Smooth triaxial weaving with naturally curved ribbons
Changyeob Baek, Alison G. Martin, Samuel Poincloux, Tian Chen, Pedro M. Reis
SSmooth triaxial weaving with naturally curved ribbons
Changyeob Baek,
1, 2
Alison G. Martin, Samuel Poincloux, Tian Chen,
2, 4 and Pedro M. Reis ∗ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, USA Flexible Structures Laboratory, Institute of Mechanical Engineering,École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland Independent Artist, Fivizzano, Italy Computer Graphics and Geometry Laboratory, School of Computer and Communication Sciences,École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
Triaxial weaving is a handicraft technique that has long been used to create curved structuresusing initially straight and flat ribbons. Weavers typically introduce discrete topological defects toproduce nonzero Gaussian curvature, albeit with faceted surfaces. We demonstrate that, by tuningthe in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be variedcontinuously, which is not feasible using traditional techniques. Further, we reveal that the shapeof the physical unit cells is dictated solely by the in-plane geometry of the ribbons, not elasticity.Finally, we leverage the geometry-driven nature of triaxial weaving to design a set of ribbon profilesto weave smooth spherical, ellipsoidal, and toroidal structures.
Traditional basketmakers have long been employingthe handicraft technique of triaxial weaving to fabri-cate intricate shell-like structures by interweaving ini-tially straight ribbons into tri-directional arrays [1, 2].Beyond basketry, triaxial weaving is also encounteredin textiles [3], composite materials [4], molecular chem-istry [5, 6], and biology [7]. While weaving with straightribbons in a regular hexagonal pattern yields a flat sur-face, topological defects ( e.g. , pentagons or heptagons)induce local out-of-plane geometry [1, 8–10]. Basketmak-ers have extensive empirical know-how on how and whereto place these defects, and recent research has investi-gated their optimal placements to approximate targetsurfaces [2, 11]. The strategy to achieve shape by defectsis also akin to the concept of topological charge [12] incurved two-dimensional (2D) crystals such as the buck-minsterfullerene [13], colloidal crystals [14–16], confinedelastic membranes [17, 18], and dimples on curved elasticbilayers [19, 20]. However, the curvature attained fromthese defects is discrete, which limits the range of re-alizable shapes. Even if previous studies [21, 22] havesuggested a polygon-based combinatorial design proce-dure that includes weaving with initially curved ribbons,a predictive understanding of the effect of the ribbon ge-ometry on the shape of the weave is lacking.Here, we investigate how triaxial weaving with natu-rally curved (in-plane) ribbons can yield smooth three-dimensional (3D) shapes. We make use of a combinationof rapid prototyping, X-ray micro-computed tomography( µ CT), and finite element methods (FEM) to perform adetailed characterization of the geometry of our wovenstructures. First, we take a unit-cell approach to sys-tematically explore how the original 2D geometry of theribbons dictates the 3D shape of the weaves and regardthese cells as building blocks to construct more complexwoven objects. Fig. 1 shows representative units cellswith different topological characteristics and with rib-bons with different in-plane curvatures. Excellent agree- ment is found between the experiments ( µ CT) and simu-lations (FEM). These unit cells comprise n identical rib-bons that are woven to form an n -gon surrounded by atotal of n triangles. Each ribbon has three segments (in-dexed by j = { , , } ), with rivets placed at the crossingpoints to fix the segment length ‘ j . The in-plane cur-vature, k j , of each segment can be varied continuously(see red solid lines in Fig. 1). Traditional weaving cor-responds to the case of straight ribbons, k j = 0 mm − (Fig. 1, middle column). By considering ribbons thatare naturally curved in-plane (examples in Fig. 1 with k = 0 ), we demonstrated that the curvature of the re-sulting surface of the unit cells can be tuned smoothly,in a way not possible through the traditional approach.A purely geometric analysis is performed to rationalizethe integrated Gaussian curvature of the physical unitcells, revealing that geometry is at the core of settingthe shape of our physical triaxial weaves. This geomet- N o r m . e rr o r ,
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Figure 1. Representative family of triaxially woven unit cellsfor different numbers of ribbons (rows, n = { , , } ) andfor different values of the in-plane curvature in their middlesegment (columns, k = {− . , − . , , . , . } mm − ). Allribbons have three segments, each of arc length ‘ = ‘ = ‘ = 15 mm ; only the middle one is curved ( k = 0 mm − ),while those in the two extremities are naturally flat ( k = k = 0 mm − ). Tomographic µ CT scans (grey images) arejuxtaposed on FEM simulations, color-coded by the distancebetween their respective centerlines locations, e/‘ . a r X i v : . [ c ond - m a t . s o f t ] N ov ric reasoning forms the basis of a set of design principles,which are then leveraged to construct a variety of smoothcanonical structures, including spherical, ellipsoidal, andtoroidal weaves.Before turning to the general case of curved ribbons,we first focus on the ‘ traditional weaving ’ of unit cellswith naturally straight ribbons ( k ◦ = k ◦ = k ◦ = 0 );hereon, the superscript ( · ) ◦ denotes quantities pertain-ing to straight ribbons. In Fig. 2(a), we present thephotograph of a physical unit cell for a representativecase with n = 5 ribbons. The specimens were fab-ricated by, first, laser-cutting ribbons of width from a polymer sheet and, then, hand-weaving themto produce a 3D structure, which was imaged tomo-graphically using a µ CT scanner ( µ CT100, Scanco Med-ical; a = 29 . µ m voxel size). The original poly-mer sheet was a bilayer of a polyethylene terephthalate(PETE; Young’s modulus E ≈ GPa) plate of thickness t = 0 .
25 mm , coated with an elastomer-metal compos-ite ( E ≈ MPa) of thickness t = 0 .
35 mm . The lat-ter comprised vinylpolysiloxane (VPS-16, Zhermack) in-fused with a metal powder (NdFeB, 30065-089, neo Mag-nequench; ≈ µ m particle size) mixed at 2-to-1 weightratio. Given the disparity in bending stiffnesses of thetwo layers, E ( t ) / [ E ( t ) ] ≈ O (10 ) , the mechanicalstiffness of the ribbons was provided by the PETE, witha width-to-thickness ratio of 16. The radiopacity of theelastomer-metal served in detecting of the ribbons us-ing X-ray tomography to extract their framed center-lines from the volumetric data [23, S1] (Fig. 2b). Thecorresponding framed centerlines extracted from FEM(see [23, S2] for procedure) are in excellent agreementwith the experiments, as demonstrated in Fig. 2(c).Since the n -gon in the woven unit cell does not have awell-defined surface (its inner region is void of material),it is impossible to define a pointwise Gaussian curvature.However, the n -gon does have a well-defined boundaryset by the ribbons centerlines, from which we define the integrated Gauss curvature of the unit cell, K n . The re-markable Gauss-Bonnet theorem [24] states that K n canbe determined solely by the n -gon boundary: K n = (2 − n ) π − n X i =1 κ i g + n X i =1 θ i , (1)where κ i g = R i k g d s is the integrated geodesic curva-ture of the i -th edge and θ i is the i -th interior angleof the n -gon at each crossings (see schematic definitionsin Fig. 2b). In Fig. 2(d), we plot experimental and sim-ulated averages of both the integrated geodesic curva-ture, h κ ◦ g i = n P ni =1 κ i g , and the interior angles, h θ ◦ i = n P ni =1 θ i , for representative unit cells with ≤ n ≤ .Within the same cell, we find that all ribbons share thesame values of κ i g and θ i (their standard deviation issmaller than the symbol size), as expected from rota-tional symmetry. Also, h θ ◦ i ≈ π/ independently of (a) (b) I n t e g . c u r v ., -3 -2 -1 0 1 2 3 Exp. FEM Eq. (2)
Topological charge, (e)
Number of ribbons, A ng l e [r a d ] (d) Exp. FEM
Centerline n -gon (c)
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Figure 2. Units cells woven with straight ribbons. (a) Pho-tograph of a unit cell with n = 5 straight ribbons. (b) Ex-perimental data of the framed centerlines of the cell in (a)extracted from µ CT [23, S2]. (c) FEM-computed version of(b). (d) Average interior angles of the n -gon, h θ ◦ i , and aver-age integrated geodesic curvatures, h κ ◦ g i , vs. n . (e) IntegratedGaussian curvature of the unit cells with straight ribbons, K ◦ n ,computed through Eq. (1), versus q ◦ = 6 − n . The solid lineis the prediction from Eq. (2). n , indicating that the exterior triangles remain devel-opable and the Gaussian curvature concentrates at the n -gon. Moreover, the vanishing integrated geodesic cur-vature h κ ◦ g i ≈ is a direct consequence of the mechanicsof elastic ribbons; ribbons favor out-of-plane (instead ofin-plane) deformation [25]. Based on these observations,we simplify Eq. (1) for the integrated Gauss curvature ofa unit cell with straight ribbons: K ◦ n = π − n ) , (2)where the integer q ◦ = 6 − n is analogous to the topo-logical charge in curved crystallography [12]. The ex-perimental and FEM data for K ◦ n plotted in Fig. 2(e) asa function of q ◦ , onto which we superpose the predic-tion from Eq. (2), evidences how the discrete nature of q ◦ constrains strongly the possible values of integratedcurvature of the unit cells in traditional weaving [1, 2].Next, we turn to the non-traditional case of weavingunit cells with naturally curved ribbons. In Fig. 3(a),we show schematic diagrams of an individual curved rib-bon (top), as well as the planar representation of thecorresponding unit cell (bottom); each segment with j = { , , } is color-coded as red, green, and blue,respectively. We seek to evaluate the effect of the ini-tial in-plane curvature, k j , on the integrated curvatureof the cell, K n , as a function of n . For convenience,we normalize the segment curvature by its arclength; κ j = k j ‘ j . The (2 π/n ) -fold rotational symmetry is en-sured naturally by the definition of the unit cell when n is even, and enforced when n is odd by further im-posing ‘ = ‘ and κ = κ . Motivated by our find-ings for unit cells with straight ribbons (Fig. 2d), wemake the following remarks. First, we assume thatthe ribbons keep their in-plane curvature when woven;hence, κ j g = R j k g d s = R j k j d s = k j ‘ j = κ j for ev-ery segment of the n -gon [25]. Second, for straight rib-bons, we found that the outer triangles remained equi-lateral, thereby enclosing a surface of vanishing inte-grated curvature; a statement that we now assume toremain valid for unit cells woven curved ribbons. Third,given that these triangles sit on an isomeric surface,the interior angles (opposite to the arc ‘ j ; see the in-set of Fig. 3) are evaluated using Euclidean trigonom-etry: φ j = cos − (cid:18) P m = j ( g m − g j )2 Q m = j g k (cid:19) − P m = j κ m , where g j = 2 sin( κ j / · ‘ j /κ j . In turn, the interior angles of the n -gon, θ i , are the supplementary angle of either φ or φ ;such that P ni =1 θ i = { n ( π − φ )+ n ( π − φ ) } / . Thus, us-ing the Gauss-Bonnet theorem stated in Eq. (1), the inte-grated curvature of a unit cell reads K n = π [6 − n ( f + κ ∗ )] ,where f = π cos − (cid:0) g − g − g g g (cid:1) , and κ ∗ = 34 π ( − κ + 2 κ − κ ) . (3)The arc length ‘ j and the curvature κ j are coupledthrough the nonlinear term f . Noting that g j ≈ ‘ j in therange of in-plane curvatures considered, | κ j | ≤ . ( e.g. , g j ( κ j = 0 . ≈ . l j ), we take the asymptotic limit of | κ j | (cid:28) . We further impose ‘ = ‘ = ‘ to quantifyonly the effect of the in-plane curvatures. Ultimately, weobtain f = 1 and Gauss-Bonnet reduces to K n ( κ , κ , κ ) = π − n (1 + κ ∗ )] . (4)From the similitude between Eqs. (4) and (2), we define q ∗ = 6 − n (1 + κ ∗ ) as the modified topological charge ofthe unit cell with curved ribbons. We highlight that q ∗ can be varied smoothly using curved ribbons, yieldinga continuous range of K n , whereas K ◦ n in Eq. (2) wasdiscrete and restricted to multiples of π/ .In Fig. 3(b), we plot experimental and FEM datafor K n vs. q ∗ , while fixing ‘ j = 15 mm, to system-atically explore the parameter space n = { , , } and κ = {− . , · · · , . } (in steps of 0.1). Also, the experi-ments had κ = κ = 0 (total of 33 configurations, withtwo experiments per configuration) and the simulationshad κ = ± κ = {− . , − . , , . , . } (total of 352configurations). As above for the unit cells with straightribbons, we measured θ i and κ i g of the n -gon and usedEq. (1) to compute K n [23, S1-2]. Remarkably, we findthat the data in Fig. 3 collapse over the full range of − π ≤ K n ≤ π . This continuous variation for curvedribbons contrasts to the analogous result for traditionalweaving (Fig. 2e), where K ◦ n was limited to discrete stepsof π/ ( cf. Fig. 2e). Importantly, our geometric predic-tion for K n from Eq. (4) is in excellent agreement withthe data, demonstrating that in-plane ribbons geometryis at the heart of our triaxial weaving problem. -3 -2 -1 0 1 2 3 I n t e g r a t e d G a u ss c u r v a t u r e , Modified topological charge,
Exp. FEM Eq. (4) (d1)(c2)(c1) (d2) N o r m . d e v ., -5 (b)(a)
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Figure 3. Weaving with curved ribbons. (a) Schematics of theribbons (top) and the planar representation of a typical unitcell. The ribbons have 3 distinct curved segments of arclength ‘ j and normalized curvature κ j ( j = 1 , , ; color-coded as red,green, and blue, respectively). (b) Integrated Gaussian cur-vature of the unit cells versus q ∗ = 6 − n (1+ κ ∗ ) , with κ ∗ fromEq. (3). (c1)-(c2) Photographs of spherical weaves with (c1)straight and (c2) curved ribbons. (d1)-(d2) Reconstructed µ CT images of the weaves in (c1) and (c2), respectively. Thecolorbar indicates the normalized voxel-wise radial deviation, δ/R , from a sphere of radius R = 42 mm. Thus far, we have followed a unit-cell approach todemonstrate that smooth weaving can be physically re-alized with curved ribbons, purely from geometric designprinciples. We now seek to assemble these unit cells intoa spherical weave, adopting the topology of the rectifiedtruncated icosahedron [26, 27] for the layout of our de-sign. As an example, we fix the segment length to ‘ s =15 mm and inject a (nondimensional) segment curvature κ s = k s ‘ s into the pentagonal cells. The resulting weavecomprises 12 pentagonal cells with ( κ , κ , κ ) = ( κ s , , and 20 hexagonal cells with ( κ , κ , κ ) = (0 , κ s , . InFigs. 3(c1)-(c2), we present photographs of two sphericalweaves: one with straight ribbon ( κ s = 0 ), the traditionalcase, and the other with curved ribbons ( κ s = 0 . ). Thecorresponding µ CT images are shown in Figs. 3(c3)-(c4),color-coded by the radial distance between the scans anda sphere of radius R = 42 mm, δ [ mm ] . Negative values of δ indicate voxels located inside the targeted sphere. Forthe weave with straight ribbons (Fig. 3c2), the pentagonsprotrude from the reference sphere, with 5 % maximumradial deviation. This faceted geometry is a signature ofthe localized curvature intrinsic to the discrete nature oftraditional weaving; Eq. (4), predicts K = π/ for thepentagons and K = 0 for the hexagons. By contrast, theweave with curved ribbons ( κ s = 0 . ) shown in Fig. 3(c4)exhibits a significantly smoother shape, with a radial de-viation within 1 % of the perfect sphere; Eq. (4) predicts (b)(a2)(a1) (a3)
10 mm 10 mm
Figure 4. Design of nonspherical weaves with initially curved ribbons. (a1)-(a3) Ellipsoidal weaves of aspect ratios, a/b = { . , . , . } , respectively. (b) Toroidal weave of inner radius r i = 35 mm and outer radius r o = 105 mm. The planargeometries of the underlying curved ribbons for each of these weaves are provided in [23, S3.2-3] K = 0 . and K = 0 . .Our unit cells with curved ribbons are rotationallysymmetric. Hence, the possible design space availableby their tessellation is limited to shapes with local sym-metry ( e.g. , the sphere in Fig. 3c). We do not expectthis approach to be, in general, viable to design weaveswith more complex or arbitrary geometries. To overcomethis limitation, we expanded our framework to design theinitial shape of piecewise-circular ribbons that are to bewoven into a given target surface. Similar to what we didfor the unit cells, the injection of geodesic curvature intothe weave through the in-plane curvature of the ribbonsis at the core of the procedure. However, the curvature ofthe ribbons is now highly heterogeneous and dictated bythe target surface. Out design protocol (detailed in [23,S3.1]) consists of inputting a target surface , onto whichwe project a graph representing the triaxial weave topol-ogy. This graph contains nodes (corresponding to thecrossing points of the ribbons) and edges for their con-nectivity. At each node, a geodesic turning angle betweenconsecutive nodes is computed with respect to the targetsurface. The shape of piecewise-circular segments of theribbon is then obtained by averaging the two geodesicturning angles from its neighboring crossing points.In Figs. 4(a1-a3), as a first example of nonspheri-cal designs, we present a series of ellipsoidal weavesof an equatorial radius, b = 40 mm, and polar radii, a [ mm ] = { , , } . The ellipsoid in Fig. 4(a1) isoblate, whereas those in Figs. 4(a2-a3) are prolate, con-veying the generality of the approach. The graph forthese weaves was obtained by adopting the topology ofthe rectified truncated icosahedron [26, 27] and linearlyexpanding it by a factor a along the x, y -axes, and a fac-tor b along the z -axis [23, S3.2]. Within the frameworkof curved ribbons, their initial shape can be adapted toaccommodate a variation in the aspect ratio of the el-lipsoidal target surface, thus not requiring a variation inthe weave topology. The remarkably smooth shape ofthe weaves presented in Figs. 4(a1-a3) would not havebeen trivial to achieve through traditional weaving; inan ellipsoid, only the meridians and the equator are the closed geodesics [24]. As a second example, a smoothtorus (genus-zero surface with zero total curvature [24])cannot be achieved through traditional weaving; usingstraight ribbons inevitably requires the placement of pen-tagonal and hexagonal defects, albeit with a localizationof curvature that leads to faceted geometry. By contrast,as demonstrated by the physical realization in Fig. 4(b),our design with curved ribbons yields a toroidal weavewith hexagonal cells alone (the in-plane curvature of theribbons distributes the total curvature). The presentedsmooth toroidal weave has an inner radius r i = 35 mmand an outer radius r o = 105 mm. The topology of thetoroidal weave was obtained by considering a map froma regular triaxial pattern in 2D parameter space to a 3Dtoroidal target surface [23, S3.3].Our work demonstrates that the discrete nature of tra-ditional triaxial weaving can be circumvented by usinginitially curved, piecewise-circular ribbons. The shapeof the weaves can be decoupled from their topology, withmultiple topological layouts and ribbons geometries lead-ing to the same weaved shapes. However, when woven,these geometrically identical solutions store elastic en-ergy differently [23, S2]. This observation calls for a fulloptimization problem, which we hope future work willaddress, where both the distance from the target surfaceand the associated elastic energy are minimized in tan-dem by changing the geodesic curvatures and the segmentlengths of the ribbon as design parameters. Beyond artand architecture, future implementations of our designframework may include morphing structures in which thein-plane curvature of ribbons would be pre-programmedinto the ribbons and actuated upon stimuli to attain de-sired target shapes.We thank Yingying Ren, Julian Panetta, Florin Isvoranu,Christopher Brandt, and Mark Pauly for fruitful discus-sions. ∗ Correspondence email address:pedro.reis@epfl.ch[1] A. G. Martin, in
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Bridges: Mathe-matical Connections in Art, Music and Science , editedby R. Sarhangi and S. Jablan (Bridges Conference,2001) pp. 21–28, available online at http://archive.bridgesmathart.org/2001/bridges2001-21.html . Smooth triaxial weaving with naturally curved ribbons– Supplemental Information –
Changyeob Baek, Alison G. Martin, Samuel Poincloux, Tian Chen, and Pedro M. Reis
S1 Image processing of the unit cells
In this section, we detail the image-processing algorithm that we developed to process the volumetric images of theunit cells obtained through X-ray micro-computed tomography ( µ CT). The goal is to construct a centerline-baseddescription of each of the ribbons within the. Our image-processing procedure involves the following four steps: (i)acquisition of the µ CT volumetric image of the unit cell, (ii) segmentation of the voxels corresponding to each ribbon,(iii) extraction of the centerline, (iv) refinement and smoothing of the centerline, and (v) quantification of the materialframe of the ribbon. As a result, we obtain a centerline-based description of the ribbons composing the unit cell. Thiscenterline data is then used to calculate the geodesic curvature of the ribbons and interior angles of the unit cell.The framed-centerline of the i -th ribbon is denoted as Γ i = { γ i ; t i , n i , b i } , where the superscript i corresponds theindex of each ribbon in the unit cell. Specifically, γ i is a three-dimensional (3D) discrete curve containing N i vertices; γ i = { γ ij } ( j = 1 , , · · · , N i ) The subscript j denotes the index of the vertices. Each point along the centerline γ ij is assigned with the material frame ( t ij , n ij , b ij ) where t ij is the tangent vector, n ij is the normal vector, and b ij isthe binormal vector. Ultimately, based on the complete description of the centerlines of the ribbons, we compute thefollowing two quantities: (a) the i -th interior angle of the n -gon, θ i , and (b) the integrated geodesic curvature of the i -th segment of the n -gon, κ i = R i κ g d s , of the unit cell. With the quantification of θ i and κ i at hand, we can thenreadily compute the integrated Gauss curvature of the unit cells thanks to the Gauss-Bonnet theorem presented inEq. (1) of the main text.Next, in Sec. S1.1, we will first describe the experimental fabrication of the ribbons and image acquisition protocolusing the µ CT. The fabrication protocol was developed specifically so as to facilitate the µ CT imaging and theextraction of the centerlines of the ribbons. In Sec. S1.2, we then detail an algorithm that we developed in-house toquantify Γ i . Finally, in Sec. S1.3, we compute θ i and κ i based on Γ i . S1.1 Acquisition of the volumetric µ CT images
In the main text, we introduced the fabrication procedure of the unit cells, which comprise n composite ribbons.These bilayer ribbons consist of a layer of metal-infused vinylpolysiloxane (VPS-16, Zhermack) and a layer of polyethy-lene (PETE, Plastic Shim Pack DM1210, Partwell Group). We assembled the composite ribbons using nylon rivets(Snap Rivet 4.2 mm, Distrelec AG, Switzerland). In Fig. S1(a) and (b), we present a photograph of the unit cell anda magnified view of the crossings. In Fig. S1(c), we present a cross-sectional view of the µ CT image, zooming in thevicinity of the crossing of two ribbons. As stated in the main text, the higher radiopacity of the metal-infused VPSlayers enables us to segment the VPS layer. In Fig. S1(d), we present an image whose brightness has been adjustedmanually to highlight only the VPS regions. (a)
10 mm
Composite Nylon rivet (c)
PETE sheetVPS +Metal powderNylon rivetVPS + metal powderNylon rivetPETE (b)
VPS +Metal powder (d) sheetribbon
FIG. S1. (a) Photograph of a representative unit cell woven with composite ribbons, which consist of a bilayer of polyethyleneterephthalate (PETE) and metal-infused vinylpolysiloxane (VPS). The ribbons were assembled into the unit cell using nylonrivets. (b) Magnified view of a crossing region between two ribbons. (c) Cross-sectional view of the crossing region obtainedfrom a µ CT scan. In the X-ray images, the metal-infused VPS layer appears brighter than any other object in the image due toits high radiopacity. (d) The µ CT image can be brightness-adjusted to only shows the metal-infused VPS layer on an otherwisedark background. a r X i v : . [ c ond - m a t . s o f t ] N ov S1.2 Quantifying the centerlines of the ribbons
Since the PETE layers physically separate the metal-infused VPS layers (see Fig. S1d), we were able to isolatevolumetric images of every single ribbon from the rest of the image using the command bwconncomp in MATLAB.Fig. S2(a) shows a reconstructed image of a single ribbon, from which we extract the framed centerline, Γ i , using theprocedure described in this section.We start by manually inputting an ansatz of the centerline, γ = { γ j } , depicted by the red dashed line in Fig. S2(a).We will temporarily omit the superscript i for the ribbon index, for the sake of brevity. To set this ansatz, we firstpicked a few seeding points (the number of manual input is approximately ten) on the volumetric image using theMATLAB command datacursormode and interpolated those points using the MATLAB command fit . Note that γ = { γ j } is not yet the final centerline; the components of γ are still to be iterated for a more accurate representationof the actual centerline. The ansatz γ is a 3D polygonal curve comprising N vertices; γ = { γ j } ( j = 1 , , · · · , N ).The initial value of N is determined from a combination of the resolution of the µ CT tomographic imaging ( a ) andthe total length of the ribbon ( ‘ ). Throughout our study, we used a = 29 . µ m and ‘ ’
30 mm (thus N ’ , γ along the ribbon.Having identified the ansatz of the centerline ( γ ), we updated γ with a refined centerline that is obtained as follows.At each j -th point on the ansatz γ j , we extracted a slice-view of the volumetric image along a plane normal to thetangent of the ansatz at the j -th point, t j ≡ γ j +1 − γ j ( j = 1 , , · · · , N −
1) (see inset of Fig. S2a). By definition,this cutting plane lies on the null-space of t j . For convenience, we denote the basis of this 2D null-space as { u j , v j } .We then obtained the centroid of the slice-view image in ( u j , v j ), using the MATLAB command regionprops (redpoint in the inset of Fig. S2a), and updated the centerline as γ j + u j u j + v j v j → γ j . (S1)The resulting centerline of the ribbon is depicted by the red solid lines in Fig. S2(b). During this stage of refinementof the centerline, the number of discretization points, N , remains unchanged from that of the centerline ansatz.The refined centerline γ = { γ j } would serve as the ‘ exact ’ centerline of the ribbon in the limit of infinitely highresolution of the X-ray scan. However, due to the finite, even if excellent, resolution of our µ CT, the refined centerline isstill too noisy to be used for computing derivative quantities ( e.g. , curvatures). Specifically, the number of voxels alongthe thickness of the ribbon is approximately t c /a = 350 µ m / . µ m ’
10 for the unit cells that we investigated, whichis not sufficiently high to prevent numerical errors from the voxelization process. Therefore, additional smoothing ofthe centerline is necessary. As such, we perform a window-average γ in every m steps:1 m mj X k = m ( j − γ k → γ j . (S2)During this averaging step, the level of discretization of the centerline was decreased from N to N/m . In Fig. S2(c),we present an example of the final centerline, depicted by the black dotted lines. For the analysis of the unit cells, weset m = 15; thus, a total of N ’
70 data points represent each centerline.Having obtained the final centerline γ = { γ j } , we computed the corresponding material frame at each point on thecenterline. The first components of the frame, t j (tangent of the centerline), is computed from the numerical tangentof γ via the relation t j = ( γ j +1 − γ j ) / | γ j +1 − γ j | . In order to obtain the other two components of the material (c) x yz (b) j -th plane (a) j -th plane FIG. S2. (a) A reconstructed µ CT image of an isolated ribbon. We investigate a cross-sectional cut at γ j and quantify itscentroid. (b) Plot of the refined centerlines of the full unit cell. (c) Having obtained a smoothed centerline (black dots), wemarch through γ , along its tangent ( t ; green arrows), and investigate its cross-sectional cut (inset) to obtain the material frame. frame ( n j , b j ), we investigated a cross-sectional view of the image along a plane of normal t j with a null-space of thebasis ( p j , q j ) (see inset of Fig. S2c). We then obtained the angle between the p -axis and the principal orientation ofthe j -th cross-sectional view, ω j , using the MATLAB command regionprops . The material frame ( n j , b j ) is thendetermined by rotating the basis ( p j , q j ) with ω j . Finally, the material frame at the j -th point of the centerline, γ j ,is expressed as t j = γ j +1 − γ j | γ j +1 − γ j | , n j = − (sin ω j ) p j + (cos ω j ) q j , b j = (cos ω j ) p j + (sin ω j ) q j . (S3)A representative example of the final centerline, together with the corresponding material frame, was presented inFig. 2(b) of the main text. S1.3 Measurement of θ i and κ i g Having obtained the framed centerlines Γ i = ( γ i ; t i , n i , b i ) from the µ CT images, we can now quantify both the i -th interior angle of the n -gon, θ i , and the integrated geodesic curvature of the i -th edge of the n -gon, κ i g = R i k g d s .These quantities will be used directly for the calculation of the integrated Gauss curvature of the unit cells throughthe Gauss-Bonnet theorem in Eq. (1) of the main text. First, θ i is obtained by measuring the angle between a pairof centerlines involved in the i -th crossing. Second, κ i g is measured by summing up the discrete geodesic curvatureof the i -th centerline along the i -th edge of the n -gon. The integrated geodesic curvature along the framed centerline { γ ij } between the points j = a and j = b (1 < a < b < N i ) can be expressed as [1, 2]: κ i g = b X j = a (cid:20)(cid:18) t ij − × t ij | t ij − || t ij | + t ij − · t ij (cid:19) · n ij (cid:21) . (S4)Specifically, the term inside the parenthesis in Eq. (S4) quantifies the discrete curvature vector of the j -th vertex.The component of the discrete curvature along the normal vector n j is the discrete geodesic curvature. Since it isalready an integrated quantity (hence, dimensionless), we obtain the integrated geodesic curvature by summing theterm inside the square brackets in Eq. (S4), from j = a to j = b .In Fig. 2 of the main text, we reported the average interior angle, h θ i = n P ni =1 θ i , and the average integratedgeodesic curvature, h κ g i = n P ni =1 κ i g , of representative units cells constructed with initially straight ribbons. InFig. S3(a), we plot h θ i for unit cells constructed with curved ribbons as a function of the nondimensional segmentcurvature, κ . We focus on three specific cases of weaves with n = { , , } ribbons. In that same figure, we also plotthe data obtained from the corresponding series of FEM simulations (dashed lines), which are in excellent agreementwith the experiments. More detail on the FEM simulations is provided below, in Sec S2. The average interior angleswere obtained by averaging the angles in a single unit cell, { θ , θ , · · · , θ n } , and the associated standard deviationwas found to be negligible (smaller than the symbol size in the plot). We recall that, for traditional weaving (straightribbons), the only possible value of the interior angle was h θ i = 2 π/
3, regardless of the value of n (see Fig. 2 of themain text). By contrast, the unit cells with naturally curved ribbons exhibit a continuous range of interior angles bytuning the in-plane geometry of the ribbons.In Fig. S3(b), we plot the average integrated geodesic curvature along the n -gon, ¯ κ g , as a function of the injectedin-plane curvature along the middle segment ( κ ) of each ribbon. As stated in the main text, κ contributes directlyto the geodesic curvature of the segments of n -gon: the amount of injected curvature κ = k ‘ is equal to themeasured integrated geodesic curvature κ i g = R i k g d s . Again, this finding confirms that the geodesic curvature ofthe ribbon ( k g ) remains unchanged from that of the reference configuration ( k ) during deformation, thus yielding R i k ig d s = R i k d s = k ‘ (black solid line in Fig. S3b). The FEM measurements of ¯ κ g do not have any observablestandard deviation since the rotational symmetry of the problem is well imposed in the FEM mesh. However, theexperimental measurements do have some level of uncertainty, presumably due to imperfections originating from thefabrication process and partly due to the finite resolution of the µ CT scan. The error bars in Fig. S3(b) represent thestandard deviation of κ i g ( i = 1 , , · · · , n ), measured from two independent scans per configuration. In any case, theuncertainty of the experimental measurements remains relatively small (of the order of π/ -0.5 0.50-0.25 0.25 Norm. segment curv.,Exp. FEM A vg . g e od . i n t e g . c u r v ., A vg . i n t e r i o r a ng l e , (a) (b) Exp. FEM -0.5 0.50-0.25 0.25
Norm. segment curv.,
FIG. S3. (a) Average interior angle, h θ i = n P ni =1 θ i , and (b) Average integrated geodesic curvature, h κ g i = n P ni =1 κ i g , asfunctions of the nondimensional segment curvature, κ , for woven unit cells with n = { , , } . For the FEM simulations, onlythe results with n = 6 are presented. The error bars for h κ g i represent the standard deviation of the κ i g measurements withina single unit cell. S2 FEM simulations of the representative unit cells
We followed the Finite Element Method (FEM) to perform simulations of the woven unit cells using the commercialpackage Abaqus 6.14. Quadratic shell elements were used to model the ribbons with a linear elastic material model( E = 3000 MPa , ν = 0 . n edges (seeFig. S4a). The vertices of this polygon are the overlapping inner rivet holes. Note that the edge length of this polygondecreases as the absolute value of the curvature of the center segment increases. Each rivet holes (both inner andouter) was kinematically tied in all degrees of freedom (DOFs) to a reference point at the center of the hole. For theoverlapping inner rivet holes, the two respective reference points were then kinematically tied in all DOFs such thatthey preserve planarity.The n pairs of outer rivet holes do not, in general, overlap at the start of the simulation. Each pair of twocorresponding rivet holes was connected to a string (purple solid lines in Fig. S4a). The length of these stringswas then progressively reduced during the course of the simulation. The pairs of reference points (corresponding tothe rivet holes) were connected in a way so that their rotational degrees of freedom were constrained to maintainplanarity. For the initiation of the weaving simulation, the first loading step perturbed one vertex within the mesh in n-gonRivetholes Wire (a) (b) von Mises Stress (MPa)Centerline Edge line z yxzy x FIG. S4. (a) Numerical setup and (b) representative results of the FEM simulations of a unit cell with n = 7 ribbons, with ‘ = 15 mm and κ j = [0 . , . , . n edges, and strings are used to attach the end points. (b) As the length of the strings is reduced to 0, the initiallyflat shape becomes curved and exhibits a 3D shape. Von Mises stress were plotted over the deformed meshes. the z -direction to break the x - y plane symmetry. The second loading step disabled this perturbation. A displacementwas then applied to the reference points (by shortening the length of the connecting strings) to bring them closerto each other. Eventually, the length of each string (and hence the distance between two reference points within amatching pair) was identically zero, at which point the unit cell was deemed as woven. This procedure is analogousto the sequence of physical weaving performed by hand during the experiments.Each ribbon was partitioned along the arc length such that the discrete material centerline, γ ij , could be extractedfrom the mesh nodes. As the quadratic mesh edge length was fixed to 0 .
75 mm, the mesh nodes along the arc wereapproximately equidistant. We follow the same notation introduced in Sec. S1. An analogous set of mesh nodes alongthe edge of each ribbon was used to recreate the edge curve (see Fig. S4a). These nodes were sorted in a clockwisefashion. The edge-based Cosserat frame was constructed at the midpoint between every two consecutive nodes. Thetangent vector t i was defined similarly to what we did in the image processing procedure described in Sec. S1.1. Thebinormal vector b i was constructed between each mid-point and its closest point on the edge curve. The normalvector n i was then obtained from the cross product t i × b i . With this centerline-based description, the integratedgeodesic curvature and the interior angles could be readily obtained in the same way described in Sec. S1.3. Thetotal strain energy accumulated at the completion of the weaving process could also be computed directly from theFE simulations.For n = { , , , , } , a sampling of the possible parameterizations of κ ∗ assuming κ = ± κ , and κ , , ≤ ± . . n -gon (see Fig. S4b).In Fig. S5(a), we plot numerical results for the relationship between the integrated curvature of the unit cell, K n ,and its injected in-plane curvature, κ ∗ . We observe that multiple parametrizations of the unit cell (multiple values of κ ∗ ) can lead to the same value of K n ; e.g. , an integrated Gauss curvature of K n = π/ n, κ ∗ ) = (4 , . , (5 , , (6 , − .
16 ˙6) , (7 , − . , (8 , − . K n , one can either increase the number of ribbons n or decrease the value of κ ∗ . (a) (b) -0.5 0.50-0.25 0.25 I n t e g r a t e d G a u ss c u r v a t u r e , Integrated Gauss curvature, E l a s ti c e n e r gy p e r r i bbon , (c) Integrated Gauss curvature,
FIG. S5. (a) Integrated Gaussian curvature measured from FEM for unit cells woven with n = { , , · · · , } , for the differentparameterizations of κ ∗ = π ( − κ + 2 κ − κ ) (same set of simulations as shown in Fig.3b of the main text). (b) Elastic energyper ribbon of the same unit cells plotted in (a). (c) Elastic energy of unit cells normalized by the number of ribbons times bendingstiffness of the ribbon, EI , as a function of the measured integrated curvature. Three different combinations of E and I areconsidered (with E = { , } MPa and t = { . , . } mm; see legend), for unit cells of n = 6, κ = {− . , − . , . , . } and κ , = 0. The elastic energy of each simulated system once fully woven is plotted Fig. S5(b) for each parametrization set ofthe unit cell. While a desired K n can be achieved using multiple parameterizations, Fig. S5(b) shows that the elasticenergy stored by each ribbon between these parameterizations can vary significantly. Therefore, in addition to satis-fying topological constraints (Euler characteristics), the elastic energy of the woven ribbons can inform the designerregarding which parameter set to use when weaving towards a target shape; i.e. , one would choose the parametrizationwith the lowest energy that still results in the desired K n . This observation begets a formal optimization, outside thescope of this paper, where κ , , are tuned globally such that the target surface is attained while minimizing the totalstrain energy. We hope that future work will address this (far more) complex optimization problem.To further justify that the shape of the weaves is controlled by κ ∗ , independently of the shape and rigidity of theribbons, we conducted two additional sets of simulations for n = 6, κ = {− . , − . , . , . } and κ , = 0. Forthe first set of four simulations, Young’s modulus of the ribbons was set to E = 3000 MPa and the thickness was t = 0 .
25 mm. Then, the bending modulus of the ribbons is varied by changing the material properties by taking E = 300 MPa, and the moment of inertia by considering two thicknesses values, t = 0 .
25 mm and t = 0 .
50 mm.In Fig. S5(c), we plot the elastic energy of the unit cells ( U ) normalized by the bending stiffness of the ribbons, EI = Ewt − ν ) , and the number of ribbons, n , as a function of the measured integrated curvature K . The three setsof simulations collapse onto the same values. The collapse along the horizontal axis, K , shows that the shape ofweaves is independent of both Young’s modulus and the cross-sectional profile of the ribbons. The collapse along thevertical axis, U/ ( nEI ), shows that most of the energy of the ribbons is stored into bending and not into stretchingnor twisting energies. S3 Design of (simple) non-spherical weaves
In this section, we detail the design procedure that we developed to weave simple non-spherical shapes by usingpiecewise circular ribbons. We shall focus specifically on ellipsoidal and toroidal weaves, noting that more complexgeometries and topologies would require a level of numerical optimization beyond the scope of the present work.Fig. S6(a) shows the required inputs for our design procedure: the surface S of the target shape (taking the exampleof an ellipsoid) and a graph on S representing the topological layout of the weave. Each vertex of the graph representsthe crossing points between two ribbons, and each edge represents the connectivity of the crossing points. Note thatthe edges of the graph do not represent the actual ribbons. From the topological layout in Fig. S6(a), we label thelocation of each vertex as γ ij (1 ≤ j ≤ m i ), where the superscript i is the index of the ribbons, the subscript j is theindex of the crossings comprising the i -th ribbon of the weave, and m i is the number of the crossing points on the i -th ribbon. In short, { γ ij } (1 ≤ j ≤ m i ) is an ordered array of the crossing points representing the i -th ribbon of theweave. (a) (b) FIG. S6. (a) Topology of the ellipsoidal weave of a polar radius a = 60 mm, and an equatorial radius b = 40 mm. The curvatureand arclength of the j -th arc of the i -th ribbon, ( κ j , ‘ j ), is obtained by calculating the geodesic turning angle at the adjacentcrossing points, γ ij and γ ij +1 (crossing points are denoted as white circles). (b) The discrete curve { γ j } is embedded on thetarget surface S . The curve consists of m vertices and edges, which we denote as γ j and t j , respectively. The discrete integratedgeodesic curvature that is associated to each vertex is denoted by ψ j . S3.1 Computing the shape of the piecewise circular ribbons
Our goal is to compute the undeformed geometry of the i -th ribbon from the position of its crossings points, { γ ij } .Hereafter, for the sake of brevity, we omit the superscript i (the ribbon index). In Fig S6(b), we sketch a typicalexample of { γ j } , onto which we superpose a representation of the vertices and edges. We regard the set of crossingpoints { γ j } as a discrete curve embedded on S , representing the shape of the woven ribbon. We denote the j -th edgeof the discrete curve, t j , as a vector connecting γ j and γ j +1 ; t j = γ j +1 − γ j . Since the ribbon is closed when woven,the number of edges of the discrete curve is m . Accordingly, we define t m = γ − γ m .Next, we choose to approximate the initial shape of a ribbon as a piecewise-circular curve comprising m segments.Denoting the length and curvature of the j -th segment of the undeformed ribbons by ‘ j and k j , respectively, thedesign problem is reduced to estimating ( ‘ j , k j ) (1 ≤ j ≤ m ) from { γ j } . To do so, first, we compute ‘ j by taking thelength of the j -th edge: ‘ j = | t j | . (S5)As a second step, we need to obtain k j . We compute the discrete geodesic curvature of the curve at each vertex, ψ j ,by quantifying the rotation of the adjacent edges, ( t j − and t j ): ψ j = (cid:18) t ij − × t ij | t ij − || t ij | + t ij − · t ij (cid:19) · N j ( γ j ) (S6)where N ( γ j ) is the normal of the target surface S at γ j [1, 2]. Note that, in this discrete setting, ψ j is defined at eachvertex. By contrast, the piecewise-constant curvature of the undeformed ribbon, k j , is associated with each edge.Therefore, we define the integrated geodesic curvature of the j -th segment of the undeformed ribbon, κ j = k j ‘ j , asthe average of the geodesic curvature of the vertices adjacent to the j -th edge: κ j = ψ j + ψ j +1 ψ j was defined in Eq. (S6).For a closed discrete curve, an example of which is depicted in Fig. S6(b), its last ( γ m ) and first ( γ ) points areconnected. Hence, we note that, for the cases when j = 1 and m , we need exceptions to the definition of ψ j inEq. (S6)) and κ j (Eq. (S7)). When j = 1, ψ is computed from t m and t ( t is not defined). When j = m , we define κ m = ( ψ m + ψ ) /
2, since the last edge ( j = m ) is associated to the pair of points γ m and γ . By doing so, Eqs. (S6)and (S7) are well defined in the full range 1 ≤ j ≤ m .Our design framework requires that both the location of the vertices and the topology of the weave, { γ ij } , bepre-determined as inputs. The systematic generation of the topological layout for arbitrary target shapes is beyondthe scope of our study. Still, we can design simple yet canonical shapes, such as ellipsoids and tori, whose topologicallayout can be readily obtained due to their symmetries. In Sec. S3.2 and Sec. S3.3, we will design the initial shape ofthe ribbons to construct ellipsoidal and toroidal weaves, based on Eqs. (S5) and (S7). S3.2 Ellipsoidal weave
As a first example, we consider an ellipsoidal weave with an equatorial radius, b , and a polar radius, a , such thatthe aspect ratio is a/b . The target shape of this ellipsoid, along with the underlying topological layout, were presentedabove, in Fig. S6. As a starting point of the topological layout, we obtained the coordinates of the vertices from thecoordinates of the Rectified truncated icosahedron [3, 4]. Hence, the ellipsoidal weave will contain 12 pentagons, 20hexagons, and 60 triangles, just as the spherical weave in Fig. 3 of the main text. Then, we performed a linear scalingof the layout by a factor a along the x and y axes and a factor b along the z axis so as to match the layout as closely aspossible to an ellipsoid with the specified equatorial and polar radius. In Figs. S7, we present the shape of undeformedribbons of our three ellipsoidal weaves with an equatorial radius, b = 40 mm, and polar radii, a [mm] = { , , } ,corresponding to the examples presented in Fig. 4a of the main text. These ribbons can printed (or laser cut) andwoven by the interested reader. The geometry of the ribbons at their rest configurations were computed from Eqs. (S5)and (S7). In Fig. S7, the crossing points of the ribbons are denoted as circles. The first and last circles in each ribbonare supposed to overlap in their woven configuration to form closed loops.Due to the axisymmetric shape of the ellipsoid, there are only two distinct types of ribbons (color-coded as redand black in Fig. S7) among the ten ribbons required for the ellipsoidal weave. The red and black arrows in Fig. S7and the inset of Fig. S7 are drawn to illustrate the placement of each ribbon in its planar and woven states. The five
20 mm(a)(b)(c)
FIG. S7. Planar shapes of the ribbons required to weave the three ellipsoidal weaves presented in Fig. 4a of the main text:(a) a = 30 mm , b = 40 mm, (b) a = 50 mm , b = 40 mm, and (c) a = 60 mm , b = 40 mm. Note that the ribbons form closedcurves when woven (the first and the last crossings meet in their final configuration). The weaves consist of five ribbons thatpass the pentagons at the north and south poles (red ribbons) and five black ribbons that do not (black ribbons). ribbons of the first type (red ribbons in Fig. S7) pass the pentagons at the north and south poles, and the other fiveribbons of the second type (black ribbons in Fig. S7) do not. For the given set of parameters describing the geometryof the ellipsoidal target surface, our algorithm yields ribbon designs that are highly irregular and curved. It wouldhave been unfeasible to arrive at these designs by an iterative trial and error approach. S3.3 Toroidal weave
As a second example of a simple non-spherical weave, we design a smooth toroidal weave with piecewise-circularribbons. The inner radius of the torus is r i and the outer radius is r o . We only consider the case with r i >
0, wherethe target surface does not self-intersect. The idea is to generate the topology of the toroidal weave on a rectangulardomain and then map it into a torus by considering the classical parametric expression for the torus [5]: x = (cid:18) ( r o + r i )2 + ( r o − r i )2 cos β (cid:19) cos αy = (cid:18) ( r o + r i )2 + ( r o − r i )2 cos β (cid:19) sin αz = ( r o − r i )2 sin β. (S8)We defined the toroidal angle coordinate as α (along the direction of the small circular ring around the surface) andthe poloidal angle coordinate as β (along the direction of the large circular ring around the torus, encircling the centralvoid). In Fig. S8(a), we present the rectangular domain ( α, β ) ∈ [0 , π ) × [0 , π ), where we draw the regular periodichexagonal (triaxial) pattern that repeats along the α and β directions by n α and n β , respectively. As a representativeexample, we set r o = 105 mm, r i = 35 mm, n α = 9, and n β = 4. (a) (b) x [mm] y [mm] z [mm] 00 75 -75-75 750-2525 FIG. S8. Topology of the toroidal weave with r o = 105 mm, r i = 35 mm, n α = 9, n β = 4. (a) The triaxial pattern on thetoroidal-poloidal domain ( α - β ). (b) The mapping of the triaxial pattern in (a), using Eq. (S8), yields the topology used for ourtoroidal weave design.
40 mmRepeating 9 times(a)(b)(c) (d) (e)
FIG. S9. The ribbons comprising the toroidal weave presented in the main text: r o = 105 mm, r i = 35 mm, n α = 9, and n β = 4. The weave consists of ten different ribbons; only five ribbons – from (a) to (e) – are presented due to symmetry. Notethat the ribbons shown, all share the same scale bar (40 mm). The insets illustrate placement of ribbons. β curves on the torus. We found that,when n α and n β are co-prime, both the red and blue segments form single contiguous ribbons. As a result, for ourspecific choice of parameters ( n α = 9, n β = 4), there is a total of ten ribbons – one red ribbon, one blue ribbon,and eight black ribbons – for this specific toroidal weave. Even when n α and n β are not co-prime, it is feasible togenerate a topological layout of toroidal weaves in the same way for arbitrary values of n α and n β . For example, forthe toroidal weave with n α = 10 and n β = 4, there is a total of twelve ribbons – two red ribbons, two blue ribbons,and eight black ribbons.Using the protocol introduced in Sec. S3.1, we computed the undeformed shape of the ribbons comprising thetoroidal weave presented in Fig. S9; the placement of the ribbons on the weave is denoted by the black arrows on eachribbon and all the ribbons are drawn to scale. Among the ten ribbons of the weave, we only present five ribbons inFigs. S9(a)-(e), since the others are mirror-images due to the symmetry. Note that the ribbon shown in Fig. S9(a) isrepeated n α times. Although we designed a toroidal weave for a specific set of parameters, r o = 105 mm, r i = 35 mm, n α = 9, and n β = 4, the design of toroidal weaves presented that we have introduced can be generalized to othertoroidal weaves with arbitrary combinations of the parameters. [1] Bobenko, A. I., Sullivan, J. M., Schr¨oder, P. & Ziegler, G. Discrete differential geometry , vol. 38 (Springer, 2008).[2] Bergou, M., Wardetzky, M., Robinson, S., Audoly, B. & Grinspun, E. Discrete elastic rods.
ACM Trans. Graph. , 63(2008).[3] Barbaro, D. La pratica della perspettiva . Biblioteca di architettura urbanistica (A. Forni, 1980). URL https://books.google.com/books?id=UlNLAQAAIAAJ .[4] Kaplan, C. S. & Hart, G. W. Symmetrohedra: Polyhedra from symmetric placement of regular polygons. In Sarhangi,R. & Jablan, S. (eds.)
Bridges: Mathematical Connections in Art, Music and Science , 21–28 (Bridges Conference, 2001).Available online at http://archive.bridgesmathart.org/2001/bridges2001-21.html .[5] Stoker, J. J.