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ARTICLE

A general method for the creation of dilational

surfaces

Freek G.J. Broeren 1,2*, Werner W.P.J. van de Sande1,2, Volkert van der Wijk1& Just L. Herder 1

Dilational structures can change in size without changing their shape. Current dilational

designs are only suitable for speciﬁc shapes or curvatures and often require parts of the

structure to move perpendicular to the dilational surface, thereby occupying part of the

enclosed volume. Here, we present a general method for creating dilational structures from

arbitrary surfaces (2-manifolds with or without boundary), where all motions are tangent to

the described surface. The method consists of triangulating the target curved surface and

replacing each of the triangular faces by pantograph mechanisms according to a tiling

algorithm that avoids collisions between neighboring pantographs. Following this algorithm,

any surface can be made to mechanically dilate and could, theoretically, scale from the fully

expanded conﬁguration down to a single point. We illustrate the method with three examples

of increasing complexity and varying Gaussian curvature.

https://doi.org/10.1038/s41467-019-13134-0 OPEN

1Department of Precision and Microsystems Engineering, Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology,

Mekelweg 2, 2628 CDDelft, The Netherlands.

2

These authors contributed equally: Freek G. J. Broeren, Werner W. P. J. van de Sande. *email: f.g.j.

broeren@tudelft.nl

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Expandable structures are of signiﬁcant relevance in nature

and engineering and come in a variety of forms. Natural

examples include the stowing of the precious wings of

beetles1or the ﬁtting of young leaves into buds2. Numerous

engineering examples can be found as well, including satellite

antennas and solar panels that need to be compact for launch and

expand for operation3–6, and medical stents that need to be

moved through arteries7or the esophagus8in compacted form

and deploy at the target position.

Most expandable structures rely on an underlying mechanism

to allow them to be reversibly compacted. One well-known

example of an expandable structure in which the mechanism is

clearly visible is the Hoberman Sphere9,10. This mechanism

dilates, i.e., its envelope changes in size without changing its

shape11,12.Wedeﬁne dilational structures as structures composed

of mechanisms whose only degree of freedom corresponds to

dilation. Other examples of dilational structures are dilational

polyhedral linkages13,14.

Dilational structures have also been studied in the ﬁeld of

mechanical metamaterials15,16, particularly for auxetic behavior

where the Poisson’s ratio is negative17–19. A Poisson’s ratio of

exactly -1 corresponds to dilation.

Currently, several limitations in dilational structures exist.

Firstly, most dilational structures have been designed for a spe-

ciﬁc shape or curvature, making these mechanisms applicable to a

very limited set of shapes; for example, the buckliball20, which is

based on a polyhedral linkage that resembles a sphere. Secondly,

linkages such as the Hoberman Sphere use mechanisms that

move perpendicular to the described surface, making them pro-

trude into the enclosed volume, which, for instance, could be a

problem in stent design. Thirdly, to the authors knowledge, no

examples of spatial mechanical metamaterials exist that can be

sculpted to thin, curved surfaces. The unit cell that governs the

behavior of such structures inherently has a ﬁnite volume, making

the construction of thin dilational surfaces with only motion

tangent to the plane impossible. Also, the current planar auxetic

metamaterials can not, in general, be used, since the underlying

kinematics are not valid for arbitrarily curved surfaces.

In this paper, we present a general method to create

mechanism-based dilational structures ﬁtting to any spatially

curved surface, by which we mean 2-manifolds with or without

boundary. Our method improves on existing work on two key

points. Firstly, the resulting mechanism structure is placed on the

surface, with no parts of the mechanism moving into the enclosed

volume and normal to the surface, unlike for instance the

Hoberman mechanism. Secondly, the method is applicable to

surfaces with any curvature and can even be applied to non-

closed surfaces, i.e., surfaces containing holes or cuts. Enabling

these properties in dilational structures makes them of use in, for

instance, structures that grow with a person such as medical

braces for children and expandable furniture, medical devices that

require stowability or compression but need to be stiff otherwise,

or implants that need to accommodate some motion but main-

tain their shape, such as aortic stents.

In the following, we describe the method, where we ﬁrst tri-

angulate the surface and then place pantograph mechanisms on

each of the faces of the triangulation. We prove that this method

can be used for any spatially curved surface and comment on the

maximum scaling factor possible for these structures. Finally, we

apply our method to three surfaces of increasing complexity,

illustrating its versatility.

Results

Dilation. Dilation is a homothetic transformation that relates two

similar shapes with respect to a homothetic center21. Any two

ﬁgures related by a dilation are similar and have the same

orientation (see Fig. 1). This transformation preserves the shape

and orientation of the ﬁgure, but changes the size of each of

the elements of the structure by the same factor. In a dilational

structure the distances between a representative set of points on

the structure, typically corner joints, all scale by the same factor

during actuation.

Triangulation. The ﬁrst step of the presented design method is to

triangulate the curved surface from which we want to create a

dilational structure. Triangulation is a common strategy to

approximate curved surfaces by a mesh of triangular faces and lies

at the basis of the STL ﬁle format used in 3D printing22,23. Tri-

angulation is illustrated in Fig. 2for a sphere. It is observed that

the accuracy of the approximation increases with the number of

triangular faces in the resulting mesh.

The triangulation results in a polyhedral surface with only

triangular faces. It can be shown that every surface (by which we

mean a 2-manifold with or without boundary) can be

triangulated such that at most two triangular faces share an

edge24. If the resulting polyhedron undergoes dilation, the

number of triangles, their shapes and their respective relations

must stay constant. Only the size of the triangles is allowed to

change, their aspect ratio and orientation are preserved.

Pantograph linkage. To transform the polyhedral surface into a

movable linkage with dilational motion, we employ the skew, or

Sylvester’s, pantograph mechanism25–27.

The skew pantograph is a four-bar mechanism of which two

adjacent links are extended into triangles. The mechanism has

revolute joints at p,q,Cand r, as illustrated in Fig. 3. The link Cq

is equal in length to side rp, as are Cr and qp, which makes pqCr a

parallelogram. Also, the triangles Apq and pBr are similar. A

resulting feature from these properties is that in any pose of the

mechanism the triangle ABC is similar to triangles Apq and pBr.

The proof can be found in Hall (1961)28. When the mechanism

moves, the distances AC,AB, and BC become smaller as the

parallelogram decreases in area. As a result, the mechanism has a

single degree of freedom and during motion the striped red

triangle described by its three similarity points (indicated A;B,

and C) changes in size but remains similar of shape, as is

illustrated for three poses. These three joints will be referred to as

the similarity points of the pantograph linkage.

The maximum scaling that can be achieved with a skew

pantograph depends on the placement of the joints on the edges

of the spanned triangle in the neutral position. We place the joint

in the middle of the sides of the spanned triangle in the neutral

H

Fig. 1 Dilation of structures. Under dilation, a structure scales with respect

to a homothetic center (point H in this ﬁgure), preserving its size and

orientation

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position, such that Aq ¼qC. In this way, the rigid triangles are

sized down by a factor 2 relative to the spanned triangle, which

allows the spanned triangle to scale down to a single point.

Coupling the pantographs. Each of the faces obtained by tri-

angulating the curved surface is replaced by a skew pantograph

mechanism. In this way, we ensure that each individual face can

only deform by scaling, keeping its original shape.

The pantograph mechanisms ensures the proper scaling of

each individual face. However, in order for the whole structure to

dilate, it is also required that each face simultaneously scales by

the same factor and that the faces do not rotate with respect to

each other. We achieve this by connecting neighboring

pantograph mechanisms by means of compound universal joints,

whose description follows.

Two adjacent faces of the triangulated surface share a single

edge and two vertices. In order to maintain the mobility of the

neighboring pantograph mechanisms, they can only be connected

at the vertices. At these points, we connect the pantograph

mechanisms with universal joints (two consecutive revolute

pairs), of which the axes are parallel to the normals of the

respective faces, as is illustrated in Fig. 4. This conﬁguration

constrains the rotation about the shared edge of the faces and

therefore preserves the relative orientation of the two faces.

Because the two neighboring pantograph mechanisms are now

connected along the shared edge, their degrees of freedom are

also coupled. When one of the pantograph mechanisms moves,

the length of the shared edge will change, causing a movement in

the other pantograph. In this way, it is ensured that both

pantographs are scaled simultaneously by the same factor and

maintain their relative orientation.

Each set of neighboring pantograph mechanisms is connected

in this way, creating compound universal joints at the corners of

the faces. This preserves the relative angles of all faces and ensures

the same scale factor for each of the faces. Therefore, the total

resulting motion will be dilation.

Range of motion of the pantograph. Kinematically, a structure

constructed from the proposed pantograph mechanisms can scale

down to a point. In reality, the range of motion of the planar

pantographs is limited because of collisions among the rigid

bodies that make up the pantographs. In this section, we highlight

the factors limiting the range of motion of the dilational

mechanisms constructed from pantographs and discuss how to

minimize their effects.

n = ∞n = 20 n = 12 n = 8

Fig. 2 Triangulation of surfaces. Any curved surface, in this case a sphere, can be approximated by a mesh of triangles in a process called triangulation. A

larger mesh of triangular faces gives a better approximation of the original surface. Shown are polyhedral triangulations of a sphere with ntriangular faces

AB

r

q

C

p

Fig. 3 The skew pantograph. The skew pantograph is a one-degree-of-freedom mechanism that scales the spanned triangle (i.e., the red-striped area)

determined by three of its joints; the similarity points, indicated A;B, and C. This mechanism has revolute joints at points p;q;C;and r. For the neutral

position, shown in the middle, the spanned triangle equals the triangular shape of the pantograph, this is a useful pose for constructing dilational surfaces

Fig. 4 Connections between pantographs. The relative orientation of

adjacent pantograph mechanisms is maintained by compound universal

joints. This ﬁgure shows part of a dilational surface, the triangular faces are

shown in gray, and the rigid parts of the pantograph mechanisms are shown

in blue. At each of the corners of the pantographs, they are coupled to their

neighbors by universal joints. A universal joint consists of two revolute

joints (shown as white cylinders) in series. A compound universal joint is

created at an intersection with three (or more) triangular faces

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We have described the pantograph mechanism used to make

the triangular faces of the polygons dilational. The motion of the

mechanism can be described by a single parameter θand the sides

of the spanned triangle are scaled by a factor λ¼cosðθÞwhen the

mechanism is actuated29. The area of the spanned triangle is then

scaled by a factor of λ2¼cos2ðθÞ, see Fig. 5.

Starting at the neutral position, where θ¼0, the mechanism

can move in two directions: θcan increase or decrease,

corresponding to a counter-clockwise or clockwise rotation of

the lower left (green) rigid triangle. In both cases, the effect on the

scaling of the spanned triangle will be the same, since cosðθÞis

symmetric around θ¼0.

In the case where θincreases, the mechanism will protrude out

of the spanned triangle at two edges, while it will open up free

space at the third edge. Conversely, when θdecreases, the

mechanism protrudes out of the spanned triangle at one edge,

and opens up space at the other two edges.

When the rigid bodies of the pantograph mechanism are

allowed to overlap and cross each other, the minimum area of the

spanned triangle is obtained at θ¼±π

2for the two different cases,

both resulting in a scaling factor λ¼0. When collisions are

considered, these values of θcan no longer be reached and often,

the scaling factor λdiffers between the two motion directions.

If the pantographs are designed to be planar and therefore move

within a single plane, the range of motion is limited by internal

collisions of the links. All pantograph mechanisms in the dilational

surface are linked to have a single degree of freedom. Therefore,

when one pantograph is actuated such that, for that mechanism, we

obtain a rotation θin its triangular faces, all other pantograph

mechanisms in the dilational surface will have a rotation of ± θ.

This causes the complete mechanism to reach the end of its motion

as soon as self-collision occurs in any pantograph of the structure.

Therefore, this is the main limiting factor on the maximum scaling

factor of the assembled dilational mechanism.

Looking at Fig. 5, we can see that the pantograph mechanism

reaches the end of its range of motion when θ¼α

2for one

direction of motion, or when θ¼πα

2for the other direction, where

αis the top angle of the spanned triangle. At these points, the binary

links of the pantograph mechanism become collinear. In this

calculation, we have considered the links as lines of zero width. In

reality, the links and joints from which the pantograph mechanisms

are constructed will have ﬁnite width. This will cause collisions to

happen earlier and the range of motion to be limited further.

The total range of θis π

2radians, because the internal angles

between the links at two adjacent corners of a parallelogram four-

bar linkage always add up to π. For the case where the links have

zero width, the optimal scaling factor would be found when α¼0

or when α¼π, allowing only for motion in one direction.

However, in both of these cases, the pantograph degenerates to

a line, in which case no feasible mechanism would be possible.

For simplicity and ease of tiling, it is beneﬁcial when both

motions directions have the same range from the neutral position.

This is the case for α¼π

2; i.e., when the pantograph mechanisms

are right-angled. In this case, the maximal scaling factor is

λ¼cos π

4

¼ﬃﬃ

2

p

20:71. So, when self-collisions are considered,

the distances between points on the dilational surface can be

scaled down by at most 29% relative to the neutral position.

Placement of the pantographs. When the pantograph mechan-

ism moves, some parts of the mechanism protrude out of the

spanned triangle, while other parts move into the spanned tri-

angle. When all the faces of a triangulated surface are replaced by

pantograph mechanisms, two neighboring pantographs could

have parts moving out of the respective spanned triangles at their

shared edge. This will cause neighboring pantographs to collide,

locking the motion of the structure and thereby no longer

allowing the scaling of the structure. To remedy this, we have

created an algorithm that places the pantographs on a triangu-

lated surface such that neighboring pantographs move along with

each other, i.e., when one side of a pantograph has parts that

move out of the spanned triangle, the corresponding side of the

neighboring pantograph will have parts moving inwards. The

algorithm consists of the following procedure.

We ﬁrst construct the dual graph to the triangulated surface. In

this graph, there is one node for each triangular face and two

nodes are connected if the two corresponding faces share an edge.

Such a graph is shown in Fig. 6a for the octahedron. Note that, as

was mentioned in Section “Triangulation”, at most two faces

share an edge since the original surface is a 2-manifold. The dual

graph to an octahedron is shown in Fig. 6a. For closed surfaces,

this creates a simple, connected, 3-regular graph. We assign a

direction to each of the edges in the graph to represent the

motions of the pantograph mechanisms placed on the triangu-

lated surface. A directed dual graph for an octahedron is shown in

Fig. 6b. Since each edge in the dual graph can only have a single

direction, the sides of the triangles are enforced to move along

with each other.

The movement of the pantograph mechanisms is such that

either the links on two sides move out of the spanned triangle and

the links on the other into it or vice-versa, as shown in Fig. 3.

Therefore, we require for each vertex of the graph that its

AB

C

θ

A

A

= 0

B

B

COS2()

C

C

2

2

Fig. 5 Range of motion of the skew pantograph. The limits of scaling of a skew pantograph. The area of the spanned triangle (shown as a red-striped area)

scales with cos2ðθÞ. The red dots illustrate the values of θin each of the drawings. The end of motion is reached when the bars of the linkage become

collinear (left and right drawings). Between these states, the rotation angle of θis always π

2. The distribution of this range over the left and right motion

directions depends on the top angle α

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indegree and outdegree are larger than zero. In Supplementary

Note 1, we show that for each simple connected graph where

every node has a degree of at least 2, it is always possible to ﬁnd

an orientation of the graph such that both the indegree and

outdegree of every node are larger than zero. The dual graph to

every feasible triangulated surface always has nodes of degree at

least 3; nodes with degree larger than three correspond to holes in

the surface. Therefore, there must exist an orientation of the

pantographs on each surface such that movement without

collisions between the pantograph mechanisms is possible.

To ﬁnd a suitable orientation, we use an algorithm that searches

for ﬂows through the representing graphs. A ﬂow through a vertex

ensures that the difference between the indegree and outdegree is

one. This algorithm is further discussed in Supplementary Note 2.

Once this orientation is found, the pantographs can be placed

accordingly, see Figs. 6c, d for an example.

Examples. In this section, we will show the application of our

method to three surfaces: an octahedron as a simple example29,a

cardioid with both positive and negative Gaussian curvature, and

the Stanford Bunny as an advanced example. For all three examples,

the reported maximum scaling is based on the pantographs in the

resulting structure with the largest and smallest top angle, as

described in subsection “Rangeofmotionofthepantograph”.

We start with the octahedron. This polygonal surface can be

viewed as a very rough triangulation of a sphere, comprised of

only 8 triangular faces. The eight faces of the octahedron are

replaced by pantograph mechanisms, see Fig. 7and

Supplementary Movie 1. In this way, a dilational surface with

only equilateral faces is obtained (α¼60); this gives them a

range of ½30;60. Any placement of the pantograph linkages

will include pantographs with opposite motion directions, the

maximum scaling can therefore be calculated to be

λ¼cosð30Þ¼0:866.

As a second example, we look at the cardioid. The cardioid is a

planar curve obtained by tracing a point on a circle, which is

rolled around a second circle with equal radius. This curve can be

parameterized as follows:

xðtÞ¼að2 cosðtÞcosð2tÞÞ

yðtÞ¼að2 sinðtÞsinð2tÞÞ:ð1Þ

By revolving this curve around the x-axis, we obtain a spatial

surface, as is shown in Fig. 8.

We have triangulated this shape by taking a planar map of the

surface and performing a Delaunay triangulation30,31 on this

map. The points of the triangulation have been chosen to

minimize the number of sharp angles in the triangulation. This

triangulation is shown in Fig. 9a.

On this triangulated surface, we apply our method. First, the

dual graph of this surface is determined and we apply

the placement algorithm on that graph to determine a suitable

placement of the pantograph mechanisms. When the mechan-

isms are placed on the surface according to this placement, we

obtain the shape shown in Fig. 9b and Supplementary Movie 2.

For the cardioid we have constructed here, the rotation angle θ

can lie in the range ½20:5;20:3, resulting in a maximum

scaling factor of λ¼0:937. A movie of the resulting dilational

surface moving between its extremal points is included in the

supplementary materials.

As a ﬁnalexample,wehavetakentheStanfordBunny

32,shown

in Fig. 10a. For the bunny, we took an available triangulation33,and

edited the triangulation manually to remove the triangles with the

sharpest angles in order to increase the maximum scaling factor.

The resulting model is shown in Fig. 10b. This mesh was then fed

into our algorithm, which computed a suitable placement of the

pantograph mechanisms. The resulting dilational mechanism is

shown in Fig. 10c and Supplementary Movie 3.

For this mechanism, the rotation angle θcan lie in the range

½15:0;13:1, resulting in a maximum scaling factor λ¼0:966.

This scaling factor is not limited by the shape of the Stanford

Bunny, but rather by the speciﬁc triangulation used to

approximate it and the placement of the pantographs on the

triangulation. The maximum scaling factor could be increased

further by triangulating the Stanford bunny such that the

triangles are close to equilateral, thereby removing even more

sharp angles from the polyhedron. Even so, the linear scaling of

3.4% obtained here is already similar to the diametric expansion

of human arteries during the cardiac cycle34.

ab c

Fig. 7 Dilation of an octahedron. Eight pantograph linkages are placed on the octahedron. ashows the wireframe of an octahedron. bshows the dilational

structure in the neutral position, (θ¼0) and (c) shows a compacted position (θ¼25)

ab

cd

Fig. 6 Tiling method. Steps in obtaining a correct tiling pattern of

pantographs: adual graph of the net, where the vertices are the faces of the

triangulated surface, bdirected dual graph with correct orientation, c

correct tiling pattern, dcorrect tiling pattern with θ¼20

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Discussion

In this work, we introduce a comprehensive strategy to achieve

dilation of any surface. We do this by triangulating the surface

and replacing the triangular faces with Sylvester’s pantographs.

The similarity points of this pantograph always span a similar

triangle. We constrain these triangles in such a way that their

orientation is preserved, this preserves the shape of the triangu-

lated surface while allowing it to scale.

Kinematically, a structure constructed using this strategy can

be scaled to a single point from its original size. In practice,

however, the range of motion of the pantographs is limited by

both collisions between links within a pantograph, and collisions

between neighboring pantograph mechanisms.

Collisions within the pantograph linkage cause the pantograph

with the smallest range of motion to limit the motion of the whole

structure, since all pantographs share a single degree of freedom.

This could be improved by changing the triangulation strategy and

optimizing the placement of the pantographs such that the top-

angle of the triangles comes out more favorable (possibly favoring

one motion direction over the other). Better pantograph placements

might be found, since our pantograph placement algorithm yields

non-unique solutions to the pantograph placement problem.

We avoid collisons between neighboring pantograph mechan-

isms by tiling them in a speciﬁc manner. We used a graph-based

approach to generate suitable placemements of mechanisms and

showed that this approach works for any triangulated surface.

We have illustrated our method with three examples: an

octahedron, a cardioid and the Stanford bunny. These surfaces

increase in complexity and have varying Gaussian curvature. For

the octahedron, the maximal scaling and suitable tiling can be

determined by hand. For the cardioid and the Stanford bunny,

there are many more triangular faces and the faces are more

irregular, for which we present computational methods to gen-

erate dilational structures for these surfaces.

The planar kinematics of the pantographs ensure that the

resulting dilational mechanism stays close to the described surface

throughout the range of motion. This leaves the encompassed

interior entirely empty.

A interesting side-effect is that our implementation of the

method is directly compatible with the often used STL ﬁle format

for 3D objects. As such, our strategy could be implemented as a

one-click solution to create dilational models.

Methods

STL preparation. The STL ﬁles for the octahedron and the cardioid were created

using OpenSCAD and Matlab, respectively. For the Stanford Bunny Model, the

original STL ﬁles were prepared using OpenSCAD and Matlab. For the Stanford

Bunny model, the original triangulated model by Thingiverse user johnny633 was

adjusted in Blender to decrease the number of sharp corners in the triangular faces.

The used STL ﬁles have been made available in the Supplementary Data.

Algorithm implementation. The algorithm described in Supplementary Note 2

was implemented in Python3, making use of the NetworkX35 and numpy-stl

modules.

First, the prepared STL ﬁles were read using the numpy-stl module and the

faces, points and edges were extracted. From this information, the dual graph of the

triangulated structure was constructed. On this graph, the tile placement algorithm

was run to obtain a suitable placement of the pantographs.

The STL ﬁles were processed using Python3, using the STL module, the faces,

points and edges were extracted. This information was then used to create dual

graph of the triangulated structure. On this graph, the triangle placement

algorithm, as described in Supplementary Note 2, was run to obtain a suitable

placement of the pantographs. Using this placement, two lists were created, one

containing the three vertex positions for each face, always starting with the top

vertex of the pantograph, the other indicating the motion direction for the face.

These lists were ﬁnally used to create 3D models of the tiled structures in

OpenSCAD.

ab c

Fig. 10 Dilation of the Stanford bunny. ashows the original Stanford bunny32.bshows our adaptation of the triangulated version by Thingiverse user

johnny633, which was used to create the dilational surface shown in cby replacing each triangular face with skew pantograph mechanisms. The resulting

surface has a scaling factor of λ¼0:966. A movie of the ﬁnal mechanism is included in the supplementary material

Y

X

ab

Fig. 8 The cardioid surface. The cardioid surface is constructed by revolving

the planar cardioid curve. ashows the curve, bshows the complete

revolved cardioid

ab

Fig. 9 Dilation of a cardioid. The surface of the cardioid is ﬁrst triangulated

(a), after which each triangular face is replaced by a skew pantograph (b)

to obtain a dilational surface. A movie of the ﬁnal mechanism is included in

the supplementary material

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Movie generation. Using the models of the dilational structures, movies were

created by rendering the models for a range of rotation angles in OpenSCAD. The

rendered frames were then combined into a movie using FFMPEG.

Data availability

The datasets generated during and/or analyzed during this study are available in the 4TU.

ResearchData repository, DOI: 10.4121/uuid:36cfec67-aa04-469f-ac13-64e7d95a0c18.

Code availability

Algorithms used in this study are included in this published article (and its

supplementary information ﬁles). Code will be made available from the corresponding

author upon reasonable request.

Received: 4 June 2019; Accepted: 21 October 2019;

Published online: 15 November 2019

References

1. Brackenbury, J. H. Wing folding in beetles. In IUTAM-IASS Symposium on

Deployable Structures Theory and Applications 37–44 (Springer, Dordrecht,

2000).

2. De Focatiis, D.S.A. & Guest, S. Deployable membranes designed from folding

tree leaves. Philos. Trans. R. Soc. London A Math. Phys. Eng. Sci. 360, 227–238

(2002).

3. Fernandez, J. M., Schenk, M., Prassinos, G., Lappas, V. J. & Erb, S. O.

Deployment mechanisms of a gossamer satellite deorbiter. 15th European

Space Mechanisms Tribology Symposium (Noordwijk, The Netherlands,

2013).

4. Schenk, M. Kerr, S. G., Smyth, A. M. & Guest, S. D. Inﬂatable cylinders for

deployable space structures. In Proceedings of the First Conference

Transformables (eds Escrig, F. & Sanchez, J.) (Seville, 2013).

5. Petroski, H. Engineering: deployable structures. Am. Sci. 92, 122–126 (2004).

6. You, Z. & Pellegrino, S. Foldable bar structures. Int. J. Solids Struct. 34,

1825–1847 (1997).

7. Kuribayashi, K. et al. Self-deployable origami stent grafts as a biomedical

application of Ni-rich TiNi shape memory alloy foil. Mater. Sci. Eng. A 419,

131–137 (2006).

8. Ali, M. N., Busﬁeld, J. J. C. & Rehman, I. U. Auxetic oesophageal stents: structure

and mechanical properties. J. Mater. Sci. Mater. Med. 25,527–553 (2014).

9. Hoberman, C. Reversibly expandable doubly-curved truss structure. United

States Patent and Trademark Ofﬁce US4942700 (1990).

10. Hoberman, C. Retractable structures comprised of interlinked panels. United

States Patent and Trademark Ofﬁce US6739098 (2002).

11. Milton, G. W. Composite materials with poisson’s ratios close to 1. J. Mech.

Phys. Solids 40, 1105–1137 (1992).

12. Milton, G. W. New examples of three-dimensional dilational materials. Phys.

Status Solidi 252, 1426–1430 (2015).

13. Kiper, G. & Söylemez, E. Polyhedral linkages obtained as assemblies of planar

link groups. Front. Mech. Eng. 8,3–9 (2013).

14. Gosselin, C. M. & Gagnon-Lachance, D. Expandable polyhedral mechanisms

based on polygonal one-degree-of-freedom faces. Proc. Inst. Mech. Eng. Part C

J. Mech. Eng. Sci. 220, 1011–1018 (2006).

15. Bertoldi, K. Harnessing instabilities to design tunable architected cellular

materials. Annu. Rev. Mater. Res. 47,51

–61 (2017).

16. Zadpoor, A. A. Mechanical meta-materials. Mater. Horizons 3, 371–381

(2016).

17. Evans, K. E. Auxetic polymers: a new range of materials. Endeavour 15,

170–174 (1991).

18. Evans, K. E. & Alderson, A. Auxetic materials: functional materials and

structures from lateral thinking! Adv. Mater. 12, 617–628 (2000).

19. Greaves, G. N., Greer, A. L., Lakes, R. S. & Rouxel, T. Poisson’s ratio and

modern materials. Nat. Mater. 10, 823–837 (2011).

20. Shim, J., Perdigou, C., Chen, E. R., Bertoldi, K. & Reis, P. M. Buckling-induced

encapsulation of structured elastic shells under pressure. Proc. Natl Acad. Sci.

USA 109, 5978–5983 (2012).

21. Meserve, B. Fundamental Concepts of Geometry. (Addison-Wesley,

Cambridge, 1955).

22. Lennes, N. J. Theorems on the simple ﬁnite polygon and polyhedron. Am. J.

Math. 33, 37 (1911).

23. Grimm, T. The rapid prototyping process. In User’s Guide to Rapid

Prototyping. Chapater 3, 49–84 (Society of Manufacturing Engineers (SME),

2004).

24. Moise, E.E. Geometric Topology in Dimensions 2 and 3, Vol. 47 of Graduate

Texts in Mathematics, Chapter 4, 38 (Springer-Verlag, New York, 1977).

Lemma 4.

25. Sylvester, J. J. On the plagiograph aliter the skew pantigraph. Nature 12, 168

(1875).

26. Sylvester, J. J. History of the plagiograph. Nature 12, 214–216 (1875).

27. Dijksman, E. Motion Geometry of Mechanisms. (Cambridge University Press,

Cambridge, 1976).

28. Hall, A. S. Kinematics and Linkage Design. (Prentice-Hall Inc., Englewood

Cliffs, 1961).

29. Broeren, F. G. J., van de Sande, W. W. P. J., van der Wijk, V. & Herder, J. L.

Dilational triangulated shells using pantographs. 2018 International

Conference on Reconﬁgurable Mechanisms and Robots (ReMAR) 1–6 (Delft,

The Netherlands, 2018).

30. Delaunay, B. Sur la Sphère Vide. Bull. Acad. Sci. USSR 12, 793–800 (1934).

31. Lee, D. T. & Schachter, B. J. Two algorithms for constructing a Delaunay

triangulation. Int. J. Comput. Inf. Sci. 9, 219–242 (1980).

32. Stanford University Computer Graphics Laboratory Stanford bunny. http://

graphics.stanford.edu/data/3Dscanrep/ (1994).

33. johnny6. Low poly stanford bunny. https://www.thingiverse.com/thing:151081.

Licensed under CC Attribution-Non-Commercial (2013).

34. Weissman, N. J., Palacios, I. F. & Weyman, A. E. Dynamic expansion of the

coronary arteries: Implications for intravascular ultrasound measurements.

Am. Heart J. 130,46–51 (1995).

35. Hagberg, A. A. Schult, D. A. & Swart, P. J. Exploring network structure,

dynamics, and function using network X. In Proceedings of the 7th Python in

Science Conference (eds Varoquaux, G., Vaught, T. & Millman, J.), 11–15

(Pasadena, 2008).

Acknowledgements

We thank M.P. Noordman (University of Groningen) for assistance with the formulation

of the graph theory proof. The work was part of the Nanoscience Engineering Research

Initiative of TU Delft. The authors would like to acknowledge NWO-TTW (HTSM-2012

12814: ShellMech) for the ﬁnancial support of this project.

Author contributions

F.G.J.B. and W.W.P.J.S. proposed and designed the research, performed the numerical

calculations and wrote the paper. F.G.J.B. wrote the algorithm. J.L.H. proposed the

cardioid example. V.W. and J.L.H. supervised the project and reviewed the paper.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information is available for this paper at https://doi.org/10.1038/s41467-

019-13134-0.

Correspondence and requests for materials should be addressed to F.G.J.B.

Peer review information Nature Communications thanks Jun-Hee Na and the other,

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