Spatial Structures ,50(3):173–179, 2009.[5] Thomas A. Evans, Robert J. Lang, Spencer P. Magleby, and Larry L. How-ell. Rigidly foldable origami gadgets and tessellations.
Royal Society openscience , 2(9):150067, 2015. doi: 10.1098/rsos.150067.[6] Robert J. Lang and Larry Howell. Rigidly foldable quadrilateral meshes fromangle arrays.
Journal of Mechanisms and Robotics , 10(2):021004, 2018. doi:10.1115/detc2017-67440.[7] Tomohiro Tachi. Freeform rigid-foldable structure using bidirectionally flat-foldable planar quadrilateral mesh.
Advances in architectural geometry 2010 ,pages 87–102, 2010. doi: 10.1007/978-3-7091-0309-8_6.[8] Hellmuth Stachel. Remarks on flexible quad meshes. In
Proc. BALTGRAF-11, Eleventh. Internat. Conf., Tallin, Estonia , 2011.[9] Niek Vasmel.
Rigidly Foldable 2D Tilings . B.S. Thesis, Leiden University,2016.[10] Robert J. Lang.
Twists, Tilings, and Tessellations: MathematicalMethods for Geometric Origami . AK Peters/CRC Press, 2017. doi:10.1201/9781315157030.[11] Peter Dieleman.
Origami Metamaterials: Design, Symmetries, and Com-binatorics . Ph.D. Thesis, Leiden University, 2018.2412] Ivan Izmestiev. Classification of flexible Kokotsakis polyhedra with quad-rangular base.
International Mathematics Research Notices , 2017(3):715–808, 2016. doi: 10.1093/imrn/rnw055.[13] Erik D. Demaine and Joseph O’Rourke.
Geometric folding al-gorithms . Cambridge university press Cambridge, 2007. doi:10.1017/CBO9780511735172.[14] Tomohiro Tachi. Design of infinitesimally and finitely flexible origami basedon reciprocal figures.
Journal of Geometry and Graphics , 16(2):223–234,2012.[15] Zeyuan He and Simon D. Guest. Approximating a Target Surface with1-DOF Rigid Origami. In
Origami 7: Seventh International Meeting ofOrigami Science, Mathematics, and Education , pages 505–520, 2018.[16] Jessica Ginepro and Thomas C. Hull. Counting Miura-ori foldings.