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O igami F us a ion and I s In luence on Ene gy
Landscapes o O igami Assemblies
Shixi Zanga, Tuo Zhaoa, Diego Misse onib,2, and Glaucio H. Paulinoa,c,2
This manusc ip was compiled on Sep embe 25, 2025
Ha nessing ins abili ies o mul icomponen mul is able s uc u al assemblies can po en ially
lead o scalable and e e sible unc ionali ies, which can be enhanced by explo ing us a ion.
Fo ins ance, s anda d K esling o igami cells exhibi non- unable in insic ene gy landscapes
de e mined by hei geome y and ma e ial p ope ies, limi ing hei adap abili y a e
ab ica ion. To o e come his limi a ion, we in oduce us a ion o enable ine- uning o
he ene gy landscape and esul ing de o ma ion s a es. By p es essing he K esling cell by
means o special sp ings wi h indi idual con ol, we induce ei he global o localized (i.e.,
c ease le el) us a ion, which allows changing he ene gy ba ie (cell o assembly). We
in es iga e he mechanical beha io o us a ed K esling assemblies, bo h heo e ically
and expe imen ally, unde a ious loading and bounda y condi ions. Ou indings e eal
ha changing he us a ion s a e leads o p ecise con ol o olding sequences, enabling
p e iously inaccessible olding pa hs. The p oposed concep pa es he way o inno a i e
applica ions in mechanical me ama e ials and o he ields equi ing highly p og ammable and
econ igu able sys ems.
Geome ical us a ion |O igami |K esling pa e n |Ene gy landscape
R
econ igu able assemblies consis o enginee ed mac oscopic nonlinea s uc u es
unde going la ge de o ma ions. The beha io o he assemblies depends
on he ma e ial p ope ies and geome ical nonlinea i y o he local uni cells.
Classic nonlinea cells, such as snap- h ough beams (
1
,
2
), and buckling-d i en
con inuum elemen s(
3
,
4
), ha e been explo ed in applica ions, including ene gy
abso p ion (
5
,
6
), so obo ic ac ua o s (
7
,
8
), non-commu a i e esponse (
9
), wa e
p opaga ion (
10
), acous ic me ama e ials (
11
,
12
), and so ma e unde going
d ama ic shape changes (
13
,
14
). Mo e ecen ly, o igami-inspi ed geome y has
en iched he design space o he nonlinea uni cells such as K esling (
15
–
20
), squa e-
wis (
21
,
22
), Wa e bomb (
23
,
24
), and Miu a-O i a ia ions (
25
–
27
), and cu ed
olds (
28
,
29
). Fu he mo e, he non- igid o igami assemblies ha e enabled ini e
de o ma ion o applica ions in ol ing shape mo phing (
30
–
32
) and con ollable
ene gy landscape (
32
–
34
). The a o emen ioned econ igu able assemblies assume a
p e-de ined de o ma ion pa h; hus, he nonlinea p ope ies, e.g., he shape o he
ene gy landscape and he ins abili y beha io , a e non- unable a e ab ica ion.
On he o he hand, ep og ammable s uc u es enable con inuously a iable elas ic
modulus ia changing he con igu a ional s a e o local uni s (
35
), e.g., heigh s
o he elas ic shells (
36
), o a ion angles o he gea s (
37
), and olling mo ion o
he cams (
38
). A limi ed numbe o s udies ha e applied his ep og ammabili y
concep o assemblies wi h unable ins abili y beha io s (
39
,
40
), e.g., swi ch
be ween monos able and bis able esponses. The swi ching beha io is achie ed by
ac ua ing wo dis inc opological s a es o a local uni . Howe e , he limi ed local
s a es es ic he numbe o global de o ma ion pa hs, which makes i di icul o
ep og am he ene gy ba ie o he assembly con inuously.
He e, we in oduce geome ical us a ion (
41
) in o he o igami-inspi ed
assemblies, oge he wi h no el expe imen al ix u es (e.g., ee- o a ing and ee-
ansla ing), o achie e con inuous ene gy landscape ep og ammabili y. The
geome ical us a ion is embedded wi hin he o igami cells by means o h ee
mechanisms: global s e ch, global o a ion, and c ease (local) s e ch (Fig. 1A).
Each mechanism in eg a es shell-based o igami wi h special sp ing elemen s,
which in oduce p es ess in o he us a ed o igami cell. The p es ess le el
o he us a ed model is con inuously adjus able by con olling he sp ing
p ope ies, i.e., s e ching/ o a ing di ec ion and magni ude. The us a ed
assemblies, wi h unable p es esses, enable one o enginee he ene gy landscape
o mul iple s able s a es on he ly. We achie e unp eceden ed olding pa hs,
o he wise in easible (Fig. 1B, Mo ie S1). This inding pa es he way o po en ial
Signi icance S a emen
F us a ion: de imen al o desi -
able? Some imes de imen al, bu
some imes desi able o achie e
new unc ionali ies o non igid mul-
is able o igami s uc u es. I p o-
ides he means o augmen he
ene gy landscape o such s uc-
u es as i can be ailo ed o he
ea u es o he geome y o he
o igami uni cell and he us a-
ion ype. By equipping he cell
wi h special, con ollable, elas ic
sp ings, ine- une con ol o e en-
e gy ba ie s is enabled, acili a ing
p ecise olding sequences. Expe -
imen s demons a e ha ac i a ion
o deac i a ion o he us a ion can
enhance he p og ammabili y o a
mul icell o igami a ay, unlocking
o he wise un easible olding pa hs.
Wi h po en ial impac in ields such
as mechanical compu ing and non-
commu a i e s a e ansi ion, his
app oach o e s possibili ies o scal-
able and adap able s uc u es wi h
high unabili y.
Au ho a ilia ions:
a
Depa men o Ci il and En i onmen-
al Enginee ing, P ince on Uni e si y, P ince on, NJ,
08544, USA;
b
Depa men o Ci il, En i onmen al
and Mechanical Enginee ing, Uni e si y o T en o,
I aly;
c
P ince on Ma e ials Ins i u e (PMI), P ince on
Uni e si y, P ince on, NJ 085444, USA
Au ho con ibu ions: S.Z., T.Z., D.M., and G.H.P.
designed and pe o med esea ch; S.Z., T.Z., and
D.M. pe o med expe imen s; S.Z. implemen ed he
compu a ional app oach; and S.Z., T.Z., D.M., and
G.H.P. analyzed da a, de eloped he heo y, and w o e
he pape .
The au ho s decla e no compe ing in e es .
2
To whom co espondence should be
add essed. E-mail: diego[email p o ec ed]
o [email p o ec ed]
www.pnas.o g/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS — Sep embe 25, 2025 — ol. XXX — no. XX — 1–12
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unable ene gy
landscape
in insic ene gy
landscape
u
φ
sp ing sp ing
U
u o φ
l
η
l
e
e
e
U
u o φ
ene gy
ba ie (ΔU)
base
ene gy
U
u o φ
U
u o φ
S anda d K esling cell assembly
F us a ed K esling cell assembly
o sional
sp ing
global o a ion
global s e ch c ease s e ch
in easible
ΔU1
ΔU1
ΔU2
ΔU1
ΔU2>
easible
ΔU1
ΔU1
ΔU u
ΔU1
ΔU u <
0 80
0
T (N•mm)
-20
40
φ (deg)
φ
u
F=0
T
A B
=ΔU2-Usp
b
δ
F ee-
ansla ing
ix u e
a
b
c
δa
c
δ
Fig. 1. S anda d and geome ically us a ed o igami assemblies wi h unable ene gy landscapes and olding pa hs. (A) Top: Schema ic o he s anda d K esling o igami
(b own) and i s in insic ene gy landscapes. The symbols
u
and
φ
deno e displacemen and wis angle, espec i ely. Bo om: Schema ics o he us a ed models (o ange)
wi h h ee ypes o p es ess: (i) global s e ch, (ii) global o a ion, and (iii) c ease s e ch (local) and hei unable ene gy landscapes. The symbols
ℓe
and
ηe
deno e he leng h
o he axial sp ing and he o a ing angle o he o sional sp ing, espec i ely. (B) Top: an in easible olding pa h using he s anda d K esling assembly. He e, ∆
U1
and ∆
U2
deno e he in insic ene gy ba ie s o he o igami cells made o di e en ma e ials. Bo om: us a ed assembly achie ing an unp eceden ed olding de o ma ion. He e, ∆
U u
and
Usp
deno e he ene gy ba ie o he us a ed model and he elas ic ene gy s o ed in he o sional sp ing elemen , espec i ely. Inse s show he o a ional es se up wi h
he ee- ansla ion ix u e and expe imen al da a o wis angle φ e sus o que T.
applica ions in econ igu able mechanical me ama e ials and
non-commu a i e s a e ansi ions.
Resul s
Theo y o geome ical us a ion. S a ing om he heo-
e ical modeling o he K esling o igami, which desc ibes
i s mechanics h ough an elas ic ene gy dependen on wo
independen a iables, displacemen
u
and wis angle
φ
,
we de elop an enhanced model o he us a ed sys em.
We w i e he o al elas ic ene gy o he us a ed model,
U u(u, φ), as ollows:
U u(u, φ) = Usp (u, φ) + U(u, φ),[1]
whe e
U
(
u, φ
) is he elas ic ene gy o he s anda d K esling
cell (
42
), and
Usp
(
u, φ
) deno es he elas ic ene gy s o ed
in he p es essed sp ings, which embed us a ion in o he
o igami cell. The i e- e m elas ic ene gy o he s anda d cell
is exp essed as:
U(u, φ) = 1
2nbks,b(b(u, φ)−b0)2
+1
2ncks,c(c(u, φ)−c0)2
+1
2nak ,a(δa(u, φ)−δa0)2[2]
+1
2nbk ,b(δb(u, φ)−δb0)2
+1
2nck ,c(δc(u, φ)−δc0)2.
We o e a de ailed explana ion o he ma e ial pa ame e s
ks,i
(
i
=
band c
) and
k ,i
(
i
=
a, b, and c
), he geome ic
pa ame e s
b
,
c
,
δi
(
i
=
a, b, and c
), and he pa ame e s
ni
(
i
=
a, b, and c
) in he SI Appendix, sec ion 1, Fig. S1,
and Table S1.
Using he p inciple o minimum o al po en ial ene gy, he
equilib ium condi ions o axial and o que loading can be
de i ed wi h Eq.1. Speci ically, he axial o ce and o que
can be calcula ed as ollows:
F u(u, φ) = ∂Usp (u, φ)
∂u +∂U(u, φ)
∂u ,[3]
T u(u, φ) = ∂Usp (u, φ)
∂φ +∂U(u, φ)
∂φ .[4]
He e, he exp ession o
Usp
(
u, φ
) depends on he p es essed
sp ing mechanism o in e es . We p esen h ee ypes
o us a ed models, i.e., global s e ch, global o a ion,
and c ease s e ch, in he ollowing sessions (see de ails o
heo e ical o mula ion in SI Appendix, sec ion 1).
Global s e ch. This model in ol es a single de o med sp ing
elemen inse ed in he o igami and aligned wi h i s cen al
axis. In he heo e ical analysis, he ini ial s a e o he
sp ing can be ei he ex ended o comp essed, p o iding he
o igami wi h unable p es ess p ope ies. The ex ended
sp ing de o ms he o igami cell in he olding di ec ion, while
he comp essed sp ing s e ches he uni in he deploying
di ec ion. The p es essed sp ing is coupled wi h he o igami
2— www.pnas.o g/cgi/doi/10.1073/pnas.XXXXXXXXXX Zang e al.
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cell o achie e a new equilib ium s age, deno ed as he
us a ed mode. The esul ing ene gy landscape o he
us a ed mode is p og ammable by adjus ing he elas ic
ene gy s o ed in he sp ing elemen , de ined as ollows:
Usp (u, φ) = 1
2ks,e(∆ℓe−u)2,[5]
whe e
ks,e
and ∆
ℓe
a e he s i ness and leng h change o he
sp ing elemen embedded in he us a ed K esling cell.
Global o a ion. This model akes ad an age o he o a ional
deg ee o eedom o he K esling o igami and embeds
o sional p es ess in o he uni cell. The p es ess le el
is con olled by a o sional sp ing in eg a ed wi h he o igami
cell. The sp ing o a es he unde o med cell and eaches a
new equilib ium s a e wi h p es esses. The sp ing enables
wo ypes o p es esses, which a e de ined as posi i e and
nega i e. The posi i e p es ess o a es he cell in he same
di ec ion as i s in insic wis ing di ec ion, while he nega i e
p es ess o a es he uni in he opposi e di ec ion. The
elas ic ene gy unc ional o he o sional sp ings is de ined
as ollows:
Usp (u, φ) = 1
2k ,e(∆ηe−φ)2,[6]
whe e
k ,e
and ∆
ηe
a e he s i ness and o a ing angle change
o he o sional sp ing in eg a ed in he us a ed model.
C ease (local) s e ch. This model embeds p es essed sp ings
along he moun ain c eases (local) o he o igami cell o
us a e he sys em. Those local sp ing elemen s de o m
he uni in o a new s able s a e wi h a non-ze o base ene gy.
The magni ude o he base ene gy and ene gy ba ie o he
us a ed model a e unable by con olling he elas ic ene gy
s o ed in he sp ings, de ined as ollows:
Usp (u, φ) = 1
2neks,e(b(u, φ)−b0+ ∆ℓe)2,[7]
whe e
ne
is he numbe o he s e ching sp ings along he
moun ain c eases (
ne
= 3 in his pape ),
b
(
u, φ
) is he leng h
o he moun ain c ease,
b0
is he ini ial leng h o he moun ain
c ease a he unde o med s a e o he K esling cell, and ∆
ℓe
is he leng h change o he sp ing elemen .
Pa ame ic s udy. The us a ed cells in eg a e h ee ypes o
p es essed models in o he s anda d K esling o igami. Eqs. 5-
7show ha he elas ic ene gy s o ed in he p es essed sp ings
is con olled by i s de o ma ion. Thus, we can na iga e
he ene gy landscape by a ying he leng h change ∆ℓeand
o a ing angle change ∆
ηe
o he sp ing elemen s, espec i ely
(Fig. 2). We e e o SI Appendix, Table S2 o he selec ion
o pa ame e s. Recall ha we deno e he posi i e p es ess
as de o ming he o igami cell in he olding di ec ion, while
he nega i e p es ess de o ms he uni in he deploying
di ec ion. Bo h posi i e and nega i e p es ess d i e he
unde o med o igami in o new equilib ium s a es wi h non-
ze o base ene gy, and he co esponding ene gy landscapes a e
unable. Fo ins ance, he global s e ch model enables wo
us a ed modes: one wi h nega i e p es ess and he o he
one wi h posi i e p es ess (Fig. 2A). Bo h modes ha e non-
ze o base ene gy a he ini ial s able s a e; howe e , in he
nega i e mode, he ene gy a he second s able s a e always
inc eases, while o he posi i e mode, he base ene gy o
he 2nd s able s a e can ei he inc ease o emain unchanged
depending on he gi en leng h change in he p es essed
sp ing. No e ha he base ene gy o he 2nd s able s a e o
he posi i e mode is simila o he base ene gy o he s anda d
K esling. This beha io is due o he us a ing se up used
in he posi i e mode, whe e he elonga ed sp ing becomes
in alid a e e u ning o i s o iginal leng h. By con as , he
sp ings a e always comp essed in he nega i e mode. Thus,
he esul ing ene gy landscapes o he wo modes, nega i e
and posi i e, can be qui e di e en . This inding holds o
esul s ob ained using wo independen loading condi ions:
comp ession wi h ee- o a ion (Fig. 2B-le ) and o sion wi h
ee- ansla ion (Fig. 2B- igh ). No ably, by con olling he
le el o p es ess in he nega i e mode, we can swi ch he
ins abili y beha io om bis able o monos able, as shown in
Fig. 2B (Top: wo o ange cu es). Ano he unique ea u e is
he capabili y o con inuously p og am he ene gy ba ie o
he cell wi h us a ion. He e, he ene gy ba ie is de ined as
he di e ence be ween he local maximum on he landscape
and he ini ial base ene gy. Fig. 2C shows he heo e ical
ene gy ba ie as a unc ion o he s e ching leng h o he
p es essed sp ing. Mo eo e , he cu e shows a smoo h
ansi ion be ween he nega i e mode and he posi i e mode.
This esul highligh s he capabili y o he us a ed model
o achie e ine- uned con ol o e ene gy ba ie s.
Ano he us a ion model we in es iga e is he global
o a ion (Fig. 2D), which also has wo modes simila o
he global s e ch, i.e., nega i e mode and posi i e mode.
The nega i e mode in eg a es a p es essed o sional sp ing
o a ing he o igami opposi e o he wis ing di ec ion o he
cell while i olds. The posi i e mode embeds a o sional
sp ing ha o a es he o igami along he same di ec ion
while he cell olds. By con olling he p es ess le el o
he o sional sp ings, we can achie e non-ze o bases o he
ini ial s able s a es o he cell. Fu he , he shape o he
ene gy landscape can be p og ammed as shown in Fig. 2E.
As a esul , he global o a ion-induced us a ed cell has
swi chable ins abili ies, i.e., monos able o bis able beha io .
No ably, he ene gy ba ie be ween he local maximum and
he ini ial base ene gycan be con inuously unable as shown
in Fig. 2F.
The las us a ion model we in es iga e is he c ease
s e ch shown in Fig. 2G. The p es essed sp ings a e
loca ed along some moun ain c eases due o ab ica ion
conside a ions, as shown in he la e expe imen al alida ion
sessions. The c ease s e ch has wo p es essed modes, i.e.,
posi i e mode wi h elonga ed sp ings and nega i e mode
wi h comp essed sp ings. The wo p es essed modes enable
he us a ed cell wi h p og ammable ene gy landscapes
(Fig. 2H). The s o ed elas ic ene gy is con ibu ed by he
s e ch o p es essed sp ings and he de o ma ion o he
o igami panels and c eases. The moun ain c eases o he
K esling cell sho ens when olding is ini ia ed, bu he c eases
e u n o he ini ial leng h a he olded s able s a e. On
he o he hand, he de o ma ion o he p es essed sp ings
beha es di e en ly unde he wo modes. Fo he nega i e
mode, he p es essed sp ing has simila kinema ics o he
moun ain c ease. The comp essed sp ing is u he sho ened
while he cell is olding. Then, he sp ing e u ns o i s ini ial
comp essed leng h a he olded s able s a e. Fo he posi i e
mode, he sp ing is elonga ed a he ini ial s able s a e. As
Zang e al. PNAS — Sep embe 25, 2025 — ol. XXX — no. XX — 3
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he cell olding ini ia es, he amoun o elonga ion educes.
A he olded s able s a e, he sp ing e u ns o i s ini ial
elonga ed s a e. We can see ha he de o ma ion his o y o
he p es essed sp ing in he posi i e mode and nega i e mode
is qui e di e en . As a esul , he wo modes lead o dis inc
shapes o he ene gy landscape. This obse a ion en iches
he p og ammabili y o he us a ed cells by means o local
c ease con ol. Mo eo e , we p esen an addi ional local
us a ed model, i.e., c ease o a ion, in he SI Appendix,
sec ion 1, Fig. S2, and Table S3.
The a o emen ioned heo e ical analysis conside s linea
sp ing mechanisms wi h cons an s i nesses, i.e.,
ks,e
=
cons an
and
k ,e
=
cons an
. Howe e , he heo e ical
amewo k (Eqs. 5-7) can be gene alized o inco po a e
nonlinea sp ings in he us a ed model. We can eplace
ks,e
and
k ,e
wi h exp essions desc ibing he nonlinea beha io o
he sp ing elemen s. Mo e de ails o he heo y o nonlinea
sp ing modeling a e shown in SI Appendix, sec ion 2, Fig.
S3, and Table S4.
Expe imen al s udy on uni cells. Expe imen s in ol e he
h ee us a ed models heo e ically in es iga ed in he
p e ious session. We de elop a modula ab ica ion solu ion
o ealize he global s e ch model, including he sp ing
elemen , he wi e connec o , 3D-p in ed ames and handles
(Fig. 3A, Mo ie S2). The handle con ols he le el o p es ess
in he sp ing elemen . The sp ing ex ension ∆
ℓ
is a unc ion
o he adius o he handle
handle
and he numbe o in e al
u ns
n
, de ined as ollows: ∆
ℓ
=
n
(2
π handle
)
/
12. The
p es essed sp ing de o ms he s anda d K esling in o a new
equilib ium s a e con igu a ion. This new s able s a e is
us a ed as i s o es elas ic ene gy in bo h panels and
he sp ing. The amoun o ene gy s o ed in he us a ed
model is unable by con olling he sp ing ex ension. In
he comp ession expe imen s wi h he ee- o a ing ix u e,
we es h ee di e en sp ing ex ensions and compa e he
esul s wi h hose o he s anda d o igami cell (Mo ie S2).
Figu e 3B epo s he expe imen al esul s and heo e ical
p edic ion. Fo he s anda d K esling, we conduc es s on
i e specimens and calcula e he mean alue (solid cu es)
and he co esponding s anda d de ia ion (shaded egions)
using he o mulas in SI Appendix, sec ion 3. Fo he
us a ed K elsing, we conduc es s on h ee specimens
o each p es essed model. The e o ba a he i s s able
s a e is he s anda d de ia ion calcula ed om he measu ed
displacemen .
The s ain ene gy plo s (Fig. 3B op-le ) con i m ha he
us a ed models ha e non-ze o base ene gy a he ini ial
s able s a es, and he amoun o he base ene gy depends on
he sp ing ex ensions. No e ha we assume ha he sp ing
elemen s do no con ibu e o he ene gy o mula ion anymo e
i hey a e deac i a ed (i.e., ze o p es ess). Consequen ly,
he beha io o he us a ed cell becomes iden ical o ha
o he s anda d cell (Fig. 3B bo om-le ). The plo s in
Fig. 3B ( igh ) illus a e he o ce-displacemen ela ionship
ob ained om bo h expe imen and heo y, espec i ely. We
zoom in on he ini ial loading egion o show he shi o
o ce cu es. Bo h expe imen al da a and heo e ical analysis
e i y ha he s a ing poin o he o ce cu e shi s as
he p es ess inc eases. The amoun o shi co esponds o
he heigh change a he i s s able s a e in he us a ed
models. No e ha he magni ude o he peak o ce dec eases
as mo e p es ess is applied in expe imen s. In heo e ical
analyses, he peak o ces a e simila . This disc epancy is
caused by he panel buckling a ini ial loading wi h ac i a ed
us a ion (Fig. 3B bo om- igh ), which is no conside ed in
he heo e ical analysis (see mo e discussion in SI Appendix,
sec ion 4).
We design a speci ic mechanism o apply posi i e p es ess
in he global o a ion model. This mechanism, which beha es
like a o sional sp ing, in ol es he cus omized inclined
componen , he sp ing elemen , he wi e connec o , and
3D-p in ed ames and handles (Fig. 3C, Mo ie S3). The
handle con ols he amoun o p es ess applied along he
o sional di ec ion. The o sional angle ∆
η
and o sional
s i ness
k ,e
a e de ined as: ∆
η
=
n
(2
π handle/ ame
)
/
12 and
k ,e
=
T/
∆
η
, whe e
T
is he eac ion o que, and
ame
is he
adius o he ame. Compa ed wi h he s anda d cell, he
p es essed cell is de o med in i s ini ial s able con igu a ion
wi h a non-ze o base ene gy. The elas ic ene gy is s o ed in
bo h he de o med panels and he sp ings. Gi en a cons an
sp ing s i ness, he o sional angle con ols he magni ude o
he ene gy s o ed in he ini ial con igu a ion o he us a ed
model. We es h ee di e en angles unde o sional loading
wi h he ee- ansla ing ix u e (Mo ie S3). Bo h heo e ical
and expe imen al esul s (Fig. 3D-le ) e i y he capabili y
o he global o a ion model o uning base ene gy a ini ial
s able s a es. In addi ion, he wis angle- o que cu es o
deac i a ed and ac i a ed us a ion a e di e en as shown
in Fig. 3D ( igh ). The zoomed-in plo s u he illus a e
he di e en ini ial loading poin s. The shi o hose poin s
is ela ed o he amoun o p es ess applied in he global
o a ion us a ed model.
The wo a o emen ioned global us a ed models inco -
po a e only posi i e p es esses. In con as , he c ease
(local) s e ch wi h nega i e p es ess u he enhances he
ene gy landscape p og ammabili y o he us a ed model.
The c ease s e ch p o o ype in ol es comp essed sp ings
inse ed in 3D-p in ed cases aside om he moun ain c eases
o he cell (Fig. 3E, Mo ie S4). Due o he p es essed
sp ing elemen s, he us a ed cell ge s ex ended and hen
s ays in a new equilib ium s a e. The new s a e has a
non-ze o elas ic base ene gy con ibu ed by he de o med
o igami cell and he p es essed sp ing elemen . Tuning he
magni ude o he p es ess leads o con ollable base ene gy
a he ini ial s able s a es, as shown in Fig. 3F ( op-le ).
Mo eo e , he expe imen al esul s show ha he shape o
he ene gy landscapes depends on p es ess le els o he
sp ings. The mo e he sp ing elemen is comp essed, he
highe ene gy ba ie is achie ed o he us a ed cell. This
beha io ag ees wi h he heo e ical analysis shown in Fig. 3F
(bo om-le ). Figu e 3F ( igh ) e i ies ha he ini ial loading
posi ion o samples wi h deac i a ed and achie ed us a ion
a e di e en . The ini ial loading posi ion is ela ed o he
con igu a ion o he K esling o igami a he i s s able s a e.
The nega i e displacemen
u
indica es inc eased heigh o he
o igami sample in he us a ed model. On he o he hand,
we alida e he c ease (local) s e ch wi h posi i e p es ess.
We e e o SI Appendix, sec ion 5 and Fig. S4 o mo e
de ails.
Expe imen al s udy on assemblies. Beyond he s udies a
he uni cell le el, we explo e he mechanical beha io o
o igami assemblies composed o s anda d cells and us a ed
4— www.pnas.o g/cgi/doi/10.1073/pnas.XXXXXXXXXX Zang e al.
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cells. The cells a e modula , and hey can be connec ed by
minia u e neodymium magne s embedded in he ames.
S anda d K esling assemblies. Fo an o igami a ay wi h h ee
s anda d cells, he e a e, in o al, eigh s able s a es as
shown in Fig. 4A. In heo y, i y-six olding pa hs connec
any wo s able s a es. In p ac ice, six een ou o i y-six
olding pa hs a e unachie able wi hou ansi ing h ough
o he s able s a es (Fig. 4B). Fo ins ance, s a e I (wi h
all h ee cells deployed) canno de o m di ec ly o s a e
VIII (wi h all h ee cells olded) unless passing h ough he
o he s a es whe e one o wo cells ha e been olded. No e
ha eigh addi ional pa hs in Fig. 4C a e in easible due
o he in insic ene gy ba ie ∆
U
buil in he h ee cells,
i.e., ∆
U
( ed cell)
>
∆
U
(yellow cell)
≈
∆
U
(blue cell), which a e
di e en ia ed by con olling he panel hickness (see de ails
in Ma e ials and Me hods). Mo eo e , s a e VIII canno be
pulled o s a e V because lowe -ene gy ba ie cells (blue
o yellow) mus deploy be o e he highe -ene gy ba ie cell
( ed). As a esul , only hi y- wo ou o i y-six olding
pa hs a e easible (Fig. 4D). Eigh easible pa hs ha e been
e i ied expe imen ally unde wo ypes o loading condi ions:
axial loading and o sional loading (Mo ie S5, de ails in
Mo ie S5 a e shown in SI Appendix, sec ion 6 and Fig. S5).
The axial loading condi ion in ol es wo ix u es, i.e., one
is o a ionally cons ained, and he o he one displays ee
o a ion (see mo e de ails in SI Appendix, sec ion 7). The
co esponding es ing esul s a e p esen ed in Fig. 4E. On
he o he hand, he o sional loading condi ion is equipped
wi h he axially cons ained ix u e and he ee- ansla ing
ix u e, espec i ely. The co esponding expe imen al da a
a e shown in Fig. 4F. Acco ding o he expe imen al da a,
we calcula e he ene gy landscapes o eigh easible pa hs in
SI Appendix, sec ion 8 and Fig. S6.
F us a ed K esling assemblies. The 3-cell o igami assembly
shown in Fig. 5A is p es essed such ha he op wo cells
include sp ings on he moun ain c eases (local le el) and he
bo om cell includes a o a ional sp ing (global le el). Each o
he us a ed cells has wo s able s a es wi h unable ene gy
ba ie s enabled by he p es essing le el. As a esul , he
assembly has, in o al, eigh s able s a es, like Fig. 4; howe e ,
he beha io is qui e di e en . The us a ed assembly
can be con inuously ep og ammed by means o local cell
con ol. This ep og ammabili y leads o p ecise con ol
o he olding sequences, and enables he eigh p e iously
in easible olding pa hs o Fig. 4C o be achie ed as easible
olding pa hs in Fig. 5B. Fo example, al hough he in insic
ene gy ba ie o he ed cell is bigge han ha o he blue
cell, he global o a ion mechanism ac i ely lowe s he ed
cell ene gy ba ie . The e o e, when s a e I de o ms o s a e
VII ollowing pa h
1
, he us a ed ed cell olds while
he blue cell emains deployed du ing he p ocess. The
loading condi ion o his de o ma ion is axial comp ession
wi h a o a ionally cons ained ix u e, as shown by he g een
cu e in Fig. 5C (Mo ie S6). The plo has an o ange cu e,
which co esponds o he easible de o ma ion pa h
2
in
Fig. 5B, unde axial loading wi h ee- o a ing ix u e (Mo ie
S6). O he easible olding pa hs, i.e., pa h
3
and pa h
4
, a e achie ed by o sional loading condi ions wi h he
axially cons ained ix u e and he ee- ansla ing ix u e,
espec i ely, as shown in Fig. 5D (Mo ie S6). In addi ion,
ene gy landscapes co esponding o easible pa hs in Fig. 5C
and D a e p esen ed in SI Appendix, sec ion 8 and Fig. S7.
Mo e de ails o he ep og ammabili y o he easible olding
pa hs and ene gy landscapes a e shown in SI Appendix,
sec ion 9 and Fig. S8.
The esul s in Fig. 5demons a e ha he abili y o swi ch
among di e en s a es, by con olling he p es essing le els,
enables he 8 easible olding pa hs o Fig. 5B, some hing no
achie able wi h he non- us a ed K esling a ay (Fig. 4).
While hese olding pa hs could also be enabled by designing
new cells wi h p ede ined ene gy ba ie s, h ough ma e ial
and/o geome y selec ions, his disc e e app oach ine i ably
in oduces new in easible olding pa hs and does no suppo
in-si u econ igu a ion. In con as , he con inuous app oach
o geome ic us a ion, combined wi h a inely uned sp ing
mechanism, enables he elimina ion o in easible olding pa hs.
This allows o dynamic ep og amming o olding beha io
wi hin he same a ay, leading o adap abili y and con ol.
Scope o us a ion. Though ou designs o us a ion is
c ea ed based on K esling o igami s uc u e, hey can be
used o embed us a ion in o o he o igami s uc u es. Fo
ins ance, Figu e 6A gi es an example o explo e he in luence
o p es essing on a cu ed-c ease o igami ube. The s a es
a he ed and blue poin s as well as he o ce cu es indica e
ha he mechanism wi h s e ching sp ing causes a shi o
ini ial s a e o he o igami ube. Unde axial comp ession
loading, he us a ed cu ed-c ease ube exhibi s signi ican
panel buckling (see he s a e a he blue-s a poin in Fig. 6A).
Mo eo e , de ailed compa ison be ween he us a ed and
non- us a ed cu ed-c ease o igami is shown in SI Appendix,
sec ion 10 and Fig. S9.
The second applica ion ela es o p og ammable non-
commu a i e beha io o K esling a ays (Fig. 6B). We
conside a K esling a ay consis ing o wo lowe ene gy
ba ie cells a he op and one highe ene gy ba ie cell a
he bo om, and conside wo examples, one ha does no
in ol e us a ion and one ha does. In bo h examples, we
apply coun e clockwise wis ollowed by clockwise wis , and
hen we e e se he ac ua ion sequence (i.e., clockwise wis
ollowed by coun e clockwise wis ), always esul ing in a o al
ze o ne wis a he end o each ac ua ion sequence. Fo he
i s example, he a ay shows his o y-dependen beha io
in he sense ha he de o med con igu a ion depends on
he sequence o he wis ac ua ion. We demons a e his
ea u e by expe imen s on a e e ence con igu a ion unde
he ac ua ion sequences desc ibed abo e (Fig. 6B- op). No e
ha de ails o he jagged po ion on he unloading cu e a e
elabo a ed in SI Appendix, sec ion 11 and Fig. S10. The
wo di e en end con igu a ions demons a e he ele ance o
wis ing his o y, which indica es non-commu a i e beha io .
Fo he example in ol ing us a ion, he op uni cell (blue)
is he same as be o e; howe e , he bo om wo cells a e
us a ed. The middle cell has linea (local) sp ings (wi h
induced nega i e p es ess) p o iding i wi h a highe ene gy
ba ie han he op cell. The bo om cell has an especially
designed o sional (global) sp ing (wi h induced posi i e
p es ess) p o iding i wi h he highes ene gy ba ie o
he assembly. We obse e non-commu a i e s a e ansi ions
as well (Fig. 6B-bo om); howe e , compa a i ely, he wo
inal s a es in he us a ed K esling a ay a e di e en
om hose in he non- us a ed K esling a ay, esul ing in
Zang e al. PNAS — Sep embe 25, 2025 — ol. XXX — no. XX — 5
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a p og ammable ( us a ion-dependen ) non-commu a i e
s a e ansi ion.
Fu he mo e, we c ea e a shape-mo phing me ama e ial
by embedding he sp ing mechanism in o a 3D-p in ed uss
p o o ype in ol ing mul iple K esling columns (
43
) (Fig. 6C).
The le el o p es ess in each K esling column depends on
he s i ness o sp ings. He e, he column 0 has no sp ing,
while he s i ness o he sp ings in column 1 and column 2
is 0
.
11
N/mm
and 0
.
35
N/mm
, espec i ely. All he sp ings
a e connec ed wi h a handle h ough wi es. Ro a ing he
handle applies a cons an s e ch o all he sp ings. Since
he sp ing o column 2 has he highes s i ness, i de o ms
mo e han column 1 and column 0 (see Fig. 6C-bo om). As
a esul , he a ying de o ma ion in di e en columns allows
he me ama e ial o achie e shape-mo phing beha io .
Discussion
We p esen geome ically us a ed K esling assemblies wi h
unable ene gy landscapes and olding pa hs. The assembly
is modula , and i consis s o bo h s anda d o igami cells and
us a ed cells. We in oduce us a ed modules wi h h ee
ypes o p es ess, i.e., global s e ch, global o a ion, and
c ease (local) s e ch. The p es ess o he us a ed model
is con inuously adjus able by con olling he special sp ings,
which allows o changing he ene gy ba ie o he cell. The
heo e ical analysis e i ies ha he ene gy landscape o he
us a ed cells is p og ammable and he co esponding ene gy
ba ie is con inuously unable. We p o o ype all h ee ypes
o us a ed o igami cells and use ou loading and bounda y
condi ions o alida e he beha io o he cells expe imen ally.
Expe imen s demons a e ha ac i a ing and deac i a ing
us a ion can d ama ically enhance he p og ammabili y o
he o igami assembly, unlocking o he wise in easible olding
pa hs. The p esen concep can be implemen ed widely
in econ igu able sys ems whe e in-si u p og ammabili y is
needed in he applica ion, e.g., adap able me ama e ials o
shape mo phing.
Ma e ials and Me hods
Fo mula ion o he us a ion heo y. Since he us a ed cell can
be con olled by bo h axial o ce,
F
, and o que,
T
, he wo k done
on he cell is calcula ed by
W
(
u, φ
) =
RFdu
+
RTdφ
. The o al
po en ial ene gy o he us a ed cell, Π, can be exp essed using
he o al elas ic ene gy, U u, and wo k, W, i.e.,
Π(u, φ) = U u(u, φ)−W(u, φ).[8]
No ice ha Π(
u, φ
) is a unc ion o wo independen a iables,
u
and
φ
. Based on he p inciple o minimum o al po en ial ene gy,
equilib ium is achie ed when
∂
Π
/∂u
= 0 and
∂
Π
/∂φ
= 0. Thus,
he axial o ce, F, and he o que, T, a e calcula ed by:
F(u, φ) = ∂U u(u, φ)
∂u , T(u, φ) = ∂U u(u, φ)
∂φ .[9]
Fab ica ion o K esling o igami cells. Bo h s anda d and us a ed
o igami cells we e ab ica ed by a ma e ial composed o mu i-laye
o igami pape s (Tan ) and adhesi e apes in be ween (3M 9474LE,
0.17mm- hick).The c ease pa e ns o he blue and yellow cells
include wo laye s o o igami pape s and one laye o adhesi e ape.
The c ease pa e n o he ed cell is made o h ee laye s o o igami
pape s and wo laye s o adhesi e apes. We cu he moun ain
c eases o all he cells. Addi ionally, o he c ease (local) s e ch
(Fig. 3E) design, we cu a apezoid hole on he panel o a oid
in e ac ion be ween he p es essed elemen and he o igami cell.
Mo e de ails a e shown in SI Appendix, sec ion 12 and Fig. S11.
Fab ica ion o 3D-p in ed uss. The 3D-p in ed uss model in
Fig. 6C consis s o wo componen s: ods and so join s, which a e
ab ica ed using a S a asys J55 P ime polyje p in e . The ods
a e p in ed using he Ve oWhi e ma e ial, and he join s a e p in ed
using FLXA9950 (Sho e-A 50), wi h a mix o Ve oUl aClea and
Elas icoClea .
Fab ica ion o p es essed elemen s. The 3D p in ed p es essed
componen s a e ab ica ed using a S a asys J55 P ime polyje
p in e . Fo he global s e ch model, he ames and handle a e
p in ed by he Ve oWhi e ma e ial, and he ma ke s a e p in ed by
he Ve oMagen a ma e ial. Fo he global o a ion model, he op
ame and ma ke s a e ab ica ed by he same ma e ial as he global
s e ch design, while he bo om ame and handle a e p in ed
by he Ve oUl aClea ma e ial. Fo he c ease (local) s e ch
model, he ames and ma ke s a e p in ed by he same ma e ial
as he global s e ch one, while he case assembled on he op
and bo om ames p in ed by he Ve oYellow and Ve oMagen a
ma e ials, espec i ely. The pulley used in he global o a ion model
is a ball bea ing pulley (MiSUMi, SZV3-12). The sp ing used in
he global s e ch model is an ex ension sp ing (McMas e -Ca ,
9065K566). The sp ing used in he global o a ion model is an
ex ension sp ing (McMas e -Ca , 5108N036). The sp ing used in
he c ease (local) s e ch model is a comp ession sp ing (McMas e -
Ca , 9657K641). The sp ings in he global models a e connec ed
o he 3D-p in ed componen s using a 0.3 mm diame e ishing wi e.
Mo e de ailed pa ame e s a e p o ided in SI Appendix, sec ion 12.
No e ha he sp ing sys em can be manu ac u ed by s anda d 3D
p in e s, which enhances he p ac icali y o he us a ion concep .
Fo ins ance, we show an illus a ion o in eg a ing 3D-p in ed
sp ings in o cell o igami in SI Appendix, sec ion 12 and Fig. S12.
Expe imen al se ups. In Fig. 3, we conduc bo h comp ession es s
and o sion es s on an Ins on loading ame machine (Model 68SC-
5 Single Column Tes ing Sys em), equipped wi h ee- o a ing and
ee- ansla ing ix u es, espec i ely (see mo e in o ma ion in SI
Appendix, sec ion 12 and Fig. S13). The applied axial load and
o que ha e been measu ed wi h a o ce/ o que senso (Biaxial
Load Cell ±445 N, ±5.65 Nm). The comp ession expe imen s
a e conduc ed a a speed o 0.25 mm/s, while he o sional
expe imen s a e pe o med a 0.5 deg/s. In Figs. 4and 5, we
conduc comp ession es s using bo h o a ionally cons ained and
ee- o a ing ix u es. Mo eo e , we conduc o sion es s using
bo h axially cons ained and ee- ansla ing ix u es.
Da a, Ma e ials, and So wa e A ailabili y. All da a a e included in
he a icle and/o suppo ing in o ma ion.
ACKNOWLEDGMENTS. This esea ch was suppo ed by Ma -
ga e a E. Augus ine P o esso ship o Enginee ing a P ince on
Uni e si y and he Na ional Science Founda ion unde g an no.
2323276. D.M acknowledges inancial suppo om he Eu opean
Union, ERC g an HE GA 101086644 S-FOAM (Views and opinions
exp essed a e howe e hose o he au ho (s) only and do no
necessa ily e lec hose o he Eu opean Union o he Eu opean
Resea ch Council Execu i e Agency. Nei he he Eu opean Union
no he g an ing au ho i y can be held esponsible o hem.)
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B
U (mJ)
0
50
0 0.5 1
u/um
U (mJ)
0
50
φ/φm
0 0.5 1
100 100
U/φ = 0 U/u = 0
A
0 0.5 1
0
U (mJ)
50
u/um
U (mJ)
0
50
φ/φm
0 0.5 1
100 100
global s e ch
global o a ion
c ease s e ch
h0h u
u
l
e
l
e,0
Δl
e
l
e
Δl
e
nega i e posi i e
h u
u
0
0
Δl =0
e
nega i e
posi i e
012
0
20
40
60
ΔU (mJ)
Δl
e/um
nega i e
posi i e
U u
I
U u
II
Umax
ΔU=U -
max U u
I
E
U (mJ)
0
50
0 0.5 1
u/um
U (mJ)
0
50
φ/φm
0 0.5 1
100 100
D
0 0.5 1
0
U (mJ)
50
u/um
U (mJ)
0
50
φ/φm
0 0.5 1
100 100
nega i e posi i e
nega i e
posi i e
012
0
20
40
60
ΔU (mJ)
Δη
e/φm
nega i e
posi i e
U u
I
U u
II
Umax
H
U (mJ)
0
50
0 0.5 1
u/um
U (mJ)
0
50
φ/φm
0 0.5 1
100 100
G
0 0.5 1
0
U (mJ)
50
u/um
U (mJ)
0
50
φ/φm
0 0.5 1
100 100
u
l
e
l
e,0
Δl
e
l
e
Δl
e
nega i e posi i e
Δl =0
e
nega i e
posi i e
0 0.75
0
20
40
60
ΔU (mJ)
Δl
e/um
nega i e
posi i e
U u
IU u
II
Umax
Δη
e
Δη
e
Δη
e=0
θ u θ0
φ
0
θ u
φ
0
b0b
u
b
80
global s e ch ( -0.6 -0.3 0.5 1Δle/um):
c ease s e ch ( -0.12 -0.06 0.3 0.6Δle/um):global o a ion ( -1 -0.5 0.6 1.2):Δηe/φm
1 s able s a e (I)
s 2 s able s a e (II)
nd
s anda d K esling
-0.44
C
F
I
-0.79
-0.71
80
Fig. 2. Theo e ical model o geome ically us a ed K esling o igami cells. (A) Global s e ch ea u e. S anda d K esling o igami ( op-middle) and wo us a ed models ( op-le
and op- igh ). The le model has a comp essed sp ing (nega i e p es ess), while he igh one has an ex ended sp ing (posi i e p es ess). He e,
h0
deno es he heigh o
he s anda d K esling cell,
h u
is he heigh o he us a ed cell, and
u0
is he heigh di e ence. (B) Rep og ammable ene gy landscapes. Top: in insic ene gy landscape
(black) e sus unable landscapes (o ange) wi h he nega i e p es ess model. Bo om: in insic ene gy landscape (black) e sus unable landscapes (o ange) wi h he posi i e
p es ess model. Le : he elas ic ene gy
U
e sus he no malized axial displacemen
u/um
unde axial loading wi h ee- o a ion. Righ : he elas ic ene gy
U
e sus he
no malized wis angle
φ/φm
unde o sional loading wi h ee- ansla ion. (C) Con inuously unable ene gy ba ie wi h he global s e ch ea u e. No malized leng h changes
o he sp ing ∆
ℓe/um
e sus he ene gy ba ie ∆
U
, which is de ined as he maximum ene gy o a us a ed model
Umax
minus he ene gy a he i s s able s a e
UI
u
. (D)
Global o a ion ea u e. S anda d K esling o igami ( op-middle) and wo us a ed models ( op-le and op- igh ). The le model has a de o med o sional sp ing ha o a es he
o igami clockwise. The igh model has a de o med o sional sp ing ha o a es he o igami coun e clockwise. He e, ∆
ηe
deno es he o a ing angle o he o sional sp ing.
(E) Rep og ammable ene gy landscapes. (F) Con inuously unable ene gy ba ie s o he us a ed model wi h global o a ion. (G-I) C ease (local) s e ch ea u e and he
co esponding ene gy solu ions. The symbols b0and b u deno e he leng hs o moun ain c eases in he s anda d and he us a ed model, espec i ely.
8— www.pnas.o g/cgi/doi/10.1073/pnas.XXXXXXXXXX Zang e al.
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A
Δl
handle
n in e al
F
wi e
handle
sp ing
global s e ch (posi i e sp ing) B
u0
Δl1=5.24mm s able s a es
T=0
F
u
φ
U (mJ)
0 15 30
0
U (mJ)
20
F (N)
0
1.5
30
3
Expe imen Theo y
u (mm) 0 8
0
20
10
45
U (mJ)
u (mm)
0 8
-0.5
0
0.5
u (mm)
-1.5
0 15 30 45
F (N)
u (mm)
Δl2=6.55mm Δl3=7.85mm s anda d K esling
0
20
0 8
0
20
30
10
U (mJ)
u (mm)
0 15 30
u (mm) 45 u (mm)
0 15 30 45
F (N)
0
1.5
3
-1.5
0 8
0
0.5
F (N)
u (mm)
-0.5
s0s1
F ee-
o a ing
ix u e
40
40
C
Δl
handle
n in e al
T
Δη
ame
wi e
handle
sp ing
φ0
global o a ion (posi i e sp ing) D
U (mJ)
0 60 90
U (mJ)
0
30
φ (deg)
F=0
T
φ
u
Expe imen Theo y
φ (deg)
0
6
0 10
0
20
30
10
30
Δη1=6.86os able s a es
Δη2=11.43oΔη3=13.72os anda d K esling
U (mJ)
φ (deg) 0 10
-10
0
10
-30
-60
60
0 60 9030 φ (deg)
0
20
30
10
0 60 90
φ (deg)
30
0
30
φ (deg)
0
6
0 10
U (mJ)
φ (deg) 0 10
-10
0
10
-30
-60
60
0 60 9030 φ (deg)
T (N mm)
T (N mm)
T (N mm)
T (N mm)
s0s1
F ee-
ansla ing
ix u e
F
U (mJ)
F (N)
0 30 45
0
U (mJ)
40
F (N)
-3
0
60
3
c ease s e ch (nega i e sp ing)
Δl
F
es ic ion
sp ing
E
s0
bl
b0
b0
T=0
F
u
φ
Expe imen Theo y
u (mm) u (mm)
15 0 30 4515
20
0
40
60
20
Δl1= -2mm s able s a es
Δl2= -3mm s anda d K esling
6
-3
0
3
6
0 30 45
u (mm) u (mm)
15 0 30 4515 -2 2
0
1.5
F (N)
u (mm)
-1.5
-2 0
0
1.5
F (N)
u (mm)
-1.5
-2 2
u (mm)
0
10
U (mJ)
-2 2
u (mm)
0
10
U (mJ)
s1
2
0
0
0
F ee-
o a ing
ix u e
Fig. 3. Expe imen s in ol ing he h ee us a ed models o he p e ious igu e. (A) Global s e ch wi h posi i e p es ess. Top-le : illus a ion o axial loading wi h a ee- o a ing
ix u e. Top- igh : schema ic o he design enabling p es ess in he axial di ec ion. Bo om-le : pho o o an unde o med o igami cell. Bo om- igh : pho o o a p es essed uni in
i s ini ial s able s a e. (B) Expe imen al esul s and heo e ical p edic ions. Top: expe imen al esul s compa ing he beha io o he us a ed models wi h he s anda d o igami
cell. Solid lines ep esen he mean alue and shade egions ep esen he s anda d de ia ion o he expe imen al da a. Bo om: co esponding heo e ical p edic ions. Le :
axial displacemen
u
e sus s o ed elas ic ene gy
U
. Righ : displacemen e sus applied o ces
F
. The inse s highligh he ea ly s ages o he es . (C) Global o a ion wi h
posi i e p es ess. Top-le : illus a ion o o sional loading wi h a ee- ansla ing ix u e. Top- igh : schema ic o he design wi h o sional p es ess in he cell. Bo om-le : pho o
o an unde o med o igami cell. Bo om- igh : pho o o a p es essed cell in i s ini ial s able s a e. (D) Expe imen al esul s and heo e ical p edic ions. (E) C ease (local) s e ch
wi h nega i e p es ess. Top-le : illus a ion o he loading wi h a ee- o a ing ix u e. Top- igh : schema ic o he comp essed sp ings s a egically loca ed along he moun ain
c eases. Bo om-le : pho o o an unde o med o igami cell. Bo om- igh : pho o o a p es essed cell in i s ini ial s able s a e. (F) Expe imen al esul s and heo e ical p edic ions.
Zang e al. PNAS — Sep embe 25, 2025 — ol. XXX — no. XX — 9
