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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.
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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 specic 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 conguration 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 signicant relevance in nature
and engineering and come in a variety of forms. Natural
examples include the stowing of the precious wings of
beetles1or the tting 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 operation36, 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.Wedene 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 eld of
mechanical metamaterials15,16, particularly for auxetic behavior
where the Poissons ratio is negative1719. A Poissons ratio of
exactly -1 corresponds to dilation.
Currently, several limitations in dilational structures exist.
Firstly, most dilational structures have been designed for a spe-
cic 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 nite 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 tting 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 rst 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
gures related by a dilation are similar and have the same
orientation (see Fig. 1). This transformation preserves the shape
and orientation of the gure, 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 rst 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 le 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
Sylvesters, pantograph mechanism2527.
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 gure), 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 conguration
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 gure 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 θ¼α
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 nite 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 benecial 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 rst 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 nd
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 nd a suitable orientation, we use an algorithm that searches
for ows through the representing graphs. A ow 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 ½30;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 cosðtÞcosð2tÞÞ
yðtÞ¼að2 sinðtÞsinð2tÞÞ: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 ½20:5;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 nalexample,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
½15:0;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 specic 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 Sylvesters 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 specic 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 le 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 les for the octahedron and the cardioid were created
using OpenSCAD and Matlab, respectively. For the Stanford Bunny Model, the
original STL les 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 les 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 les 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 les 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 nally 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 nal 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 rst triangulated
(a), after which each triangular face is replaced by a skew pantograph (b)
to obtain a dilational surface. A movie of the nal 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 les). 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
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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 nancial 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-
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