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19 JANUARY 2017 | VOL 541 | NATURE | 347

ARTICLE doi:10.1038/nature20824

Rational design of reconfigurable

prismatic architected materials

Johannes T. B. Overvelde1,2, James C. Weaver3, Chuck Hoberman3,4,5 & Katia Bertoldi1,6

In the search for materials with new properties, there have been great

advances in recent years aimed at the construction of architected

materials, whose behaviour is governed by structure, rather than

composition1–3. Through careful design of the material’s architecture,

new material properties have been demonstrated, including negative

index of refraction

4,5

, negative Poisson’s ratio

6

, high stiffness-to-weight

ratio7,8 and optical9 and mechanical10 cloaking. However, most of the

proposed architected materials (also known as metamaterials) have a

unique structure that cannot be reconfigured after fabrication, making

each metamaterial suitable only for a specific task and limiting its appli-

cability to well known and controlled environments.

The ancient art of origami provides an ideal platform for the design

of reconfigurable systems, since a myriad of shapes can be achieved

by actively folding thin sheets along pre-defined creases. While

most of the proposed origami-inspired metamaterials are based on

two- dimensional folding patterns, such as the miura-ori11–17, the

square twist18 and box-pleat tiling19, it has been shown that cellular

structures can be designed by stacking these folded layers13, or

assembling them in tubes20–23. Furthermore, taking inspiration from

snapology

24,25

—a modular origami technique—a highly reconfigurable

three- dimensional (3D) metamaterial assembled from extruded cubes

has been designed26. Although these examples showcase the potential

of origami-inspired designs to enable reconfigurable architected

materials, they do not fully exploit the range of achievable deformations

and cover only a small region of the available design space. As a result,

ample opportunities for the design of architected materials with tunable

responses remain.

Here, we introduce a robust strategy for the design of 3D recon-

figurable architected materials and show that a wealth of responses

can be achieved in periodic 3D assemblies of rigid plates connected

by elastic hinges. To build these structures, we use periodic space-

filling tessellations of convex polyhedra as templates, and extrude

arbitrary combinations of the polygon faces. In an effort to design

architected materials with specific properties, we systematically

explore the proposed designs by performing numerical simulations

and characterize the mobility (that is, number of degrees of freedom)

of the systems. We find that qualitatively different responses can

be achieved, including shear, uniform expansion along one or two

principal directions, and internal reconfigurations that do not alter

the macroscopic shape of the materials. Therefore, this research paves

the way for a new class of structures that can tune their shape and

function to adapt and even influence their surroundings, bringing

origami- inspired metamaterials closer to application.

Design strategy

To design 3D reconfigurable architected materials, we start by selecting

a space-filling and periodic assembly of convex polyhedra (Fig. 1).

We then perform two operations on the tessellation. (i) We separate

adjacent polyhedra while ensuring that the normals of the overlapping

faces remain aligned. This can be achieved by imposing that for each

overlapping face pair

−=pp ndd L2(1)

jj jj

,b ,a

where dpj denotes the displacements applied to the polyhedra to

separate the jth pair of faces, and the subscripts a and b indicate to

which polyhedron the two overlapping faces belong. Moreover, L

j

is

the distance between the jth pair of faces in the separated state, and nj

is the unit normal to the faces pointing outward from the polyhedron

indicated by the subscript a. (ii) We extrude the edges of the polyhedra

in the direction normal to their faces to form a connected thin-walled

structure (Fig. 1), which we refer to as a prismatic architected material

(Supplementary Video 1).

Importantly, for the periodic space-filling tessellations considered

here, it is sufficient to focus on a unit cell that consists of only a few

polyhedra and covers the entire assembly when translated by the three

lattice vectors

li

0

(i = 1, 2, 3). While equation (1) can be directly imposed

on all internal face pairs in the unit cell, for overlapping faces that are

periodically located (that is, lie on the external boundary of the unit

cell) the constraint needs to be updated as

−+−=

ppRR n

dd L2(2)

jj

jjjj

,b ,a

0

Advances in fabrication technologies are enabling the production of architected materials with unprecedented properties.

Most such materials are characterized by a fixed geometry, but in the design of some materials it is possible to incorporate

internal mechanisms capable of reconfiguring their spatial architecture, and in this way to enable tunable functionality.

Inspired by the structural diversity and foldability of the prismatic geometries that can be constructed using the snapology

origami technique, here we introduce a robust design strategy based on space-filling tessellations of polyhedra to create

three-dimensional reconfigurable materials comprising a periodic assembly of rigid plates and elastic hinges. Guided by

numerical analysis and physical prototypes, we systematically explore the mobility of the designed structures and identify

a wide range of qualitatively different deformations and internal rearrangements. Given that the underlying principles

are scale-independent, our strategy can be applied to the design of the next generation of reconfigurable structures and

materials, ranging from metre-scale transformable architectures to nanometre-scale tunable photonic systems.

1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA. 2AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands. 3Wyss Institute for

Biologically Inspired Engineering, Harvard University, Cambridge, Massachusetts 02138, USA. 4Hoberman Associates, New York, New York 10001, USA. 5Graduate School of Design, Harvard

University, Cambridge, Massachusetts 02138, USA. 6Kavli Institute, Harvard University, Cambridge, Massachusetts 02138, USA.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

348 | NATURE | VOL 541 | 19 JA NUARY 2017

ARTICLE

RESEARCH

where

α=∑=

Rl

j

i

ji i

1

3

,

and

α=∑=

Rl

jiji i

0

1

3

,

0

denote the distance between

the two periodically located faces in the expanded and initial

configuration, respectively, li being the lattice vectors of the expanded

unit cell and αj,i ∈ {− 1,0,1}. As shown by equations (1) and (2), for a

unit cell with F face pairs the expanded configuration is fully described

by F extrusion lengths Lj (j = 1,…, F) (Fig. 1). However, for most unit

cells the extrusion lengths cannot all be specified independently owing

to the constraints introduced by equations (1) and (2). As a result, each

unit cell is characterized by F

indep

≤ F independent extrusion lengths

as illustrated in Supplementary Fig. 6. For the sake of convenience we

chose the F

indep

independent extrusion lengths to be as close as possible

to an average extrusion length

L

, by solving

∑−

... =

LLmin()(3)

LL j

F

j

1

2

F1indep

while ensuring that the constraints imposed by equations (1) and (2)

are not violated.

Finally, we note that all periodic and space-filling assemblies of

convex polyhedra tested in this study were successfully extruded

following the proposed design strategy (that is, we always found

Findep ≥ 1). As an example, in Fig. 1 we show three prismatic

architected materials based on unit cells containing two triangular

and one hexagonal prism (Fig. 1a), an octahedron and cuboctahedron

(Fig. 1b), and four triangular prisms (Fig. 1c).

Characterizing reconfigurability

Although the aforementioned design strategy represents a robust and

efficient approach to construct prismatic architected materials, it does

not provide any indication of their reconfigurability. To determine

whether, and to what extent, the meso-structure of the designed

architected materials can be reshaped, we started by fabricating

centimetre-scale prototypes from cardboard and double-sided tape

(Fig. 2a–c), using a stepwise layering and laser-cutting technique

(see the ‘Methods’ subsection of Supplementary Information)26,27.

Focusing on the three architected materials shown in Fig. 1, we find

that the structure based on triangular prisms and the one based on a

combination of triangular and hexagonal prisms can be reconfigured by

bending the edges and without deforming the faces, and are respectively

characterized by one and two deformation modes (Fig. 2d, e and

Supplementary Video 2). In contrast, the material based on a combination

of octahedra and cuboctahedra is completely rigid (Fig. 2b and

Supplementary Video 2). Furthermore, our experiments reveal that

these architected materials have fewer degrees of freedom than their

constituent individual extruded polyhedra (Supplementary Fig. 7),

indicating that the additional constraints introduced by the connections

between the polyhedra effectively reduce their reconfigurability.

Numerical algorithm

While the examples of Fig. 2a–e illustrate the potential of our strategy

to design reconfigurable architected materials, they also show that the

design of systems with specific behaviour is not straightforward. To

improve our understanding of the reconfigurability of the proposed

architected materials, we implemented a numerical algorithm that

predicts their mobility and corresponding deformation modes. In our

numerical analysis, each extruded unit cell is modelled as a set of rigid

faces connected by linear torsional springs, with periodic boundary

conditions applied to the vertices located on the boundaries. To

characterize the mobility of the structure we solved the following eigen-

problem

ω

=

−

~~

aaM K

1m

2

m

, in which

~

M

and

~

K

are respectively the mass

and stiffness matrices, which account for both the rigidity of the faces

and the periodic boundary conditions through master–slave

elimination. Moreover, ω is an eigenfrequency of the system and am is

the amplitude of the corresponding mode (see the ‘Mode analysis for

1

1

2

2

3366

4

4

5

5

Triangular prisms and hexagonal prisms

cTriangular prisms

bOctahedra and cuboctahedra

a

Select space-lling

tessellation

Identify unit cell

Expand unit cell

Identify face pairs

Prismatic architected material

Tessellate extruded unit cell

Extrude expanded

unit cell

l0

3

l0

2

l0

1

l0

1

l0

2

R0

5

R0

2

R0

1

R0

4dp1 = 0

R1

R

5

dp2

R2R4

dp3

l1

(i)

(ii)

F = 11

F

indep

= 1

F = 10

F

indep

= 4

n1L1

n5L5

l2

n2L2

n1L1

n3L3

n4L4

n2L2

n6L6

n5L5

n4L4

F = 9

Findep = 4

Figure 1 | Design strategy to construct 3D prismatic architected

materials. Space-filling and periodic assemblies of convex polyhedra

are used as templates to construct prismatic architected materials

(Supplementary Video 1). After selecting a space-filling tessellation,

we focus on a unit cell spanned by the three lattice vectors

li

0

(i = 1, 2, 3)

and identify all pairs of overlapping faces. We then separate the polyhedra

while ensuring that the normals of all face pairs remain aligned. Finally,

we extrude the edges of the polyhedra in the direction normal to their

faces to construct the extruded unit cell. Note that the architected material

can be constructed by tessellating the extruded unit cell along the three

new lattice vectors li. Using this approach, we designed three architected

materials that are based on space-filling tessellations comprising triangular

prisms and hexagonal prisms (a), octahedra and cuboctahedra (b) and

triangular prisms (c).

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

19 JANUARY 2017 | VOL 541 | NATUR E | 349

ARTICLE RESEARCH

3D prismatic architected materials with rigid faces’ subsection of

Supplementary Information).

Figure 2f, g shows the simulated eigenmodes for the two reconfig-

urable architected materials considered in Fig. 1a, c. Although the

simulations predict only the deformation for small rotations, the modes

are strikingly similar to the deformation observed in the experiments

(Fig. 2d, e). Solving the aforementioned eigenproblem therefore

provides a convenient approach to determine the mobility of the

structures and gives insight into their deformation without the need

for specific boundary conditions.

Designs based on uniform tessellations

To further explore the potential of prismatic architected materials, and

to establish relations between their reconfigurability and the initial

space-filling polyhedral assembly, we next focus on extruded materials

based on the 28 uniform tessellations of the 3D space, which comprise

regular polyhedra, semiregular polyhedra and semiregular prisms

28–30

.

Owing to their relative simplicity, these uniform templates provide a

convenient starting point to explore the design space.

Using the numerical algorithm, we first determined the number of

degrees of freedom, ndof, of the resulting 28 architected materials

(Supplementary Fig. 9). We find that the mobility of the unit cells is

affected by two parameters: the average connectivity of the unit cell,

=∑

=

zz

P

p

Pp

1

1

, and the average number of modes of the individual

polyhedra,

=∑

=

n

n

P

p

Pp

1

1

, where P is the number of polyhedra in the

unit cell and zp and np are the number of extruded faces and modes of

the pth polyhedron, respectively (Supplementary Fig. 8). The results

for the 28 architected materials reported in Fig. 3 show three key

features. First, higher values for

z

lead to rigid materials (that is, n

dof

= 0

for

z

> 8). Second, if all the constituent extruded polyhedra are rigid

(that is,

=n0

), the resulting architected material is rigid as well. Third,

only 13 of the 28 designs are reconfigurable (that is, ndof > 0).

Interestingly, we find that all of the 13 reconfigurable structures are

based on unit cells comprising only prisms, such that they recover the

relation previously demonstrated for extruded individual prisms,

=−n z 3

(ref. 31). Moreover, our results indicate that most of the

reconfigurable structures are characterized by fewer degrees of freedom

than the constituent individual polyhedra (that is,

<n n

dof

), with the

exception of the architected materials based on the cube (number 22)

and the triangular prism (number 11) for which

=n n

dof

.

Having determined the number of modes for the 28 architected

materials, we next characterize the macroscopic deformation associated

to each of them. More specifically, we determine the macroscopic

volumetric strain

δ

=∑ =ε

j

j

1

3

for each mode, where ε

j

are the macro-

scopic principal strains (see the ‘Mode analysis for 3D prismatic archi-

tected materials with rigid faces’ subsection of Supplementary

Information). Interestingly, we find that for the 13 reconfigurable

architectures all modes are characterized by δ = 0, which indicates pure

macroscopic shearing deformation, as also confirmed by visual

inspection of the modes (Supplementary Fig. 9).

To characterize the reconfigurability of prismatic architected

materials, so far we had assumed the faces to be completely rigid and

abc

de

fg

Rigid

Mode 1 Mode

2M

ode 1

Figure 2 | Deformation modes of 3D prismatic architected materials.

a–c, Prototypes of the 3D prismatic architected materials shown in

Fig. 1 were constructed using cardboard (rigid faces) and double-sided

tape (flexible hinges). d, The structure based on a combination of

triangular and hexagonal prisms can be reconfigured in two different

ways (that is, has two degrees of freedom). e, The structure based

on triangular prisms has a single deformation mode. Note that the

architected material based on the octahedra and cuboctahedra cannot

be reconfigured. f, g, Simulated modes of the reconfigurable architected

materials. The obtained deformation modes were linearly scaled to match

the experiments (scale bar in a, 10 cm).

0

1

2

3

123456 7

8910 11 12 13 14

15 16 17 18 19 20 21

23 24 25 26 27 28

22

7

6

5

4

3

2

1

00246810 12 14

z

n

ndof n=z – 3

Figure 3 | Number of degrees of freedom for architected materials based

on the 28 uniform tessellations of the 3D space. The mobility of the

structures is affected by the average connectivity,

z

, and the average

mobility,

n

. Overlapping data were separated for clarity; the small black

lines indicate the original position of the data in each cluster. The

prismatic architected materials and their deformation modes are shown in

Supplementary Fig. 9.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

350 | NATURE | VOL 541 | 19 JA NUARY 2017

ARTICLE

RESEARCH

the hinges to act as linear torsional springs. However, fabrication will

always result in deformable faces, raising the question of whether

prismatic architected materials can be reconfigured when their faces

are deformable. To explore this direction, we updated our numerical

algorithm by introducing a set of springs to account for the deformabil-

ity of the faces

12,13,21

(see the ‘Stiffness of 3D prismatic architected mate-

rials with deformable faces’ subsection of Supplementary Information).

We then deformed the extruded unit cells uniaxially and investigated

their macroscopic stiffness for different loading directions (identified

by the two angles γ and θ as shown in Fig. 4).

In Fig. 4 we report the normalized stiffness K/E as a function of

γ and θ for four prismatic architected materials characterized by

t/

L

= 0.01, where E is the Young’s modulus of the material and t is the

thickness of the faces. We find that the response of the architected

material based on template number 28, which was previously qualified

as rigid (that is, n

dof

= 0), is fairly isotropic because its stiffness does not

vary much as a function of the loading direction (that is,

3.1 × 10

−3

≤ K/E ≤ 4.0 × 10

−3

). In contrast, the stiffness of architected

materials for which ndof > 0 drops noticeably for specific directions

(that is, Kmin/Kmax = O(10−3)). Interestingly, these are the loading

directions for which the reconfiguring modes are activated, as indi-

cated by the deformed structures shown in Fig. 4. Therefore, these

results indicate that the deformation modes we found in the limit of

rigid faces still persist even when the faces are deformable. We used

the same stiffness for bending of the faces and bending of the hinges,

and from the results we can therefore conclude that the architecture

of these systems makes bending of the faces energetically costly

(because it is typically accompanied by stretching and shearing of the

faces). Finally, materials characterized by higher n

dof

are characterized

by more ‘soft’ deformation modes. As such, materials with ndof = 1

seem most promising for the design of reconfigurable architected

materials, since they can be reconfigured along a specific direction,

while still being able to carry loads in all other directions (Fig. 4 and

Supplementary Fig. 10).

Enhancing the reconfigurability

Although we have shown that by extruding the edges of expanded

assemblies of polyhedra we can construct reconfigurable architected

materials, our results indicate that the mobility of the resulting

structures is strongly reduced by their connectivity. Furthermore,

the modes of all reconfigurable designs show a qualitatively similar

shearing deformation. To overcome these limitations, we next introduce

an additional step in the design strategy and reduce the connectivity

of the materials by extruding some of the faces of the unit cell, while

making the remaining faces rigid.

As an example, in Fig. 5 we consider the architected material based

on a tessellation of truncated octahedra (number 28). When all faces

are extruded,

=z14

, leaving the structure rigid (that is, ndof = 0).

However, by making 8 of the 14 faces rigid instead of extruding them

(Fig. 5a and Supplementary Video 3) we can reduce the connectivity

to

=z6

and the resulting architected material is no longer rigid, because

n

dof

= 1. As shown in Fig. 5b and Supplementary Video 3, this response

was also confirmed experimentally. Finally, we note that by varying the

face pairs in the unit cell that are made rigid instead of extruded, a total

of 2

F

= 128 different architected materials can be designed using the

truncated octahedra as a template. However, only 82 combinations are

possible (as all the other cases will result in structures with discon-

nected parts) and of those designs only four are reconfigurable. Owing

to symmetries in the truncated octahedron, these four configurations

are identical to the one shown in Fig. 5.

Next, to determine the range of deformations that can be achieved

in the proposed structures, we apply the same brute force strategy

to the other 27 uniform space-filling tessellations depicted in

Fig. 3. For this study we considered a maximum of 216 designs per

tessellation, randomly selected from the 2F possibilities, so that for 11

of the tessellations (numbers 4, 5, 9, 10, 16, 17, 20, 21, 23, 25 and 27)

the results are not complete, but rather indicate a trend. We expanded

the number of possible designs by removing the polyhedra for which all

faces have been made rigid from the extruded unit cell, because those

would have resulted in rigid parts completely disconnected from the

architected materials.

Of the approximately 0.6 × 106 connected designs investigated here

(Supplementary Table 1), 90% are rigid (that is, n

dof

= 0) while the other

10% are reconfigurable (that is, ndof > 0). Supplementary Fig. 11a, b

shows that to achieve reconfigurability we still need

≤z8

, with the

exception of six designs based on number 5 for which

=z9

(see Supplementary Fig. 12). Moreover, fully extruded architected

materials characterized by

=n0

always remain rigid, independent of

the reduced number of connections. Finally, and perhaps more

importantly, we also find that using this design approach the mobility

of the architected materials can be greatly enhanced, as 0 ≤ n

dof

≤ 16

and for many of the structures

>n n

dof

(Supplementary Table 1).

Inspection of the modes also reveals that a variety of qualitatively

different types of deformation can be achieved. To characterize them

better, in Fig. 6 and Supplementary Fig. 11c–f we report the magnitude

Loading axis

min max

a dcb

min max min max min max

22 26 12 28

00Tπ

J

π

00Tπ

J

π

00Tπ

J

π

00Tπ

J

πK/E

5 × 10–3

y

z

x

0

T

J

K/E = 1.2

×

10–6 4.9

×

10–3 4.6

×

10–6 5.5

×

10–3 1.0

×

10–6 4.4

×

10–3 3.1

×

10–3 4.0

×

10–3

Figure 4 | Normalized stiffness K/E of prismatic architected materials.

a–d, The results for architected materials based on template numbers

22 (a), 26 (b), 12 (c) and 28 (d). To determine the stiffness in all loading

directions, the architected materials are rotated by angles γ and θ before

loading. In each contour plot we indicate the minimum and maximum

stiffness with white and black squares, respectively. We also show

the deformed architected materials for the minimum and maximum

stiffness direction. Note that the deformation is magnified to facilitate

visualization.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

19 JANUARY 2017 | VOL 541 | NATUR E | 351

ARTICLE RESEARCH

of the principal strains,

=∑

=

εε

ii

1

32

, versus the volumetric strain,

δ, for each deformation mode observed in the reconfigurable archi-

tected materials investigated here. Interestingly, we find that for many

modes

δ==ε0

. These modes do not alter the global shape of the

structure, but result only in internal rearrangements. The design

labelled a, shown in Fig. 6, is an example of a structure undergoing such

a local deformation. Here, most of the structure is rigid except for

one-dimensional tubes that can deform independently. In contrast, the

design labelled b is an example in which the whole internal structure

is deforming, while maintaining the same macroscopic shape

(Supplementary Video 4).

Besides these local modes, Fig. 6 also indicates that there are designs

capable of achieving types of macroscopic deformation that differ from

pure shear (for which δ = 0 and

>ε0

). For example, we find that

some of the structures are characterized by an effective vanishing strain

in two directions (labelled c in Fig. 6). The deformation of such

architected materials is characterized by ε

1

≠ 0 and ε

2

= ε

3

≈ 0, resulting

in

δ=ε

. Moreover, the results also reveal that there are a variety of

structures capable of uniform biaxial expansion (or contraction), for

which ε

2

= ε

3

≠ 0 and ε

1

= 0 and

δ=/ε2

. This deformation mode

is exemplified by the design labelled d shown in Fig. 6 (Supplementary

Video 4). Finally, we note that

δ=ε3

corresponds to uniform

expansion (or contraction) characterized by ε1 = ε2 = ε3, and defines a

boundary for possible combinations of δ and

ε

. In fact, none of the

designs considered here exhibits this type of deformation.

Discussion and conclusion

In this work we introduced a convenient and robust strategy for the

design of reconfigurable architected materials, and explored the design

space by considering structures based on the 28 uniform space-filling

tessellations of polyhedra. Our study uncovered architected materials

with a wide range of qualitatively different responses and degrees of

freedom, but many more designs are made possible by using different

assemblies of convex polyhedra as templates (including assemblies

based on Johnson solids and irregular polyhedra, and assemblies that

do not fill space), by considering different extrusion lengths, or by

removing faces (instead of making them rigid before the extrusion

step). Given these additional possibilities in the design of recon-

figurable architected materials, we have made our numerical algorithm,

implemented in Matlab, available for download as Supplementary

Information, to be used and expanded upon by the community. Finally,

we believe that, building on the results presented in this work, prismatic

Extrude

Make rigid

Adapt

unit cell

a

b

Recongure the architected material along the only degree of freedom

= 1

= 6

= 1

n

z

ndof

= 5

= 14

= 0

n

z

ndof

Figure 5 | Enhancing the reconfigurability of 3D prismatic architected

materials. a, To enhance the reconfigurability of the architected material

based on the space-filling assembly of truncated octahedra (number 28 in

Fig. 3), we extrude only six of its faces and make the remaining eight faces

rigid. Using this approach, the average connectivity is reduced from

=z14

to

=z6

and the resulting structure is no longer rigid, because ndof = 1.

b, Experimental validation of the numerical predictions (scale bar, 10cm).

b

a

d

c

d

a,b

c

0.5

Fig. 5

0.5

1234567

G

1

2

3

4

5

6

7

¬

¬¬¬

= 1

n

dof

1 = –2

3 = 0

1 = 0

2 = 0

3 = 0

1 = 0

2 = 0

3 = 2

1 = 2 = 3

00G

¬¬¬¬

Figure 6 | Deformation modes of 3D prismatic architected materials

with enhanced reconfigurability. Relation between the volumetric strain,

δ, and the magnitude of the principal strains,

ε

, for all the architected

materials characterized by ndof = 1. The colour of the markers refers to the

uniform tessellation that has been used as a template, as shown in Fig. 3.

Structures a–d and the one in Fig. 5 are indicated by grey circles on the

main panel. The solid and dashed lines and associated schematics and

conditions on ε1, ε2 and ε3 highlight how different choices of strains lead to

different types of deformation (see text). Structures labelled a and b

(based on tessellations 24 and 9, respectively) are characterized by

δ==ε0

and experience internal rearrangements that do not alter their

macroscopic shape. The structure labelled c (based on tessellation 16)

deforms only in one direction (that is, δ = 4.21,

=.ε476

), while the

structure labelled d (based on tessellation 14) experiences uniform biaxial

extension (or contraction) (that is, δ = 2.45,

=.ε173

). The grey shaded

region corresponds to combinations of strains that do not permit

deformation.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

352 | NATURE | VOL 541 | 19 JA NUARY 2017

ARTICLE

RESEARCH

architected materials with specific properties may be efficiently

identified by combining our numerical algorithm with stochastic

optimization algorithms such as genetic algorithms. Such optimization

algorithms could prove essential in the design of reconfigurable

architected materials capable of handling changing environments or

multiple tasks (that will probably lead to pareto optimal solutions).

To realize prismatic architected materials, in this study we used

cardboard for the rigid faces and double-sided tape for the hinges. This

fabrication process enables the realization of centimetre-scale

prototypes (for our models we used

L

= 35 mm) that closely match the

conceptual origami-inspired mechanisms, but real-world applications

depend on the ability to efficiently manufacture assemblies comprising

a large number of unit cells at different length scales using different

fabrication techniques. Taking advantage of recent developments in

multi-material additive manufacturing, we also built the proposed

architected materials using a stiff material (with Young’s modulus

E ≈ 1 GPa) for the faces and a soft material (E ≈ 1 MPa) for the hinges

(see the ‘Methods’ section of Supplementary Information).

Supplementary Video 5 shows 3D printed models for two designs

based on assemblies of truncated octahedra (for both models we used

L

= 6 mm). Although additional local deformation arises from the

finite size of the flexible hinges, the 3D printed structures exhibit the

same deformation modes predicted by our numerical analysis and

observed in the cardboard prototypes. As such, recent advances in

fabrication, including projection micro-stereolithography7, two-photon

lithography8,32,33 and ‘pop-up’ strategies34–40, open up exciting

opportunities for miniaturization of the proposed architectures. Our

strategy thus enables the design of a new class of reconfigurable systems

across a wide range of length scales.

Data availability The Matlab model used to determine the mobility and

deformation modes of the prismatic architected materials is provided

in Supplementary Information. Other models and datasets generated

during and/or analysed during the current study are available from the

corresponding author on request.

Received 24 May; accepted 22 November 2016.

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Supplementary Information is available in the online version of the paper.

Acknowledgements This work was supported by the Materials Research

Science and Engineering Center under NSF Award number DMR-1420570.

K.B. also acknowledges support from the National Science Foundation

(CMMI-1149456-CAREER). We thank M. Mixe and S. Shuham for assistance

in the fabrication of the cardboard prototypes, and R. Wood for the use of his

laboratory.

Author Contributions J.T.B.O., C.H. and K.B. proposed and designed the

research; J.T.B.O. performed the numerical calculations; J.T.B.O., C.H. and

J.C.W. designed and fabricated the models; J.T.B.O. performed the experiments;

J.T.B.O. and K.B. wrote the paper.

Author Information Reprints and permissions information is available at

www.nature.com/reprints. The authors declare no competing financial

interests. Readers are welcome to comment on the online version of the paper.

Correspondence and requests for materials should be addressed to

K.B. (bertoldi@seas.harvard.edu).

Reviewer Information Nature thanks J. Paik, D. Pasini and the other anonymous

reviewer(s) for their contribution to the peer review of this work.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

In the following, we ﬁrst describe the fabrication approaches used to make cardboard prototypes and 3D printed

prototypes. We next describe the numerical algorithms we implemented in Matlab (i) to predict the number of

degrees of freedom and corresponding deformation modes of 3D prismatic architected materials with rigid faces; and

(ii) to characterize the elastic response of 3D prismatic architected materials with deformable faces. Moreover, this

document contains the supplemental ﬁgures and table.

METHODS

Fabrication of Cardboard Prototypes

Our cardboard prototypes were fabricated from two layers of cardboard with a thickness of 0.7 mm (13001-2506,

Blick) and one layer of double-sided tape with a thickness of 0.07 mm (23205-1009, Blick), using a stepwise layering and

laser cutting technique on a CO2 laser system (VLS 2.3, Universal Laser Systems). To fabricate each of the extruded

polygons that together form the architected material, we started by cutting one of the cardboard sheets, after which

we removed it from the laser system (steps 1-2 as shown in Supplementary Fig. 1). Using a diﬀerent pattern, cutting

slits were introduced in the second cardboard sheet (steps 3-5). The double-sided tape was bonded to the second

cardboard sheet still in the laser system, to which the initially cut cardboard sheet was attached (steps 6-7). A third

cutting step was then performed to ﬁnalize the diﬀerent components (step 8), which were completely separated from

the main sheet (step 9-10). The individual components were assembled to form the extruded architected material

using the tape that was exposed during the cutting process (steps 11-15).

Fabrication of 3D Printed Prototypes

We furthermore manufactured prismatic architected materials using multi-material additive manufacturing (Con-

nex500, Stratasys). We used a rigid material (VeroWhitePlus RGD835, Young’s modulus E ≈ 1 GPa) for the faces

and a softer (TangoPlus FLX930, E ≈ 1 MPa) for the hinges. The ﬁnal structure was designed using a custom made

Matlab script, which was based on extruded polyhedra with an edge length of L0 = 6 mm and an average extrusion

length of L¯ = 6 mm. Both the faces and hinges were given a thickness of t = 1 mm. Moreover, the size of the faces

was reduced by 0.5 + 1.5|θ|/π mm on each side to account for the ﬁnite size of the rounded hinges, θ being the initial

angle of the hinges (Supplementary Fig. 2).

MODE ANALYSIS FOR 3D PRISMATIC ARCHITECTED MATERIALS WITH RIGID FACES

Here, we describe the algorithm that we implemented to predict the number of degrees of freedom and corresponding

deformation modes of 3D prismatic architected materials. Focusing on an extruded unit cell comprising 2F rigid faces

(F face pairs) connected by H hinges (i.e. torsional springs), we ﬁrst determine the elastic and kinetic energy required

to deform the structure. Then, we describe the constraints that we impose to ensure that the faces remain rigid and

the unit cell deforms in a periodically repeated manner (i.e. we model the response of an inﬁnitely large structure

without considering boundary eﬀects), after which we describe the eigenfrequency problem that we solve to ﬁnd

the characteristic deformation modes of the architected materials. Finally, we discuss the assumptions made in the

numerical model.

Energy

Elastic Energy

Assuming that each hinge acts as a linear torsional spring of stiﬀness Kh, that no energy is required to maintain

the hinges in their initial conﬁguration and that the faces are rigid, the total elastic energy of the unit cell, Eelastic,

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Supplementary Fig. 1: Fabrication of cardboard prototypes. Steps 1-10 show the cutting proces to fabricate ﬂat three layer

composites of cardboard and double-sided tape, and steps 11-16 depict the assembly process using the fabricated fabricated

pieces.

is given by

Eelastic =Ehinge =

H

X

i=1

1

2Kh

idθ2

i=1

2dθTKhdθ,(1)

where dθidenotes the change in angle for the i-th hinge, dθ= [dθ1, dθ2, . . . , dθH]T, and Kh= diag(Kh

1, Kh

2, ...., Kh

H).

Note that dθican be expressed in terms of the displacement of the Vvertices (corner points of the faces) as

dθi=

V

X

v=1 ∂θi

∂x1,v

dx1,v +∂θi

∂x2,v

dx2,v +∂θi

∂x3,v

dx3,v!,(2)

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Supplementary Fig. 2: Fabrication of 3D printed prototypes. (a) Multi-material design based on a polyhedron template. Here,

L0depicts the edge size, θthe angle between faces, and tthe thickness of the faces. (b) 3D printed prototype using stiﬀ

materials for the faces, and soft material for the hinges. (scale bar 6 mm)

where dx1,v,dx2,v and dx3,v denote the displacement components of the v-th vertex with initial coordinates x1,v,x2,v

and x3,v. Substitution of Eq. (2) into Eq. (1) yields

Ehinge =1

2uTJT

hKhJhu,(3)

in which Jhis the Jacobian matrix with entries

Jh[i, 3(v−1)+j]=∂θi

∂xj,v

, j = 1,2,3 and v= 1, ..., V, (4)

and u= [dx1,1, dx2,1, dx3,1, . . . , dx1,V , dx2,V , dx3,V ]T.

Since the angle of the i-th hinge, which connects two faces with unit normals naand nband rotates around the

axis pointing in the direction ah, is given by

θi= tan−1ah·(na×nb)

na·nb,(5)

it follows that

∂θi

∂xj,v

=

∂y

∂xj,v z−y∂z

∂xj,v

y2+z2,(6)

in which we have used y=ah·(na×nb) and z=na·nb. The derivatives can then be found according to

∂y

∂xj,v

=∂ah

∂xj,v

·(na×nb) + ah· ∂na

∂xj,v

×nb+na×∂nb

∂xj,v !,(7)

and

∂z

∂xj,v

=∂na

∂xj,v

·nb+na·∂nb

∂xj,v

(8)

Finally, since the unit normal to any of the faces, n, can be calculated as

n=v1×v2

q|v1|2|v2|2−(v1·v2)2,(9)

where v1and v2are two non-parallel vectors lying on the face (Supplementary Fig. 3), the derivatives of nin Eqs.

(7) and (8) are given by

∂n

∂xj,v

=∂v1

∂xj,v ×v2+v1×∂v2

∂xj,v c−(v1×v2)∂c

∂xj,v

c2,(10)

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Supplementary Fig. 3: Schematic of the extruded unit cell.

in which

c=q|v1|2|v2|2−(v1·v2)2,(11)

and

∂c

∂xj,v

=1

c

∂v1

∂xj,v

·v1!|v2|2+|v1|2 ∂v2

∂xj,v

·v2!−(v1·v2) ∂v1

∂xj,v

·v2+v1·∂v2

∂xj,v !

.(12)

Note that ∂n/∂xj,v = 0 for vertices not belonging to the face with normal n.

Kinetic Energy

Next, we determine the kinetic energy, Ekinetic , associated with the displacements of the vertices of the unit cell

Ekinetic =1

2

V

X

v=1

Mv∂x1,v

∂t +∂x2,v

∂t +∂x3,v

∂t 2

=1

2˙uTM ˙u,(13)

where ˙u =∂u/∂t,Mvis the mass assigned to the v-th vertex and Mis the 3V×3Vdiagonal mass matrix

diag(M1, M1, M1, . . . , MV, MV, MV). Note that each face to which the v-th vertex belongs contributes a mass M/N

to the vertex, where Mis the mass of the face (which we take equal to the area by assuming a unit thickness and

density), and Nis the number of vertices of the face.

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Constraints

Rigidity of the Faces

To ensure that all the faces are rigid and do not deform, we triangulate them (Supplementary Fig. 3) and impose

that the length of each edge of the triangulation remains constant,

(xa−xb)·(xa−xb) = L2,(14)

where xaand xbare the two vertices connected by the edge, which has initial length L. We then linearize Eq. (14)

to obtain an expressions for each constraint that depends explicitly on the displacements of the two vertices

(xa−xb)·(ua−ub)=0,(15)

in which ua= [dx1,a, dx2,a, dx3,a] and ub= [dx1,b, dx2,b, dx3,b].

Furthermore, we also ensure that all the faces remain ﬂat (i.e. each face can undergo rigid body translation and

rotations, but cannot bend). To this end, we impose that all vertices of each face remain on the same plane spanned

by the two vectors w1and w2(Supplementary Fig. 3) [41],

wi·(w1×w2) = 0,for i= 3, ..., Vf−1,(16)

in which Vfis the number of vertices of the face. Note that this constraint is automatically satisﬁed for faces that

only connect three vertices. We again linearize the constraints to obtain

u1·(wi×w2) + u2·(w2×wi) + u3·(wi×w1) + ui·(w1×w2) = 0,for i= 3, ..., Vf−1.(17)

Finally, we note that the constraints of Eqs. (15) and (17) are only valid for small displacements, since the constraints

are linearized around the initial coordinates of the vertices.

Periodic Boundary Conditions

For the inﬁnitely large periodic prismatic architected materials considered here, it is suﬃcient to focus on a unit

cell that consists of a few extruded polyhedra and covers the entire assembly when translated by the three lattice

vectors li(i= 1,2,3). To ensure that the extruded unit cell deforms in a periodically repeated manner we constrain

the deformation of each periodically located vertex pair on its boundary as

ub−ua=

3

X

i=1

αidli,(18)

where uaand ubare the displacements of the two periodically located vertices, dlidenotes the deformation of the

lattice vectors, and

xb−xa=

3

X

i=1

αili,(19)

with αi∈ {−1,0,1}. In our implementation we treat dlias additional degrees of freedom, which we include in Eqs. (3)

and (13) as

Eelastic =1

2u

dlT

Jh0TKhJh0u

dl,(20)

and

Ekinetic =1

2"˙

u

d˙

l#TM 0

0 0"˙

u

d˙

l#,(21)

in which dl= [dlT

1, dlT

2, dlT

3]Tand d˙

l=∂dl/dt.

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Master-slave Elimination

To enforce the constraints given by Eqs. (15), (17) and (18), we adopt the master-slave elimination method [42].

We start by rewriting all the constraints in matrix form as

Au

dl=0,(22)

where each row of Arepresents one constraint. Next, we rewrite Ain its reduced row echelon form, Arref. The

dependent constraints correspond to rows of all zeros in Arref and are therefore automatically satisﬁed. Moreover, all

the columns of Arref with a single entry correspond to the slave degree of freedom, ds, while the remaining degrees of

freedom are referred to as the master degree of freedom, dm. We then rewrite Eq. (22) as

I Brrefds

dm=0,(23)

where Iis the identity matrix and Brref comprise the columns of Arref that correspond to the master degrees of

freedom. It follows from Eq. (23) that

ds=−Brrefdm,(24)

so that

ds

dm=−Brref

Idm.(25)

Finally, since the vectors [dT

s,dT

m]Tand [uT, dlT]Tcontain exactly the same degrees of freedom arranged in a diﬀerent

order, we rearrange the rows of the matrix [Brref,I]Tin Eq. (25) to obtain

u

dl=Tdm,(26)

where Tis a transformation matrix.

Using Eq. (26), the elastic and kinetic energies from Eqs. (20) and (21) can be rewritten as

Eelastic =dT

mTTJh0TKhJh0Tdm,(27)

and

Ekinetic =˙

dT

mTTM 0

0 0T˙

dm.(28)

Mode Analysis

The equations of motion for the extruded unit cell are derived using Lagrange’s equations

∂

∂t ∂Epotential

∂˙

u−∂Epotential

∂u=0,(29)

where Epotential =Eelastic −Ekinetic. Substitution of Eqs. (27) and (28) into Eq. (29) yields

TTM 0

0 0T¨

dm−TTJh0TKhJh0Tdm=0,(30)

in which we assumed that T,M,Jhand Khdo not depend on the displacement and do not change in time. Next,

we assume the solution to have the form

dm=amsin(ωt +β),(31)

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and substitute into Eq. (29) to obtain the eigenproblem

˜

M−1˜

Kam=ω2am,(32)

in which

˜

M= TTM 0

0 0T!,(33)

˜

K=TTJh0TKhJh0T.(34)

Moreover, ωis an eigenfrequency of the system and amis the corresponding mode. Finally, the displacements of all

the vertices associated to each mode are obtained from Eq. (26) as

u

dl=Tamsin(ωt +β).(35)

Characterizing the deformation modes

To characterize the macroscopic deformation associated to each prismatic material, we determine the macroscopic

inﬁnitesimal strain tensor for each of the computed modes as

¯

e=1

2H+HT,(36)

where His the macroscopic displacement gradient, which can be determined from the inﬁnitesimal deformation of

the three lattice vectors, dli, by solving the following set of equations

dli=Hli,for i= 1,2,3.(37)

Note that we normalized dliby the maximum change in angle between connected faces. To characterize the type of

macroscopic deformation associated to each mode we introduce the volumetric strain

δ=

3

X

j=1

j,(38)

where jare the principal strains, which can be determined by solving ¯

e−jIvj=0,vjbeing the principal

directions.

Discussion

While our numerical analysis proved essential in the exploration of the design space for prismatic architected

materials, it is important to note that the algorithm is only valid for small rotations. However, we found that the

numerical results still provide valuable insights into the large deformations typically experienced by the structures,

as demonstrated by the excellent agreement with the experiments. Moreover, in our model we assumed that the

deformation of the architected material can be fully captured by an extruded unit cell to which periodic boundary

conditions are applied, thus neglecting boundary eﬀects. Although in our experiments we observed additional modes

that arise from the reduced connectivity of the unit cells near the boundaries (Supplementary Fig. 4), we do not

expect them to signiﬁcantly inﬂuence the behavior of the bulk material as they are conﬁned to the outer surfaces.

These assumptions signiﬁcantly reduced computation time and also removed the need for applying speciﬁc boundary

conditions, allowing us to model and compare many diﬀerent systems.

STIFFNESS OF 3D PRISMATIC ARCHITECTED MATERIALS WITH DEFORMABLE FACES

While in the analysis used to characterize the reconﬁgurability of the structures we assumed the faces to be rigid,

we now account also for their deformability by introducing a set of springs. More speciﬁcally, for each rectangular

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Supplementary Fig. 4: Deformation modes of a ﬁnite-size prototype of the architected material based on the space-ﬁlling

assembly of hexagonal prisms (#26). Besides the two bulk modes predicted by our numerical simulations, we also observe 6

boundary modes (of which 2 are shown) that arise from the reduced connectivity of the unit cells near the boundaries (scale

bar 5 cm).

face we used four linear springs placed along the perimeter to capture its stretching, two linear springs placed along

the diagonal to capture its shearing, and a linear torsional spring placed along an arbitrary diagonal to capture its

bending [43–45] (Supplementary Fig. 5). Therefore, the elastic energy required to deform an extruded unit cell is

given by

Eelastic =Ehinge +Estretch

face +Eshear

face +Ebend

face ,(39)

where Ehinge is the elastic energy as deﬁned in in Eq. (1), and Estretch

face ,Eshear

face and Ebend

face denote the contribution

to the elastic energy of the unit cell due to stretching, shearing and bending of the faces. Focusing on unit cells

comprising 2Ffaces, the energy required to stretch the faces can be determined from the extension of their edges as

Estretch

face =

8F

X

i=1

1

2Kst

i(dest

i)2=1

2deT

stKst dest,(40)

where Kst

iand dest

idenote the stiﬀness and change in length of the i-th edge, dest = [dest

1, dest

2, . . . , dest

8F]T, and

Kst = diag(Kst

1, Kst

2, ...., Kst

8F). Following the same approach used for Ehinge (Eqs. (2)-(4)), we rewrite Estretch

face in

terms of the displacement of the Vvertices. We ﬁrst note that

dest

i=

V

X

v=1 ∂est

i

∂x1,v

dx1,v +∂est

i

∂x2,v

dx2,v +∂est

i

∂x3,v

dx3,v!.(41)

Next, we substitute Eq. (41) into Eq. (40) and obtain

Estretch

face =1

2uTJT

stKst Jstu,(42)

in which Jst is the compatibility matrix with entries

Jst [i, 3(v−1)+j]=∂est

i

∂xj,v

, j = 1,2,3 and i= 1, ..., 8F. (43)

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Supplementary Fig. 5: Schematic of the extruded unit cell indicating elements used to model the deformation of the faces.

Since shearing of faces is also modeled using springs (placed along the diagonals of the face), following the same

procedure used for Estretch

face (Eqs. (40)-(43)), Eshear

face can be determined as

Eshear

face =1

2uTJT

shKsh Jshu,(44)

where Ksh = diag(Ksh

1, Ksh

2, ...., Ksh

4F) (Ksh

idenoting the stiﬀness of the i-th diagonal spring introduced to capture

shearing) and

Jsh [i, 3(v−1)+j]=∂esh

i

∂xj,v

, j = 1,2,3 and i= 1, ..., 4F. (45)

Finally, the energy associated to bending of the faces, Ebend

face , can be determined following the procedure used to

determine Ehinge (Eq. (1)-(12)), yielding

Ebend

face =1

2uTJT

bKbJbu.(46)

where Kb= diag(Kb

1, Kb

2, ...., Kb

2F) (Kb

idenoting the stiﬀness of the i-th torsional spring placed on the diagonal to

capture shearing) and

Jb [i, 3(v−1)+j]=∂ψi

∂xj,v

, j = 1,2,3 and i= 1, ..., 2F, (47)

ψbeing the angle between the two triangulated faces separated by the diagonal on which the torsional spring is placed

(Supplementary Fig. 5).

Spring Stiﬀnesses

We assume that the faces are made from a material with Young’s modulus Eand Poisson’s ratio ν= 1/3, and their

thickness tis chosen so that t/¯

L= 0.01 (¯

Lbeing the average extrusion length). For such systems the stiﬀnesses of

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the springs introduced in our model can be determined as [45]

Kst =Et

2L2

L2−νL2

⊥

1−ν2,(48)

Ksh =Et

2LLHLW

ν(L2

H+L2

W)3/2

1−ν2,(49)

Kb=Cb

Et3

12(1 −ν2)L

t1/3

(50)

in which Lis the length of the edge on which the spring is placed, L⊥is the length of the edge perpendicular to that

on which the spring is placed, LWand LHare the width and the height of the face, and Cb= 0.441. Moreover, while

in our experiments we used hinges that can bend more easily than the faces, here we consider the extreme case for

which the bendability of the hinges is similar to that of the faces and use

Kh=Cb

ELt3

24(1 −ν2)1

t1/3

.(51)

Periodic boundary conditions

Next, we apply periodic boundary conditions to the unit cell and express them in terms of the macroscopic dis-

placement gradient H. To ensure that the extruded unit cell deforms in a periodically repeated manner under applied

loading we update Eq. (18) as

ub−ua=

3

X

i=1

αiHli.(52)

Furthermore, we also introduce three ﬁctitious nodes, (v1,v2,v3), to conveniently apply Hto the unit cell [46]. The

displacement components of three ﬁctitious nodes are assigned to be the components of H. Virtual work is then used

to determine the macroscopic stress tensor as

sij =1

V0

rvi

j(53)

where V0is the initial volume occupied by the extruded unit cell and rviis the “reaction force” corresponding to the

assigned “displacement components” of the ﬁctitious nodes vi.

The periodic boundary conditions speciﬁed by Eq. (52) are then enforced using the master-slave elimination method.

Following the procedure detailed in Eqs. (22)-(28) for Ehinge , we obtain

Estretch

face =dT

mTTJst 0TKst Jst 0Tdm,(54)

Eshear

face =dT

mTTJsh 0TKsh Jsh 0Tdm,(55)

Ebend

face =dT

mTTJb0TKbJb0Tdm,(56)

where

u

h=Tdm,(57)

and h= [H11, H 12, H 13, H21 , . . . , H 33]T.

Deformation Under Uniaxial Loading

Finally, assuming that the deformation is applied quasi-statically, the equilibrium equations for the extruded unit

cell can be obtained as

∂Epotential

∂u= 0,(58)

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where Epotential =Eelastic −W,W=hTrbeing the external work [46], for which we have deﬁned r=

(rv1)T,(rv2)T,(rv3)TT. It follows from Eqs. (54)-(58) that

TTJh0TKhJh0+Jst 0TKst Jst 0+Jsh 0TKsh Jsh 0+Jb0TKbJb0Tdm=TT0

r.

(59)

Having determined the equilibrium equations for an extruded unit cell, we apply a uniaxial loading to the system

by imposing H11 6= 0 and Hij = 0 for i6=j, while leaving H22 and H33 unset (i.e. allowing the structure to freely

expand in the lateral directions, while constraining macroscopic shear deformations). Note that we also constrained

rigid body translations by ﬁxing the displacement of a single vertex of the unit cell. To determine the response of the

architected material along all directions, we rotate the unit cell about two axis according to

x0=RzRy0x,(60)

in which

Rz=

cos γ−sin γ0

sin γcos γ0

0 0 1

,(61)

Ry0= cos θI+ sin θ

0−y0

3y0

2

y0

30−y0

1

−y0

2y0

10

+ (1 −cos θ)

(y0

1)2y0

1y0

2y0

1y0

3

y0

2y0

1(y0

2)2y0

2y0

3

y0

3y0

1y0

3y0

2(y0

3)2

(62)

with y0=Rzey(see schematic in Supplementary Fig. 10). For each direction, we can determine the stiﬀness according

to

K=s11/(H11 ).(63)

[41] Lalibert´e, T. & Gosselin, C. Construction, mobility analysis and synthesis of polyhedra with articulated faces. J. Mecha-

nisms Robotics 6, 011007 (2013).

[42] Cook, R. Concepts and applications of ﬁnite element analysis (Wiley, 2001).

[43] Wei, Z. Y., Guo, Z. V., Dudte, L., Liang, H. Y. & Mahadevan, L. Geometric mechanics of periodic pleated origami. Phys.

Rev. Lett. 110, 215501 (2013).

[44] Schenk, M. & Guest, S. D. Geometry of miura-folded metamaterials. Proc. Natl. Acad. Sci. U.S.A. 110, 3276-3281 (2013).

[45] Filipov, E. T., Tachi, T. & Paulino, G. H. Origami tubes assembled into stiﬀ, yet reconﬁgurable structures and metama-

terials. Proc. Natl. Acad. Sci. U.S.A. 112, 12321-12326 (2015).

[46] Danielsson, M., Parks, D. & Boyce, M. Three-dimensional micromechanical modeling of voided polymeric materials. J.

Mech. Phys. Solids 50, 351-379 (2002).

[47] Gru¨nbaum, B. Uniform tilings of 3-space. Geombinatorics 4, 49-56 (1994).

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SUPPLEMENTAL FIGURES

Supplementary Fig. 6: For a unit cell with Fface pairs the expanded conﬁguration is fully determined by choosing Findep ≤F

extrusion lengths. As an example, here we consider four prismatic architected materials based on the space-ﬁlling assembly of

hexagonal prisms (for which F= 4 and Findep = 2) and show the eﬀect of the two independent extrusions lengths L1/L0and

L2/L0on the ﬁnal extruded shape, in which L0denotes the length of the edges of the polyhedra. All four designs considered

here have the same degrees of freedom (ndof = 2).

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Supplementary Fig. 7: Reconﬁgurability of individual extruded polyhedra. The extruded units based on a (a) tetrahedron, (c)

octahedron, (d) truncated tetrahedron and (f) truncated cube are rigid, while those based on the (b) cube, (e) cuboctahedron,

(g) truncated octahedron, (h) rhombicuboctahedron, (i) truncated cuboctahedron, and (j-m) prisms are reconﬁgurable. For

reference, we also denoted the polyhedra on which the unit cells are based by their Sch¨aﬂi symbols. Note that only a selected

number of deformation modes is shown, as it is not straightforward to determine all of them experimentally. For all the

prototypes the edges are 35 mm.

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Supplementary Fig. 8: Numerically determined modes of individual extruded polyhedra. The extruded geometries based on

the (a) tetrahedron, (c) octahedron, (d) truncated tetrahedron and (f ) truncated cube are rigid, while those based on the

(b) cube, (e) cuboctahedron, (g) truncated octahedron, (h) rhombicuboctahedron, (i) truncated cuboctahedron, and (j-m)

prisms are reconﬁgurable. Importantly, using our numerical algorithm we can easily identify the degrees of freedom, n, and the

deformation modes for the extruded units. Note that modes characterized by the same eigenvalue ω2are identical, so that we

only show one of these modes. For reference, we also denoted the polyhedra on which the unit cells are based by their Sch¨aﬂi

symbols.

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Supplementary Fig. 9: Reconﬁgurability of architected materials based on on the 28 uniform tessellations of the 3D space,

which comprise regular polyhedra, semiregular polyhedra and semiregular prisms. The tessellations used as a template are

indicated with the notation introduced in [47]. Speciﬁcally, the individual polyhedra are indicated by their Sch¨aﬂi symbol,

and the superscript shows the number of polyhedra of the given kind that meet at each vertex. Moreover, we also provide the

number of each kind of polyhedra in the unit cell. (#1-2) The architected materials based on (3.3.3)8.(3.3.3.3)6(tetrahedra

and octahedra) are rigid. Note that #2 diﬀers from #1 as it comprises reﬂected layers of tetrahedra and octahedra. (#3-4)

The assemblies based on (3.3.3)8.(3.3.3.3)3.(3.4.4)6(tetrahedra, octahedra and triangular prisms) are rigid. Note that #4

diﬀers from #3 as it comprises reﬂected layers of tetrahedra, octahedra and triangular prisms. (#5) (3.3.3)4.(3.4.4.4)3.(4.4.4)

(tetrahedra, rhombicuboctahedra and cubes) is rigid.

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Supplementary Fig. 9: (continued). (#6) (3.3.3)2.(3.6.6)6(tetrahedra and truncated octahedra) is rigid. (#7)

(3.3.3.3)2.(3.4.3.4)4(octahedra and cuboctahedra) is rigid. (#8) (3.3.3.3).(3.8.8)4(octaheda and truncated octahedra) is

rigid. (#9) (3.4.3.4).(3.4.4.4)2.(4.4.4)2(cuboctahedra, rhombicuboctahedra and cubes) is rigid. (#10) (3.4.3.4).(3.6.6)2.(4.6.6)2

(cuboctahedra, truncated tetrahedra and truncated octahedra) is rigid.

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Supplementary Fig. 9: (continued). (#11-12) (3.4.4)12 (triangular prisms) are reconﬁgurable with ndof = 2 and ndof = 1,

respectively. Note that #12 diﬀers from #11 as it comprises reﬂected layers of triangular prisms. (#13-15) (3.4.4)6.(4.4.4)4

(triangular prisms and cubes) are reconﬁgurable with ndof = 2, ndof = 2 and ndof = 1, respectively. Note that #14 diﬀers from

#13 as the polyhedra are diﬀerently arranged in-plane, and #15 diﬀers from #13 as it comprises reﬂected layers of triangular

prisms and cubes.

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Supplementary Fig. 9: (continued). (#16) (3.4.4)2.(4.4.4)4.(4.4.6)2(triangular prisms, cubes and hexagonal prisms) is reconﬁg-

urable with ndof = 2. (#17) (3.4.4)8.(4.4.6)2(triangular prisms and hexagonal prisms) is reconﬁgurable with ndof = 2. (#18)

(3.4.4)4.(4.4.6)4(triangular prisms and hexagonal prisms) is reconﬁgurable with ndof = 2. (#19) (3.4.4)2.(4.4.12)4(triangular

prisms and dodecagonal prisms) is reconﬁgurable with ndof = 2. (#20) (3.4.4.4).(3.8.8).(4.4.4).(4.4.8)2(rhombicuboctahedra,

truncated cubes, cubes and octagonal prisms) is rigid.

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Supplementary Fig. 9: (continued). (#21) (3.6.6).(3.8.8).(4.6.8)2(truncated tetrahedra, truncated cubes and truncated cuboc-

tahedra) is rigid. (#22) (4.4.4)8(cubes) is reconﬁgurable with ndof = 3. (#23) (4.4.4)2.(4.4.6)2.(4.4.12)2(cubes, hexagonal

prisms and dodecagonal prisms) is reconﬁgurable with ndof = 2. (#24) (4.4.4)2.(4.4.8)4(cubes and hexagonal prisms) is

reconﬁgurable with ndof = 2. (#25) (4.4.4).(4.6.6).(4.6.8)2(cubes, truncated octahedra and truncated cuboctahedra) is rigid.

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Supplementary Fig. 9: (continued). (#26) (4.4.6)6(hexagonal prisms) is reconﬁgurable with ndof = 2. (#27) (4.4.8)2.(4.6.8)2

(octagonal prisms and truncated cuboctahedra) is rigid. (#28) (4.6.6)4(truncated octahedra) is rigid.

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Supplementary Fig. 10: Normalized stiﬀness K/E of the 28 architected materials based on the uniform space-ﬁlling polyhedra

assemblies. To determine the stiﬀness in all loading directions, the architected materials are rotated by angles γand θprior to

loading.

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Supplementary Fig. 11: (a-b) Number of degrees of freedom, ndof, for the altered architected materials based on the 28 uniform

space-ﬁlling tessellations. Each point represents a design in which some of the faces of the unit cell are made rigid, instead of

extruded. (c-f) Relation between the volumetric strain, δ, and the magnitude of the principal strains, ||||, for all the architected

materials characterized by ndof = 2, 3, 4 and >5, respectively (Note that the results for ndof = 1 are shown in Fig. 6). The

color of the markers refers to the uniform tessellation that has been used as a template, as shown in Fig. 3.

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Supplementary Fig. 12: One of the six reconﬁgurable prismatic architected materials characterized by ¯z= 9 that we found

using the numerical algorithm. This speciﬁc architected material is based on #5, for which 18 faces of the polyhedra are

extruded and the remaining 18 faces are made rigid. Note that the faces of both tetrahedra are made fully rigid, and therefore

are not taken into account in the numerical analysis. The resulting architected material has ndof = 2.

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SUPPLEMENTAL TABLES

Supplementary Table 1: To enhance the reconﬁgurability of the proposed architected materials, we reduce their connectivity

by selectively extruding faces of the unit cell, while making the remaining faces rigid. In this table we summarize the results

obtained for the extruded structures based on the 28 uniform space-ﬁlling tessellations depicted in Fig. 3 and Supplementary

Fig. 9. For this study we considered a maximum of 216 designs per tessellation, randomly selected from the 2Fpossibilities

(where Fis the number of face pairs), so that for 11 of the tessellations (#4-5, #9-10, #16-17, #20-21, #23, #25, and #27)

the results are not complete, but rather indicate a trend. When determining the degrees of freedom, ndof, we only consider

the designs that do not contain any disconnected parts. However, we expanded the number of possible designs by removing

the polyhedra for which all faces have been made rigid from the extruded unit cell, as those would have resulted in rigid parts

completely disconnected from the architected materials.

unit cell #designs connected percentage of connected with ndof

ndof = 0 1 2 3 4 5 6 7 >7, < 17

#1 2824.7% 100.0% 0 0 0 0 0 0 0 0

#2 213/ 216 24.8% 100.0% 0 0 0 0 0 0 0 0

#3 213 16.7% 62.5% 34.0% 3.5% 0 0 0 0 0 0

#4 213/ 226 15.3% 62.1% 33.6% 4.3% 0 0 0 0 0 0

#5 213/ 220 91.6% 96.6% 3.2% 0.230% 0 0 0 0 0 0

#6 212 45.2% 100.0% 0 0 0 0 0 0 0 0

#7 211 66.1% 100.0% 0 0 0 0 0 0 0 0

#8 211 66.9% 100.0% 0 0 0 0 0 0 0 0

#9 213/ 229 94.4% 99.8% 0.249% 0 0 0 0 0 0 0

#10 213/ 222 96.4% 100.0% 0 0 0 0 0 0 0 0

#11 2512.9% 0 0 100.0% 0 0 0 0 0 0

#12 210 9.9% 0 81.2% 17.8% 0.990% 0 0 0 0 0

#13 2811.4% 0 0 75.9% 24.1% 0 0 0 0 0

#14 213/ 216 13.1% 43.7% 34.6% 19.6% 2.0% 0.070% 0 0 0 0

#15 213/ 216 3.1% 0 36.1% 43.8% 16.9% 3.0% 0.194% 0 0 0

#16 213/ 218 28.9% 36.5% 31.4% 18.7% 8.6% 3.8% 0.909% 0.048% 0 0

#17 213/ 224 11.4% 61.7% 31.1% 7.0% 0.175% 0 0 0 0 0

#18 2926.8% 27.7% 66.4% 5.8% 0 0 0 0 0 0

#19 212 52.9% 46.3% 7.3% 9.6% 7.0% 5.5% 12.5% 0 11.9% 0

#20 213/ 244 99.0% 100.0% 0.002% 0 0 0 0 0 0 0

#21 213/ 228 99.5% 100.0% 0 0 0 0 0 0 0 0

#22 2328.6% 0 0 0 100.0% 0 0 0 0 0

#23 213/ 224 60.1% 28.6% 20.0% 12.7% 9.8% 7.3% 5.2% 4.4% 3.3% 8.6%

#24 2844.3% 18.6% 42.5% 9.7% 15.0% 14.2% 0 0 0 0

#25 213/ 229 93.4% 100.0% 0 0 0 0 0 0 0 0

#26 2433.3% 0 0 40.0% 60.0% 0 0 0 0 0

#27 213/ 228 97.2% 100.0% 0 0 0 0 0 0 0 0

#28 2762.2% 94.9% 5.1% 0 0 0 0 0 0 0

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