Downloaded at MIT LIBRARIES on May 17, 2021 a design Dudte H. origami Levi for algorithm additive An e rmtopolm:tedfclyo nigago us to guess good PNAS a finding of suf- difficulty they the but problems: generic, two are from fer departure methods empirical computational an These as point. pattern folding periodic use well-known and a framework optimization multidimensional nonconvex, in a constraints developability latter qualitative nonlinear The encoding case-specific. require strong typically inevitably symmetries—and sharing are that particular designs constructions involve of exhibiting space those (31– restricted understand- similarities—i.e., typologies a comprehensive folding a of provide ing known typically optimiza- former generalize The or 34). to (22–30) methods algorithms variations example, geometric design tion canonical direct simple the either parameterize employed to being well- generally the (21) have with moun- and pattern priori, specific Miura-ori a and assumed known geometries assignments (MV) periodic tain/valley with tessellations on unfold solutions. that freeform surfaces limited finding inversely, has either or, of fold challenge actually the that but patterns quad design, their interest of scientific problem to significant the nanoscopic to attracted the has from This scale architectural. any the at promising metastructures a for origami quad platform making additional (20), of folds and consideration (19) requires symmetries then systems multistable large floppy, flat-foldable, or and in Using rigid-foldable program governed folding. to during patterns is these encountered origami frustration structures geometric if quad of by from response plane, part mechanical derived the The materials all. to at and folded isometric be configurations can may they which folded origami, degree quad isolated called ver- internal have are four-coordinated faces one quadrilateral comprising exactly and Patterns tices with (DOF). folding freedom of of atom hydrogen of technological their by (10–18). inspired promise scientists and folding engineers of attention limits and the and (5–9) draw patterns the to by begun fascinated have mathematicians of they recently, around More origami world. of the traditions pedagogical and ceremonial, rative, F origami has rule design. local metastructure a origami-based to transform search to our global potential a Overall, the from approximants. perspective folded sur- in shift with fitting simple curvature and given origami ordered, of growing curved-folded faces by and approach straight, our sheets. of disordered, flat flexibility from devel- the folded of be illustrate space can We that complete additive, patterns the origami or of a quad marching, design opable of complete inverse boundary the a the for yields algorithm at This front seed. compatible folded given geometrically us allows a foundation grow this how to showing com- then developable the and single vertex, a for into fourfold folds conditions separate geometric two of two the completion in patible problem identifying this first solve guar- We is by plane? it steps: the that to so isometric seed be a to from grow anteed dimensions we three can in How surface 2020) perspective: folded 11, different a September a problem review from the for design pose (received we origami 2021 fabrication, of 5, additive March of allure approved the and by PA, Inspired Philadelphia, Pennsylvania, of University Liu, J. Andrea by Edited 02138 MA Cambridge, University, Harvard Biology, Evolutionary 02139; and MA Cambridge, Technology, of Institute onA alo colo niern n ple cecs avr nvriy abig,M 02138; MA Cambridge, University, Harvard Sciences, Applied and Engineering of School Paulson A. John rvosqa rgm einsuishv eddt focus to tended have studies design origami quad Previous kind a folds, four with vertex single a is origami simplest The ig,lae,adgt 14 n aealn itr ndeco- in insect history long including a systems have and in (1–4) guts nature and leaves, in wings, arise patterns olding 01Vl 1 o 1e2019241118 21 No. 118 Vol. 2021 | opttoa design computational a ayP .Choi T. P. Gary , | metamaterials a,b n .Mahadevan L. and , c eateto hsc,HradUiest,Cmrde A018 and 02138; MA Cambridge, University, Harvard Physics, of Department | diiefabrication additive a,c,d,1 ulse a 7 2021. 17, May Published doi:10.1073/pnas.2019241118/-/DCSupplemental at online information supporting contains article This 1 proposed the under the Published Submission.y on Direct PNAS patent a a is article filed This have L.M. design.y origami and for L.H.D. algorithm manuscript.y statement: the interest wrote L.M. and G.P.T.C., Competing and G.P.T.C., L.H.D., L.H.D., imple- experiments; and G.P.T.C. results; numerical and the the analyzed L.H.D. conducted L.M. results; and L.M. theoretical algorithms and the G.P.T.C., the derived L.H.D., mented L.H.D. research; the research; conceived the L.M. designed and L.H.D. contributions: Author strips folded compatible, identifies of that family framework continuous unified complete a the providing by approaches a of boundary the augment model. to bulk modules folded discrete prede- of a set from selected termined are units wherein strat- folded (38), compatible design combinatorial geometrically origami modular a to artistic (36), from method borrowed “jigsaw” origami egy known a quad origami as rigid-foldable such quad (35–37), design developable flat-foldable of and subclass rigid- as a configurations flat-folded DOF, allows to can one folded This plane, to with flat frustration. the a geometric from design to deployment no developing for to with to developed rigidly—i.e., addition in deploy been that, have recently, origami stages, quad algorithms intermediate marching in typically several this frustration pattern Although geometric the energy surface. global implies transform folded second designed, can a the process origami, to minimum, of folding configurations case intermediate a the through In that sheets? implies flat from this capa- fabricated thus being and of plane surface the ble the new to the ask that isometric of such developable—i.e., boundary outward can remains the surface origami extend one quad we folded can fabrication, a How problem: additive inverse and following design most putational the for has, form. science art origami the the quad followed and admit- part, of solutions, has variety piecemeal design origami wide only quad ted a of problem on general expound of the patterns, to lack used the strategies current been and many while have solution Thus, local problems. large desired to scalability a to convergence ensure owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To pbeqarltrlsrae,ealn h eino novel scale. any of at sheets design flat devel- the from all metastructures enabling of foldable design surfaces, additive quadrilateral the for opable discovery algorithm this how simple the show a We at theo- yields seed. directions folded constructive growth given a compatible a of of boundary to geometry simple leads the a on that proposing rem approach by marching design origami local of or and paradigm computational, typologies mathematical, physical the to reverse origami difficult we Here, are of solve. that problems geometric spaces optimization constrained simple of limited global on design for platform focused the constructions emerging on has an work structures is Prior origami-based folding, metamaterials. paper mechanical of for art the Origami, Significance ee edvaefnaetlyfo hs previous these from fundamentally deviate we Here, com- from operation edge-extrusion simple the by Inspired NSlicense.y PNAS b eateto ahmtc,Massachusetts Mathematics, of Department https://doi.org/10.1073/pnas.2019241118 d eateto Organismic of Department . y https://www.pnas.org/lookup/suppl/ | f7 of 1

APPLIED PHYSICAL SCIENCES that can be extruded directly from the boundary of a folded the entire design space of generic quad origami surfaces. This model. This lays the foundation for an algorithm that allows us constructive algorithm (39) is enabled by the following: to explore the entire space of developable quad origami designs, Theorem. The space of new interior edge directions along the not limited to just rigid- and/or flat-foldable designs. We begin entire growth front in a quad origami is one-dimensional—i.e., by exploring the flexibility in angles and lengths associated with uniquely determined by a single angle. fusing two pairs of folded faces at a common boundary, which Proof: Our proof primarily consists of three parts: single-vertex yields the geometric compatibility conditions for designing a construction, construction of adjacent vertices, and the growth of four-coordinated, single-vertex origami. We then apply the the full growth, with details given in SI Appendix, section S1. single-vertex construction to determine the space of compatible quad origami strips at the boundary of an existing folded model. 1. Single-vertex construction: Suppose we are given a vertex Critically, we establish that the new interior edge directions and along the growth front with position vector xi (Fig. 1A), with design angles along the growth front form a one-dimensional the two adjacent growth-front vertices denoted by xi−1, xi+1. family parameterized by the choice of a single face orientation in We denote the two boundary-design angles incident to xi in space along the growth front. The result is an additive geometric the existing surface by θi,3 and θi,4 and the angle in space algorithm for the evolution of folded fronts around a prescribed at xi along the growth front denoted by βi = ∠{−ei , ei+1} ∈ seed into a folded surface, establishing the means to characterize (0, π) (Fig. 1B), where ei = xi − xi−1 and ei+1 = xi+1 − xi . To

A B

C D E

F G

Fig. 1. Construction of quad origami. (A) A quad origami surface, the Miura-ori pattern, with boundary vertex xi along a growth front (green). (B) Focusing on xi, shown from a different vantage point than that of A, its adjacent growth-front vertices xi−1 and xi+1. The two design angles along the boundary θi,3 and θi,4 are shown in blue, and the angle in space βi between the two growth-front edges ei and ei+1 is in green. (C) The plane of action for new design angle θi,1 is determined by a flap angle αi (red), which sweeps from the βi face clockwise about ei.(D) As θi,1 (yellow) sweeps through its plane of action, it determines possible growth directions ri and θi,2 (dashed), the angle between ri and ei+1. These must satisfy θi,1 + θi,2 = ki (D, Inset) to create a developable vertex xi.(E) This constraint gives an ellipse γi of spherical arcs θi,1 and θi,2, which forms a closed loop around the line containing ei. The value of θi,1 that 0 satisfies the constraint is given by the unique intersection of its plane of action and γi, so αi parameterizes γi.(F) The secondary flap angle αi at xi sweeps from the βi face clockwise about ei+1 and is determined by αi. The flap angle αi+1 at xi+1 sweeps from the βi+1 face clockwise about ei+1 to the same 0 plane as measured by αi .(G) Two adjacent growth directions ri and ri+1 must be coplanar, so ri+1 is determined by the intersection of this plane and γi+1; thus, αi+1 parameterizes γi+1.

2 of 7 | PNAS Dudte et al. https://doi.org/10.1073/pnas.2019241118 An additive algorithm for origami design Downloaded at MIT LIBRARIES on May 17, 2021 Downloaded at MIT LIBRARIES on May 17, 2021 nadtv loih o rgm design origami for algorithm additive An al. et Dudte .Cntuto fajcn etcs enwso httenew the that show now We vertices: adjacent of Construction 2. r w e einage mle by implied angles design new two are where gives opposite angle spherical θ ovn Eqs. 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SI flap or to the chosen solution and is unique exist, angle a flap vertex, what neighboring matter no so bounds inequality angle interval θ θ by given foci aso ie n oie give cosines and sines of laws ,1 i i , ,1 ,1 θ θ = + π i i 2 ,2 θ ,2 π θ i and θ cos and , e xs n r nqefraygiven any for unique are and exist ,1 i γ sin = k i i ,2 ,1 n h sum the and θ i cos i = i rmthe from 2 = θ = θ ,1 fshrclarcs spherical of tan = β i 1 i ∠{−e r ,1 ,2 α α ∈ β i β i and β π < i i i 0 −e i cos = [0, i and and ∈ = − when cos omashrcltinl with triangle spherical a form k sin −1 r i i 0 2π [0, x 2 i π θ cos i , i < i +1 eeial.Mroe,teshrcltri- spherical the Moreover, generically. ], and for r α ,3 α sin aifistelclagesmdevelopability sum local-angle the satisfies θ e β θ i 2π i i 0 [x i i ∈ } i , θ i ,2 − hc ilsauiu ouinif solution unique a yields which , θ ,1 θ θ = cos i θ ) i θ β X i r j i .W oeta h single- the that note We 1C). (Fig. i ,1 ln otepaeo h e quad new the of plane the to plane e , − i ,2 θ ,1 ,1 =1 i cos i 4 i .Tekygoercintu- geometric key The 1 E). (Fig. and ,1 i etelf-adoine a angle flap left-hand-oriented the be β ,2 x (0, r −1 sin sin (α i sin α +1 cos ,4 i = i 0 + 0 = β i k , ota h peia a fcosines of law spherical the that so , +1 ] i θ i θ < θ h muto nua mate- angular of amount the , i i β o n a angle flap any For . π i k θ θ θ ) θ aaeeie h ellipse the parameterizes nteagetdqa origami quad augmented the in i θ − at ,j oi S1 Movie (see i ,2 α i i i i i ) − i given ,2 ,2 ,1 ,1 sin + ,2 − and 2 = i i cos and yields ,3 x (α (α , cos (α − θ spstv oooi nthe on monotonic positive is θ i θ + +1 i i i eto S1 section Appendix, SI π i i sin ,1 β i ,1 ,2 ) ) θ sin ) r θ , r α e θ i r , θ θ , , i i i i sin r i i i θ i i ,1 htstse Eq. satisfies that i ,2 +1 ,4 k .Furthermore, 1D). (Fig. i ,1 x ,2 ihu oso gener- of loss Without . Fg 1F (Fig. i θ htgvstedirection the gives that i i ,2 (α o ute eal and details further for + i a hnb obtained be then can < −1 sin = , β α ,2 rmthe from and i θ i θ i θ eern gi to again Referring . 2π ∠{e cos ) , i i θ i β ,2 ,1 r louniquely also are ,1 i ,1 i θ − 2 = β 6 = cos 2, =π/ i i .Denote ). , ,3 i +1 θ β β α k h spherical the , 1 ehave we 2, π/ , i i i i π α i ,2 , α α , and θ generically, β i r ninterior an − ensan defines i i i i i h sum the , oEqs. to ,4 . ∈ } teflap (the β ln to plane i γ and 2 1 i (0, when α See . must with i 0 [1] [6] [5] [3] [4] [2] for β π β as i 1 ) i 6 .Got fteetr rn:Fnly oetbihbijection establish to Finally, front: entire the of Growth 3. x in,tobudr-einage tteedonso h strip, direc- the flap of edge one endpoints and the strip: angles at and new angles design boundary-design the interior two of the tions, determine geometry to the angle determine with to boundary DOFs a to strip and narity designated of front growth strip of a strip with exist- a (seed) an by origami For quad surfaces. origami regular quad ing generic designing for rithm of development directions a give fold of set dimensional curve, discrete a For and front, growth a angles with a.Anwcmail ti of strip the compatible along new construction geo- additively, A by following landscape way. the constraints this explore origami, developability to quad satisfying us generic allows algorithm of metric space design the of vertices interior the to to fold that patterns planar of .Cluaetenweg-eghbud n choose and bounds edge-length new the Calculate 4. the Propagate 2. any from Start 1. steps. following .Cos onaydsg angles boundary-design Choose 3. in of tions solution unique a yielding α yasito h ethn-retdangle g left-hand-oriented the the of to shift a by h niegot rn r aaeeie yasnl a angle flap single a by parameterized along is α are directions front composition edge growth interior their entire new the bijective, all Therefore, is bijective. function also transfer each Since i i hspofsget meitl nefiin emti algo- geometric efficient an immediately suggests proof This Proof: Corollary. ∈ i i ecnie h olwn composition following functions the consider we angles flap the between ewe onson points between r that by see eterized to by easy determined uniquely is It plane. about left-hand-oriented measured as e uwr-aigegsi ahnwqa antintersect cannot quad new each in edges outward-facing new (m 2B). α r 2C). culating and +1 0 2 [0, : . 0 i i R m −1 and soitdwt h rwhfotedge growth-front the with associated 3 1 + r esrdaotacmo xsadaetu related thus are and axis common a about measured are 1 + o emti nuto,osreta hr r bijections are there that observe intuition, geometric For . , x i m m β π osdrassigning Consider 0, = r i α ) −1 Os.Bud r ie yteosrainta the that observation the by given are Bounds DOFs). +1 m β delnts owieormi hoe establishes theorem main our while So lengths. edge α i → ud has quads α m , i g j α nopsto toDF)(i.2D). (Fig. DOFs) (two position into i ie eei curve generic a Given m = yieaieysligfor solving iteratively by α −1 : = . . . ae hsgvsteflpagetase function transfer flap-angle the gives This face. 0 2π [0, i − +1 i 0 α ∠{−e f ,w oeta new a that note we 2A), (Fig. quads boundary , 1 i sas ucinof function a also is i , α α i ,j = m ,2, {1, ∈ i i einagecntans fw d new a add we If constraints. angle design (α .Asmlragmn ple for applies argument similar A 1G). (Fig. +1 ): g hieaogtegot rn from front growth the along choice k k i γ i i and i i (α n oigo otenx etx(Fig. vertex next the to on moving and , , C 3(m = ) i a ecoe reyt eemn one- a determine to freely chosen be can e sgvnb h xsigoiaisurface. origami existing the by given is n h afpae about half-planes the and oteplane. the to i i +1 = ) g α m m . . . α 1) + j α C ∈ } i (g C nteaoeoiaiproof, origami above the In . i 0 i mod(α and n both and , k ∈ ud,w aeattlof total a have we quads, , edges j is i −1 m (0, 0 2π [0, ∈ https://doi.org/10.1073/pnas.2019241118 Osin DOFs m α − (g g (β m π C -dimensional. r j θ j 1} i ), i 0 i tarbitrary at e sbjcie hence, bijective; is −2 0,2 iceie by discretized (α ∈ 2π , i As ). ud sdsge ythe by designed is quads i = n hoeteflpangle flap the choose and α θ · · (· R , 1, = i i ) θ i γ ,1 3 x bev that Observe . − m − e i , i , R i θ i ,1 g − and θ +1 β τ i . . . τ 3 i 1, = f e i ,1 i i (α ∈ i i ,2 x i ), 2π , ,j ujc to subject i triga the at starting and rmthe from rotating , (0, oeDF (Fig. DOF) (one , −1 i γ i i m )))). ftetransfer the of , . . . i 1, = ), +1 j , e m π − where , i θ i PNAS ) , i 1, = and 1 + r param- are ,2 1, m n rotate and . . . l j r func- are h space the − o all for x β , α . . . vertices r e m r | i m m i 1 i i i i 0 i +1 +1 cal- , to f7 of 3 face that < C pla- , and 4 + − [8] [7] m .  β x  j is is 1 1 j i , ,

APPLIED PHYSICAL SCIENCES A

B

C

D

Fig. 2. Additive algorithm. (A) To grow an existing folded quad origami model at a boundary having m quads, m + 1 new vertices must be placed in space, subject to m planarity constraints (dashed squares) and m − 1 angle sum constraints (dashed circles), for a total of 3(m + 1) − (2m − 1) = m + 4 DOFs, generically. (B) The additive strip construction begins by choosing the plane associated with any one of the quad faces in the new strip (one DOF). (C) Consecutive single-vertex systems propagate this flap-angle choice down the remainder of the strip, determining uniquely the orientations in space of all quad faces in the new strip. (D) Edge directions at the endpoints of the strip can be chosen freely in their respective planes (two DOFs), and all transverse edges in the new strip can be assigned lengths (m + 1 DOFs) for a total m + 4 DOFs.

each other, which occurs when the two interior angles of a new configurations at the boundaries given by equality (Fig. 3B). quad sum to less than π (SI Appendix, section S3). Sweeping α from zero to 2π parameterizes the ellipse such that 5. Calculate the new vertex positions given rj and lj for all j . θ1(α = 0) = (k + β)/2 and θ1(α = π) = (k − β)/2, and new edge 6. Repeat the above steps at any boundary front to grow more directions r(α) tend to cluster in space around growth-front new strips. directions, where dθ1/dα has smaller magnitude. Special single- The algorithm also applies to discrete curves not associated vertex origami (40) are recovered by identifying their flap angles with an existing folded surface via the corollary. In this case, we (Fig. 3A; see SI Appendix for formulae). Three of these vertex can design the shape of the development of the curve by choosing types are given by rearranging Eq. 3 and plugging in a desired k values in step 2, rather than calculating them from an existing value for the new design angle: the continuation solution αcon, surface. where the new pair of quads can be attached without creating Having established generic connections from single vertices a new fold; the flat-foldable solution αff, where the vertex can to quad strips to origami surfaces, we now analyze the flap- be fully folded such that all faces are coplanar; and αeq, which angle parameterization at each scale of this hierarchy in more creates equal new design angles. These each admit two solu- detail. The design space of the growth front of a pair of folded tions related by reflection over the plane of the growth front, faces, a proto-single-vertex origami, is described fully by the pair recovering a duality noted in ref. 5. Two more special vertices of scalars β, the angle in space formed by the growth front, are identified by αll and αlr, which produce locked configura- and k, the shape parameter of γ—i.e., the amount of angular tions with the left pair of faces (θ1, θ4) and the right pair (θ2, θ3) material required to satisfy developability (Fig. 3A). A generic folded to coplanarity, respectively. A vertex is trivially locked single-vertex origami can be constructed in the interior of the with coplanar (θ1, θ2) faces when α = 0, π. The continuation flap triangular region 0 ≤ β ≤ π and β ≤ k ≤ 2π − β, with singular angle that does not create a new fold along the growth front and

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APPLIED PHYSICAL SCIENCES either side of the seed (and continuing on the growing patch) that tion in the additive algorithm, the flap angle and edge lengths can approximately reflect the origami surface back and forth between be chosen randomly, thereby leading to a crumpled sheet that the upper and lower target bounds, inducing an additional cor- does not follow a prescribed MV pattern (Fig. 4E). To model rugation in a transverse direction to that of the corrugation in the physically realizable crumpled geometry, we have chosen flap the seed. Fig. 4A shows a high-resolution generalized Miura- angles according to the self-intersection bounds given by special ori sandwich structure of constant thickness obtained by our vertex solutions along the entire front, so that the growth of the approach that approximates a mixed-curvature landscape, which sheet is a locally self-avoiding walk (see SI Appendix, section S4D would be very difficult to obtain using current techniques. As our for more details). second example, using a different DOF selection setup with no Finally, to emphasize the flexibility of the corollary, we con- reference target surface, we grow a conical seed with a series of struct a quad strip that forms a folding connection between a straight ridges with fourfold symmetry via facets that are created canonically rough structure, a random walk in three dimensions by reflections back and forth between a pair of rotating planes, (3D), and a canonically smooth structure, a circle in two dimen- shown in Fig. 4B. As our third example, we turn to designing sions (2D). Fig. 4F shows a single folded strip generated by surfaces that have curve folds, folds that approximate a smooth sampling a discrete Brownian path in 3D to form one bound- spatial curve with nonzero curvature and torsion (5). Fig. 4C ary of a folded strip, choosing k values such that its development shows a twisted version of David Huffman’s Concentric Circular falls on a circle and forms a single closed loop for any choice of Tower (41) obtained by our method, which uses a similar DOF flap-angle value and choosing edge lengths such that the other selection setup as Fig. 4B, but begins with a high-resolution cone boundary of the strip develops to another, smaller concentric segment as the inner-ring seed and follows with progressively circle (SI Appendix, section S4E). Indeed, the corollary allows thicker tilted cone rings added transversely (see SI Appendix, sec- for the freeform design of both folded ribbons and their pattern tion S4B for more details on both of these models). Fig. 4D shows counterparts independently. See Movies S4–S9 for 3D anima- another curved-fold model that uses the corollary to create a tions of the models in Figs. 4 and SI Appendix, Figs. S11, S12, seed from a corrugated discrete planar parabola and proceeds S14, S16, and S17 for a gallery of other surface-fitting, curved- to add new strips with constant flap angles and edge lengths, fold, disordered, and Brownian ribbon results obtained by our growing in a direction along the folds (see SI Appendix, section additive approach. S4C for more details). As our last surface example, we use our Since the developability condition in Eq. 1 is always satis- approach to create a disordered, crumpled surface that is isomet- fied in our marching algorithm, all physical models created by ric to the plane, again a structure that would be very difficult to it that do not self-intersect can transform from a 2D flat state obtain using current techniques. For each step of strip construc- to the isometric 3D folded state, typically through an energy

Fig. 4. Additive design of straight, curved, ordered, and disordered origami. (A) A generalized Miura-ori tessellation fit to a target surface with mixed Gaussian curvature generated. Lower and upper bounding surfaces are displaced normally from the target surface, and a seed strip of quads is initialized in between the two with one growth front on each surface. New strips are attached on either side of the seed by reflecting the growth front back and forth between bounding surfaces in their interstice. (B) A low-resolution conical seed with fourfold symmetry grows by reflecting between the interstices of two rotating upper and lower boundary planes. New strips form closed loops with overlapping endpoint faces. (C) A high-resolution conical seed grows by attaching progressively tilted cone rings to reproduce a curved-fold model. New strips form closed loops with overlapping endpoint faces. (D) A curved-fold model grows from a seed created by using the corollary to attach a strip of quads to a corrugated parabola. (E) A self-avoiding walk away from a Miura-ori seed strip with noise added to the boundary growth front produces a crumpled sheet. New strips are added by sampling flap angles within bounds that prevent local self-intersection. (F) A Brownian ribbon whose development approximates a circular annulus is created by using the corollary. The seeds are highlighted in yellow, and the arrows indicate the growth direction. The fold pattern for each model is shown at the bottom right of each image. The Upper Right Insets in A–C and F, along with the small square next to the pattern in F, are zoomed-in views to highlight details. See SI Appendix, Figs. S8 and S9 for higher-resolution versions of the fold patterns.

6 of 7 | PNAS Dudte et al. https://doi.org/10.1073/pnas.2019241118 An additive algorithm for origami design Downloaded at MIT LIBRARIES on May 17, 2021 Downloaded at MIT LIBRARIES on May 17, 2021 0 .Ce,L aaea,Rgdt eclto n emti nomto nfloppy in information geometric and percolation Rigidity Mahadevan, L. Chen, S. 20. Pinson in B. M. origami” rigid 19. 1-DOF with surface target and a Origami “Approximating geometries: Guest, curved D. to S. sheets He, flat Z. From 18. Zadpoor, A. A. Callens, P. J. S. 17. panels. polyhe- any thick folding for of algorithm practical Origami A “Origamizer: You, Tachi, T. Demaine, Z. D. E. Peng, 16. R. Chen, reconfig- Y. yet stiff, 15. into assembled tubes Origami Paulino, H. G. Tachi, T. Filipov, origami. T. E. Self-folding 14. mechanics: Extreme Santangelo, of D. C. mechanics Geometric 13. Mahadevan, L. Liang, Y. Liu H. of B. Dudte, mechanics 12. L. Geometric Guo, Santangelo, V. Z. D. C. Wei, Y. Mahadevan, Z. L. 11. Dudte, H. L. Dias, A. M. 10. nadtv loih o rgm design origami for algorithm additive An al. et Dudte h ai o acigagrtmta elcstesolu- the scalable, replaces a with that problem optimization algorithm global marching difficult a a of forms dis- for theorem tion simple and basis A patterns growth. the origami via surfaces quad developable developable crete generic of design Eq. in that side note opens right also this We the unique, metastability. replacing not program by are also to folding motions the routes folding of future which geometry simulations). the for folding on for pattern depends S10 landscape Movie and the S18 Because Fig. and (see S5 frustration section geometric includes that landscape 4,4) ihsltosuiul xsigi h aewywhen way same the origami in existing non-Euclidean uniquely of K solutions design with 43), the (42, additive to our generalizes formulas, approach trigonometric subsequent the modifying .R .Lang, J. R. O’Rourke, 9. J. Demaine, D. E. 8. Leb A. folded geometry 7. discrete of limit smooth and Invariant Paulino, H. G. Tachi, paper. T. Liu, on K. primer A 6. creases: and Curvature Huffman, A. D. 5. Shyer E. origami. A. Self-organized Rica, 4. S. Mahadevan, L. Spontaneous Whitesides, 3. M. G. Hutchinson, W. J. Evans, G. A. Brittain, S. Bowden, leaves. N. tree unfolding of 2. geometry The Vincent, V. F. J. Kresling, B. Kobayashi, H. 1. vrl,orsuypoie nfidfaeokfrteinverse the for framework unified a provides study our Overall, origami. United Albans, St Educa- (2017). Group, and (Tarquin Eds. Mathematics, You, 505–520. pp. Z. Science, 2, Bolitho, vol. in M. 2018), Lang, Origami Kingdom, J. on R. (7OSME), Meeting tion International 7th The pp. 34, approaches. vol. kirigami 2017), Germany, Wadern, Dagstuh, (Schloss Eds. Katz, 1–5. J. M. Aronov, B. in dron” (2015). (2015). metamaterials. and structures urable Phys. Matter (2018). origami. pleated periodic origami. crease curved 2011). 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(CRC 10, , 6 .Deea,N aml .Wiuats .vnHce iswpzl einof design and rigidly Fuse, puzzle of T. deformations Jigsaw and 38. designs The Hecke, Plucinsky, P. van James, D. M. R. Dang, X. Waitukaitis, Feng, F. S. 37. Vasmel, N. Dieleman, P. 36. structures. Curve-pleated Pottmann, H. Wallner, J. Rist, F. Mundilova, K. Jiang, C. 34. origami using Kilian M. curvature Programming 33. Mahadevan, L. Tachi, T. Vouga, E. Dudte, H. L. origami. 32. quadrilateral-mesh rigid-foldable of origami. Generalization Miura Tachi, Helical T. James, 31. D. structures R. origami Plucinsky, axisymmetric P. and Feng, cylindrical doubly F. of rigid-foldable Design 30. Duan, of H. Design Liang, Zhong, H. Y. Hu, Y. Wang, 29. H. Zang, S. generalized Zhou, A X. surfaces: Song, curved K. to Folding 28. Chen, Q. C. Chen, X. Miura-ori. Gong, the H. of Wang, descendants F. symmetric isomorphic 27. of Design Guest, D. a S. on Sareh, based P. structures origami 26. curved-crease three-dimensional of of Design You, Z. Parametrizations Wang, origami: H. Zhou, rigid X. Miura-base 25. You, Z. Gattas, M. J. 24. 2 .Bry .E e-rml,C .Snagl,Tplgcltastosi the in transitions origami. Topological Non-Euclidean Hecke, Santangelo, Van M. D. Dieleman, P. C. Waitukaitis, S. Lee-Trimble, 43. E. M. Berry, http://www.theiff.org/oexhibits/ origami. M. four- crease special 42. Curved and Figuring, Generic for blocks: Institute building The Origami 41. thesis, Hecke, Van PhD sheets,” M. flat in Waitukaitis, cuts S. and folds 40. using shape of design “Inverse Dudte, H. L. 39. arrays. angle from meshes quadrilateral foldable Rigidly Howell, L. Lang, J. R. 35. first-level of Parameterizations origami: rigid Miura-base You, Z. Wu, W. Gattas, M. J. sweep. rotational on 23. based origami 3D for method design A Mitani, J. 22. Miura, K. 21. n ..;adNFGat M-015 t ..,DR1231(oL.M.), (to DMR-1922321 L.M.), (to L.M.). (to G.P.T.C. DMR-2011754 EFRI-1830901 (to Grants 1764269 and NSF Award Mathemati- and Harvard, at L.M.); for Biology and Center of Analysis NSF-Simons Statistical the and cal and Initiative Biology Quantitative ACKNOWLEDGMENTS. https://github.com/garyptchoi/additive-origami. dis- Availability. Data in advances creations artistic for and curved-folded alike. applications, promise and engineering computational geometry, substantial straight-, prototype crete holds disordered, rapidly ordered, to geometries us of boundary allows designs and constrained rigidity that a bulk between of flexibility interplay evolution This front. the folded for scheme easy-to-implement a-odbeqarltrlms origami. mesh quadrilateral flat-foldable origami. pluripotent tessellations. Struct. Spat. Shell (2020). cell. Miura-ori generalized on based patterns. (2017). crease 20170016 trapezoidal on based tessellations origami curved structures. cylindrical origami-based of (2016). mechanics 33312 and method design Struct. Mater. 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