# The delicate memory structure of origami switches

Théo Jules1,2,3 0000-0002-0245-3754<EMAIL_ADDRESS>Austin Reid4
0000-0003-1741-2223 Karen E. Daniels5 0000-0001-6852-3594 Muhittin Mungan6
0000-0002-0352-5422 Frédéric Lechenault2 0000-0001-6432-5240 1Université de
Lyon, Ecole Normale Supérieure de Lyon, Université Claude Bernard, CNRS,
Laboratoire de Physique, F-69342 Lyon, France  2Laboratoire de Physique de
l’École Normale Supérieure, ENS, PSL Research University, CNRS, Sorbonne
University, Université Paris Diderot, Sorbonne Paris Cité, 75005 Paris, France
3 Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv
University, Ramat Aviv, Tel Aviv, 69978, Israel  4Center for Exploration of
Energy and Matter, Indiana University  5Department of Physics, North Carolina
State University  6Institut für angewandte Mathematik, Universität Bonn,
Endenicher Allee 60, 53115 Bonn, Germany

(August 27, 2024)

###### Abstract

While memory effects emerge from systems of wildly varying length- and time-
scales, the reduction of a complex system with many interacting elements into
one simple enough to be understood without also losing the complex behavior
continues to be a challenge. Here, we investigate how bistable cylindrical
origamis provide such a reduction via tunably-interactive memory behaviors. We
base our investigation on folded sheets of Kresling patterns that function as
two-state memory units. By linking several units, each with a selected
activation energy, we construct a one-dimensional material that exhibits
return-point memory. After a comprehensive experimental analysis of the
relation between the geometry of the pattern and the mechanical response for a
single bit, we study the memory of a bellows composed of 4 bits arranged in
series. Since these bits are decoupled, the system reduces to the Preisach
model and we can drive the bellows to any of its 16 allowable states by
following a prescribed sequence of compression and extension. We show how to
reasonably discriminate between states by measuring the system’s total height
and stiffness near equilibrium. Furthermore, we establish the existence of
geometrically-disallowed defective stable configurations which expand the
configuration space to 64 states with a more complex transition pattern. Using
empirical considerations of the mechanics, we analyze the hierarchical
structure of the corresponding diagram, which includes Garden of Eden states
and subgraphs. We highlight two irreversible transformations, shifting and
erasure of the defect, leading to memory behaviors reminiscent of those
observed with more complex glassy systems.

## Introduction

Memory is the brain function that enables us to both record past events and
remember them in the future. If we extend this definition to broader
considerations, it represents the capacity of any system – electrical,
magnetic, or mechanical – to store and recall information about past
configurations or events. Memory-like behaviors appear in many forms all
around us. We regularly experience materials that wear or even break when put
under repeated stress. In this instance, the history of the system effectively
modifies its mechanical properties, a process named _aging_ [1]. Previous
studies, recently reviewed by Keim  _et al._ [2], have observed that confined
crumpled sheets [3] and rubbers [4] have a memory of the largest loading; that
folded polymeric sheets [5, 6, 7] and frictional interfaces [8, 9] have a
protocol-dependent dynamical response. These memory effects are a direct
consequence of time-dependent, out-of-equilibrium dynamics.

These types of memory only retain a small amount of information for a short
time, with the system eventually relaxing back to an equilibrium state. This
property is in conflict with practical storage needs, for which a large amount
of data should remain for a long duration. A hard disk drive (HDD) is an
excellent storage tool for these exact reasons: its binary data is encoded in
a ferromagnetic thin films via magnetic hysteresis [10, 11]. An
electromagnetic head produces a magnetic field able to change and read the
disk’s local magnetic orientation as a binary bit: up or down. Importantly,
the amount of storable data scales inversely with the localized size of the
magnetic bit and allows for modern high-capacity information storage. In
essence, the HDD uses the availability of numerous modifiable and readable
stable global configurations to memorize a large amount of information.
Following from these ideas, we are motivated to consider mechanical memory,
since mechanical systems can also present multiple stable configurations.
Because memory is profoundly linked to the states of the system, its control
on scalable systems is also a promising way to design metamaterials, materials
with physical properties determined by their structures, with reliable
reprogrammable properties [12].

Origami seems a likely candidate for designing this kind of metamaterial.
While its origins are in the art of folding paper, origami’s definition in a
scientific context has expanded to represent all the structures obtained by
folding thin sheets, and also to include the fastening of panels (discouraged
by artistic practitioners). The degrees of freedom granted by the design of
the creases’ pattern, the variety and scalability of the resulting physical
properties, and the ease of prototyping make origami a powerful concept for
metamaterial fabrication. Some patterns allow for the folding of large
structures into more compact space, generating deployable structures [13, 14,
15]. Others exploit the mechanical interactions between folds to generate a
device with multiple stable equilibria [16, 17, 18].

Importantly for their use in developing mechanical memory, previous studies
have shown that cylindrical origamis display mechanical bistability for
specific folding patterns [19, 20, 21, 22, 23, 24]. Moreover, recent research
showed that such origamis have potential applications with damping-less impact
mitigation [25], vibration isolation [26], crawling robots [27, 28, 29],
robotic bendable arms [30], mechanical metamaterials [31] or even a property
we can exploit, binary memory [32, 33, 34, 35]. While the search for
mechanical memory using cylindrical origami is not novel, the previous
investigations limit their model and analysis to a simplified elastic system
with truss elements and forego a crucial component for effective memory, the
external non-destructive reading of the configuration.

In this work, we explore the memory capacities of bistable cylindrical
origamis generated from Kresling creases’ pattern [36], where the state,
folded or deployed, of a barrel, a pair of two mirrored unit cells, encodes
for one bit of information. In the first section of this paper, we detail the
characteristics of the pattern and remind the reader of previous geometrical
models establishing the existence of two flat-faced solutions for the barrel
[22]. Then, we concentrate on the mechanical study of a single-barrel module
produced from a thin mylar sheet, focusing on the case of uniaxial loading for
which the rotation of the ends is suppressed. We observe a clear structure in
the relation between the angles of the folding pattern and the threshold
forces for the transition between configurations. We also confirm that the
experimental measurement for the deployed origamis’ height is consistent with
the geometrical prediction. Next, we investigate the characteristics of a
stack of four modules connected in series. We describe how the Preisach model
defines a quasi-static transition diagram of reachable configurations that we
verify experimentally. We use the study on single modules to highlight how the
stack’s state is readable from its height at equilibrium and its local elastic
response. Finally, we manually prepare states that we expected to be forbidden
and investigate these hidden stable defective-configurations allowed by the
material’s elasticity. We explain how these new configurations modify the
transition diagram into a more complex hierarchical graph, presenting
properties analogous to amorphous materials, and establish a detailed
framework for future analysis.

##  I The Kresling Module as a Mechanical Bit

###  I.1 Origami Bellows

The term cylindrical origami, which we will also name origami bellows,
represents a class of cylindrical structures obtained by folding flat thin
sheets while following precise patterns of creases. For an abstract ideal
origami, the creases are straight and have no hinge bending energy, while the
faces are rigid without thickness. Given these constraints, the whole surface
of the folded origami can be divided into triangles. Consequently, the origami
bellows are constrained by Connelly’s Bellows Theorem [37]: continuous
deformations under the rigid faces constrain must conserve its contained
volume. In other words, for a cylindrical origami to be able to function as a
bellows that changes volume under actuation, some elements must be driven away
from the idealized flat-faced configuration used to derive its equilibrium
state [20]. Because origamis are folded from thin sheets, the bending
properties of the sheets enable such deformations. Thus, behaviors such as
bistability, snap-through, self-deployment, and plastic fatigue emerge from
mechanical frustrations in the overall configuration.

In order to reduce the number of adjustable parameters, a previous study by
Reid  _et al._ only considered regular patterns fully tiled by a single
triangular or trapezoidal shape [22]. Of the four types of achievable
patterns, only the Kresling [36] and the Miura-Ori [38] patterns demonstrate
two valid flat-faced configurations, an indicator of mechanical bistability.
The Kresling pattern is characterized by a single obtuse triangle (see fig. 1
A) and can naturally be obtained when torsion is added to the compression of a
thin cylinder [36]; it is closely related to the Yoshimura buckling pattern
[39], a triangular pattern observed when a thin cylinder is compressed.
Starting from the Kresling pattern, the Miura-Ori pattern is obtained by
fusing pairs of triangles with adjacent long-sides into obtuse trapezoids.

Even though the two patterns seem tightly linked, the small difference in
tessellation heavily affects the mechanical response of the resulting origami.
For instance, a notable disparity is the evolution of the polygonal base,
delimited by the horizontal folds, during the deformation. On the one hand, a
triangular tiling yields a unique cross-sectional geometry for all
configurations (see fig. 1 C). On the other hand, the polygonal base is
significantly modified during the deformation when a trapezoidal tiling is
considered [22]. Moreover, a quadrilateral tiling presents more opportunities
to make inexpensive small deflections of the faces compared to triangular
tiling. As a consequence, the Kresling bellows appear stiffer under external
loading, effectively producing a higher energy barrier between equilibrium
configurations, a property which is confirmed experimentally [22]. Owing to
its pronounced bistability and relatively unconstrained geometric demands, the
following study will exclusively feature Kresling bellows.

###  I.2 Kresling Patterned Module

The unit cell of the Kresling pattern, shown in fig. 1, is defined by four
vectors $\bm{\omega}_{i}$, with $i=$$1234$, and two angles
$\phi_{2}<\phi_{1}<\pi/2$. The lengths of both vectors $|\bm{\omega}_{1}|$ and
$|\bm{\omega}_{4}|$ are identical while both vectors $\bm{\omega}_{2}$ and
$\bm{\omega}_{3}$ are constrained to reach the same height $\frac{h}{2}$.
Given these restrictions, a unique characteristic length $l_{1}$, such that
$|\bm{\omega}_{1}|=l_{1}/2$, is enough to define the pattern’s dimensions
completely.

UnitCellGeometry

Figure 1: Geometry of cylindrical bellows produced with a Kresling pattern.
(a) Schematics of the pattern for a single barrel. The unit cell is delimited
by the dashed lines, $\bm{\omega}_{1}$, and $\bm{\omega}_{4}$. A Kresling
pattern corresponds to $\phi_{2}<\phi_{1}<\pi/2$. The adjacent unit cell,
characterized by $\bm{\omega}_{2}^{\prime}$ and $\bm{\omega}_{3}^{\prime}$, is
a mirror symmetry of the first unit cell along the horizontal fold. (b) Vector
$\bm{\omega}_{3}$ projected in the $YZ$ plane for a deployed configuration.
$\Psi$ is the angle between the projection and the z axis. (c) Cross-section
in the $XY$ plane delimited by the horizontal folds.

The kinematic conditions imposed by the rigid faces hypothesis yield

$\sin\Psi=\cfrac{1}{2\tan\cfrac{\pi}{n}\tan\phi_{2}}\left[1-\frac{\tan\phi_{2}}{\tan\phi_{1}}\pm\sqrt{\left(\frac{\tan\phi_{2}}{\tan\phi_{1}}-1\right)^{2}-4\frac{\tan\phi_{2}}{\tan\phi_{1}}\tan^{2}\cfrac{\pi}{n}}\right],$
(1)

with $n$ the total number of unit cells within a circumference and $\Psi$ the
angle between the $Oz$ axis and the projection of the $\bm{\omega}_{3}$ vector
in the $YZ$ plane (see fig. 1 b). The complete computation is available in
[40].

Equation 1 seems intricate at first glance, but the broader mechanical
behavior it predicts is straightforward. The plus or minus before the radical
indicates that for any order $n$, there are at most two solutions for this
expression and the bellows is at most bistable. Since the solution only has a
physical meaning if $\Psi$ is real-valued, the bistable region’s borders is
delineated by the constraint $\left|\sin\Psi\right|\leq 1$, which sets a lower
boundary $\phi_{1}\geq(\frac{1}{2}+\frac{1}{n})\frac{\pi}{2}$, and a non-
crossing constraint imposed as $\phi_{1}\leq\frac{\pi}{2}$. To maximize the
bellows’ deployability, defined as the difference of the angle $\Psi$ between
both solution of the eq. 1, we impose that the collapsed state must be
completely flat. This flat-foldability requires

$\phi_{2}=\phi_{1}-\frac{\pi}{n},$ (2)

for all studied bellows.

Finally, we need to account for any rotation induced by the chirality of the
unit cell. As a way to avoid this issue, we will only consider bellows formed
by pairs of mirror-stacked ring that we call a _barrel_. Since the mirror
symmetry yields opposite chirality between the two halves of each barrel, the
whole bellows is achiral and will net a zero global internal torsion during
folding.

###  I.3 Bellows’ Production and Experimental Framework

The production process for the origami bellows includes two steps. First, we
use a laser cutter to inscribe the desired folding pattern (see fig. 2) in
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick transparent A4 mylar sheets.
We chose $\ell_{1}=$32\text{\,}\mathrm{mm}$$ to create the largest bellows
with a pattern that fits inside an A4 sheet. Moreover, as a compromise between
the difficulty of producing increasingly complex bellows and the size of the
configuration space with bistable structures, we chose $n=5$. We also use the
first step to ablate a fraction of each crease’s length through a series of
perforations, diminishing its local plastic threshold as well as its effective
width. This results in a substantial decrease of the crease stiffness [41]
without eliminating it entirely and further assures that hinge-bending a fold
is easier than bending an adjacent face, resulting in a high energy barrier
separating the two stable configurations. For all the studied samples, we
removed $70\text{\,}\mathrm{\char 37\relax}$ of each crease’s length and the
width of each perforation is 0.5 mm. Then, we used double-sided tape to link
neighboring sides of the bellows with predefined tabs.

While our end goal is to study long bellows with multiple barrels and stable
configurations, this analysis requires an initial understanding of the
properties of a single barrel. Hence, we started our investigation by using
the star-like pattern presented in fig. 2. (a) to produce _modules_ , bellows
containing only a single barrel. Following the geometrical analysis
established in the previous section, a module should be bistable as long as
the pattern angle $\phi_{1}$ is between $63^{\circ}$ and $90^{\circ}$. In
practice, for $\phi_{1}<74^{\circ}$, the modules have a single stable
configuration, while for $\phi_{1}>82^{\circ}$, the deployed configuration is
too rigid and does not properly fold under uniaxial compression. As a result,
we limit our study to $\phi_{1}$ between $74^{\circ}$ and $82^{\circ}$. Please
note that the open end is closed with a pentagonal slice of mylar sheet taped
to the end tabs. Then, a simple method to produce longer structure is to stack
modules on top of each other and use double-sided tape to connect them.

In order to probe the mechanical properties of the modules and the stacks, we
mounted the ends of every sample to plates that can’t rotate, themselves
attached to a uni-axial testing machine (Instron). When the ends are allowed
to rotate, individual unit cells can actuate [32]; by providing rigid
clamping, we require that unit cells of opposite chirality collapse in pairs.
The testing device records the end to end height $H_{z}$ of the bellows and
measures the force $F$ applied during deployment-folding experiments. The
exact loading protocol will be given for each experiment.

Figure 2: Cutting pattern used to produce a single module with $\ell_{1}=32$
mm, $\phi_{1}=75^{\circ}$ and $n=5$ next to the resulting bellows after
folding. $H_{z}$ is the bellows’ end-to-end height. The drawings are available
online [42].

###  I.4 Loading a Single Module

Our initial step was to establish the mechanical response of single modules
and determine how it depends on the particular folding pattern selected by the
choice of $\phi_{1}$. To reach that goal we performed cycles of 3 deployment-
folding experiments where the module started from, and returned to, the folded
configuration. We ensured that the two limit values for the forces entirely
fold and deploy the origami at each cycle while keeping the load low enough
not to damage the sample. A typical force-displacement response is shown in
fig. 3. While the response during the first cycle of a newly folded bellows is
substantially different, subsequent cycles become roughly repeatable. We
believe that the viscoelastic properties of the creases [7] is the main reason
behind this behavior. Before each experiment, the system has sufficient time
to relax to an asymptotic equilibrium; however, this is not possible during an
experiment lasting only a few minutes. As a result, the starting state for the
first cycle is unlike any of the other cycles. This observation is reminiscent
of the Kaiser behavior where various physical systems manages to keep a memory
of the largest input [2].

The cycles present typical bistable behavior with three equilibrium
configurations corresponding to the three times the force-displacement curve
crosses the $F=0$ line during deployment or folding, where the bistability
arises from the flexibility of the faces and the existence of two flat-faces
configurations.

Figure 3: Force-displacement response of a single module with $n=5$,
$\phi_{1}=78^{\circ}$ and $\ell_{1}$ = 32mm for 3 cycles of deployment-
folding. The first two cycles are transparent while the last one is opaque.
$F^{+}=1.0$ N and $F^{-}=-2.11$ N are the thresholds forces between
configurations. The heights for folded ($H_{z}^{0}$ = 21.6 mm) and deployed
($H_{z}^{1}$ = 65.5 mm) configuration are annotated.

Notably, the system displays hysteresis with a different response whether we
fold or deploy the bellows, beyond what was reported in the purely elastic
response of a similar truss structure [32, 43]. This discrepancy comes from
the system’s mechanical properties that depend on the history of the sample.
Each crease is characterized by a _rest angle_ , an internal characteristic
which represent the angle between the faces when the crease is at rest. When
prepared in a folded configuration, every rest angle is required to be small
to reach a near-flat folded configuration. During the deployment, all the
creases open and, due to the plasticity of the material, so do the rest angles
[7]. The opposite happens during folding, where the rest angles close.
Consequently, the mechanical equilibrium for the same height $H_{z}$ will
depend on whether the module is being deployed or folded. An unwanted
consequence is that the height of the module in its stable configurations is
poorly-defined. For simplicity, we set the length $H_{z}^{0}$ (resp.
$H_{z}^{1}$), the size of the folded (resp. deployed) module, to be the
minimal (resp. maximal) height obtained for $F=0$ during the last cycle. We
will see later that this choice is in good agreement with the geometrical
predictions.

An essential characteristic of a bistable system is the load threshold
required to switch from one stable configuration to the other. The origami
bellows’ response displays a maximum value for the force during deployment,
$F^{+}$, which is the threshold for the folded to deployed transition. In the
same way, we define $F^{-}$ as the threshold for the deployed to folded
transition. The difference in amplitude between both thresholds comes from the
plasticity of the creases and the path of elastic deformation to go from one
flat-faced configuration to the other.

We tested modules with different angle $\phi_{1}$ in order to obtain designs
generating distinct mechanical response and thresholds. The measurement of the
threshold loads $F^{\pm}$ and normalized heights of the stable configurations
$h_{z}^{0}=H_{z}^{0}/l_{1}$ and $h_{z}^{1}$ are shown in fig. 4. During the
normalization, we took into account experimental approximations we made while
gluing neighbor branches together by taking an interval for the length
$\ell_{1}$ from $31\text{\,}\mathrm{mm}32\text{\,}\mathrm{mm}$. These
considerations leads to the error bars in fig. 4 (b).

Figure 4: (a) Force thresholds with respect to the pattern angle $\phi_{1}$.
The four highlighted modules are those used in section II.2. (b) Normalized
height $h_{z}=\frac{H_{z}}{\ell_{1}}$ measured for the folded and deployed
configurations. The solid green line corresponds to the geometrical prediction
of the deployed height given by eq. 3. The dashed orange line corresponds to
the mean value $h_{0}=0.70$ measured for the folded height.

The patterns of creases we consider are flat-foldable. As such, the expected
height for the folded configuration is 0. However, this is not mechanically
feasible due to the bellows’ finite wall thickness. Furthermore, we noticed
that all of the modules studied showed a similar folded height
$H_{z}^{0}\approx 0.65\cdot\ell_{1}=$21\text{\,}\mathrm{mm}$$, substantially
larger than the expected height simply emerging from material thickness
$4t\approx 0.4$ mm. This extra height comes from the small remaining rigidity
of the creases, making the stable equilibrium configuration not rigorously
flat-faced. Still, this observation is reassuring with regard to the
reproducibility of our test since we used significantly different pattern and
obtained a consistent value of $h_{z}^{0}$. For the remainder of the paper, we
set $h_{0}$ as the normalized height we expect each module to have in the
folded state.

In contrast, the deployed height changes dramatically with $\phi_{1}$. The
theoretical deployed height from the geometrical flat-faced model comes from
projecting $\bm{\omega}_{3}$ on the $z$ axis

$H_{z}^{1}(\phi_{1})=\frac{2\ell_{1}\cos\left[\Psi(\phi_{1})\right]}{\cot\left[\phi_{2}(\phi_{1})\right]-\cot\phi_{1}},$
(3)

with $\phi_{2}(\phi_{1})$ the solution of eq. 2 and $\Psi(\phi_{1})$ the
negative solution of eq. 1. Experimentally, this height is measured in our
system’s state of mechanical equilibrium, which is not rigorously in its flat-
faced ideal configuration. Even with slight internal frustration, the measured
normalized height of the deployed stable configuration $h_{z}^{1}$ stays
consistent with the theoretical prediction.

The data show a monotonic dependence of the value of each threshold force
$F^{\pm}$ on the angle $\phi_{1}$, a behavior that can be understood
qualitatively via the simple requirement that, as $\phi_{1}$ increases, the
angle $\Psi$ at the deployed equilibrium decreases. In other words, individual
bellows become more and more straight for larger $\phi_{1}$, and the
straighter a cylindrical origami is, the more the faces need to bend during
the deformation. This directly leads to the observed increase of the energy
barrier and the threshold forces.

In this section, we established the geometrical and mechanical properties of
the Kresling single modules with loading experiments. We observed that its
height in the deployed configuration is in agreement with the geometrical
model, while the folded height is independent of the pattern. Another crucial
observation is the relative ordering of both load threshold $F^{\pm}$ with
respect to the pattern.

##  II Bellows as a mechanical memory system

###  II.1 Preisach Model

Figure 5: Schematical description of the Preisach model. (a) Transition cycle
for an hysteron $\sigma$, element with two stable configurations 1 and 0, with
respect to the external loading $F$. $F^{+}$ (resp. $F^{-}$) is the loading
threshold for the ${\tt 0}\rightarrow{\tt 1}$ (resp. ${\tt 1}\rightarrow{\tt
0}$) transition. (b) Stack of $L$ independent hysterons $\sigma_{i}$ in series
symbolizing a stack of modules.

Using the observed mechanical properties of the modules, we can correctly
predict, via Preisach modeling, the quasi-static transition diagram of a stack
of modules with different $\Phi_{1}$, effectively obtaining readable and
modifiable mechanical memory. The Preisach model is a simple mathematical
model of systems with hysteresis [44, 10] which was originally used to
describe the hysteretic response in magnetic materials [11]. The model
consists of a set of $L$ hysteretic elements, called hysterons, each of which
can be in one of two states, 0 or 1. When a hysteron is in state 0, it
switches to state 1 as soon as a driving field $F$ becomes larger than a
threshold value $F^{+}$. To switch back to state 0, the driving field must
become lower than a second threshold value $F^{-}<F^{+}$. Thus for
$F^{-}<F<F^{+}$, the hysteron’s state is undetermined and depends on its
history, thereby giving rise to hysteresis. A schematic view of the hysteron
transition cycle is presented in fig. 5 (a). The theory also assumes that each
hysteron couples to the applied external field independently, meaning each
transition cycle is unaffected by the other hysterons’ states.

Consider an arrangement of $L$ bi-stable modules in series and clamped at one
end, labeled $i=1,2,\ldots,L$. The hysteretic properties of such systems were
studied by Puglisi and Truskinovsky [45]. When an external force $F$ is
applied to the last bellows, an equal force is likewise exerted on each module
due to internal equilibrium. As depicted in fig. 3, each bellows $i$ switches
from the folded (0) to the deployed (1) configuration and back at threshold
forces $F^{\pm}_{i}$. These thresholds’ values depend on the pattern via the
angle $\phi_{1}$. Suppose the state of each module is independent. In that
case, we can regard this system as a Preisach model, schematized in fig. 5
(b), where each bellows is a hysteron. Therefore, we describe the
configuration or microstate of the bellows by an $L$-bit vector, where each
bit represents the state of the corresponding bellows. Finally, we consider
very slow changes of the applied driving force so that the response is
quasistatic, and we do not need to consider dynamical behaviors.

This model’s main hysteresis loop is formed by the transition from the fully
folded configuration ${\tt(00\cdots 0)}$ to the completely deployed one
${\tt(11\cdots 1)}$ under increasing force, and back when this force is
subsequently reduced. These two microstates are absorbing: starting from any
stable configuration, the fully deployed or folded configuration will be
reached for sufficiently large forces.

The modification of the bellows’ microstate due to the external loading can be
represented as a transition graph. The details of such a description have been
given in [46]. Here, each vertex corresponds to a configuration that does not
transition spontaneously to another configuration and that is compatible with
a range of applied forces, the stability interval. We call such microstate
stable. Depending on the values of the transition thresholds $F^{\pm}_{i}$ of
the individual modules, not all $2^{L}$ configurations are stable. Transitions
between stable configurations occur when the force is increased (resp.
decreased) to the upper (resp. lower) limit of a stability interval, and a
single hysteron changes its state from ${\tt 0}\to{\tt 1}$ (resp. ${\tt
1}\to{\tt 0}$). As a result, except for the two absorbing microstates, every
stable microstate has two transitions to another stable microstate: a bit
either folding or deploying. Moreover, transitions occur one hysteron at a
time, so the Preisach model does not have avalanches.

Among the set of stable states of a Preisach system, one subset is distinct:
the set of reachable microstates. It corresponds to all configurations that
can be reached from either of the two absorbing microstates under arbitrary
protocols of driving forces. We call all other, i.e. non-reachable
microstates, “Garden of Eden” (GoE) microstates, since these configurations
cannot be returned to once the system has landed in a reachable microstate
111The notion of Garden of Eden states goes back to Edward F. Moore [56] who
credited the coinage to John W. Tukey. It is based on the foundational work of
John von Neumann on cellular automata, and in particular how such automata can
self-reproduce by creating copies of their states [57]. It turns out that in
the analysis of such problems the presence of automata states that one can
transit out of but never into takes a central role: these are called Garden of
Eden configurations..

The transition graph formed by the reachable microstates alone is called the
Preisach graph [46]. It consists of the main hysteresis loop and its sub-
loops, which by virtue of the return point memory are organized in a
hierarchical structure [48]. Moreover, it turns out that the topology of the
Preisach graph does not depend on the values of the threshold fields
$F^{\pm}_{i}$ per se, but only their relative ordering. In other words, the
crucial property is the sequence in which the individual hysterons change
their state when the force is decreased relative to the order when the force
is increased [46]. In particular, if these two orders are equal, there are no
GoE microstates and all $2^{L}$ possible configurations are stable and
reachable. The Preisach graph in that case for $L=4$ is depicted in fig. 6
(a), where the folding (resp. deployment) transitions have been marked by
black/gray (resp. red/orange) arrows that point to the right (resp. left).

###  II.2 Transitions of a stack of individual modules

Figure 6: Preisach transition graph for a stack of 4 modules with – from top
to bottom – $\phi_{1}=[82^{\circ},80^{\circ},78^{\circ},75^{\circ}]$. In the
transition graph, each ordered bit 1 corresponds to a deployed module, and 0
to a folded one. Each circle is a reachable configuration while each
black/gray (or red/orange) arrow corresponds to the deployment (or folding) of
one module. Black/red (darker) arrows are a sample pathway to the 0101
configuration starting from the absorbing microstate 0000, as illustrated by
the images in (b).

In order to test the predictions of the Preisach model in an experimental
setting, we probed the response of a chain of four linked modules labeled as
$i=$ $1\text{\,}2\text{\,}3\text{\,}4\text{\,}$ with respective pattern angles
$\phi_{1}^{i}=$
$82\text{\,}\mathrm{\SIUnitSymbolDegree}80\text{\,}\mathrm{\SIUnitSymbolDegree}78\text{\,}\mathrm{\SIUnitSymbolDegree}75\text{\,}\mathrm{\SIUnitSymbolDegree}$.
From the data collected with single modules in fig. 4 (a), we expect the
modules to fold and deploy in the same order: $4\to 3\to 2\to 1$. As such, the
stack belongs to the case previously described, for which all $2^{4}=16$
configurations are stable and reachable, with the transitions between them
specified in the graph presented in fig. 6 (a). After clamping the stack to
our loading device, we explored the transition diagram through quasi-static
change of the force and we managed to observe all the transitions predicted by
the Preisach model. For instance, the ${\tt 0000}\rightarrow{\tt 0101}$
transition is shown in fig. 6 (b). Videos of complete deployment, folding, and
the ${\tt 0000}\rightarrow{\tt 0101}$ transition are available in
Supplementary Materials.

During our exploration, we examined the origami’s response to small
deformations around the rest position, corresponding to $F=0$ N, for every
configuration. More precisely, we performed a single loading-unloading cycle
between $-0.5$ and $0.5$ N which yields two mechanical characteristics for
each microstate, its height and its stiffness. A typical force-displacement
response is shown in the appendix A.

Figure 7: Mechanical properties of the stack of 4 modules. (a) Relative height
of each microstate measured experimentally (red error bar) and predicted (gray
points) through a method detailed in the text. (b) Measured stiffness $k_{eq}$
of the stack for small deformation in the vicinity of the equilibrium
position. The error bars show the difference between loading and unloading.
The boxed number $None$ gives the “magnetization” macrostate, the number of
deployed bellows for each microstate.

Extending our work on single modules, we compared in fig. 7 (a) the recorded
relative heights of the configurations at rest $h_{z}^{\text{eq}}$ to the
predicted ones $h_{z\mathrm{,\textrm{ pred}}}^{\textrm{eq}}$. The latter is
obtained by considering the sum of the height for each module

$\displaystyle h_{z\mathrm{,\textrm{
pred}}}^{\textrm{eq}}(k_{1}\;k_{2}\;k_{3}\;k_{4})=\sum_{i}h_{0}+k_{i}(h_{z}^{1}(\phi_{1}^{i})-h_{0})$
(4)

with $i$ the module number, $\phi_{1}^{i}$ the pattern angle and $k_{i}$ the
state of the $i$th module. For a given module $i$, its deployed height is
expected to be the theoretical value $h_{z}^{1}(\phi_{1})$ given in eq. 3,
while its folded height is taken as the value $h_{z}^{0}$ we measured during
single-module loading. Here again, the theoretical value is in good agreement
with the measurements and follows the same ordering. However, a precise and
unequivocal reading of the configuration does not arise from our analysis.
Note that the error bars, coming both from the uncertainty on the value of
$\ell_{1}$ and the interval for $h_{z}^{\text{eq}}$ found during the
mechanical probing near-equilibrium, do not allow for an indisputable
identification of the configuration. Still, if we consider the “magnetization”
macrostate for the system, corresponding to the total number of deployed
modules, we observe a unequivocal step in height for our stack for different
macrostates, with smaller differences among its microstates.

A way to break this degeneracy is to examine the configuration’s mechanical
properties. We identify the stiffness of each configuration using a linear
regression of the force-displacement response
$F=k^{\text{eq}}(H_{z}-H_{z}^{\text{eq}})$ between $-0.1$ and $0.1$ N. Once
again, we observe a notable difference between loading and unloading, the
latter appearing softer. We reported our measurements for all microstates in
fig. 7 (b). Compared to the height, the data shows that the stiffness is a
worse independent identification tool since we observe more overlap between
microstates, even for different numbers of deployed bellows. However, as a
complementary tool, it may help discriminate between very similar
$h_{z}^{\text{eq}}$, particularly for larger numbers of units. For instance,
let’s look at both configurations 1001 and 0110. While they both have
comparable lengths, the former appears stiffer. So a quick analysis of the
mechanical response is enough to differentiate between these two microstates.
Our analysis of a stack of individually-produced modules shows the transition
map for the collection of resulting stable configurations is well captured by
the Preisach model. Moreover, we provide a protocol to read the microstate
through their mechanical properties, more precisely their height and apparent
stiffness.

###  II.3 Hidden configurations

Figure 8: (a) Photo of the defective 11-+ configuration. (b) Partial directed
graph of transitions between stable configurations obtained experimentally.
The opaque (or transparent) section is the diagram for reachable (or Garden of
Eden, GoE) microstate. The relation between the module’s symbol and the state
of its halves is schematized in the legend. 2-GoE loops are denoted in cyan
and 4-GoE loops in purple. The black and gray (or red and orange) arrows
correspond to the deployment (or folding) of a pair of sections observed
experimentally. Thin arrows out of the GoE loops represent irreversible
“plastic” events: _erasure_ or _shift_. The dotted arrows represent expected
transitions from reachable microstates that we did not observed before the
sample’s breakdown.

During our investigations, we found that, by carefully twisting parts of a
partially compressed bellows, the entire system could be driven into
unexpected stable configurations as shown in fig. 8 (a). These configurations,
obtained by localized manual actuation, present a _defect_ : a couple of
barrels which are half-folded half-deployed. More precisely, these barrels
have their opposite half folded. While such defective configurations should be
forbidden by the global non-twisting assumption imposed by the boundary
conditions, the hidden degrees of freedom due to the material elasticity [49]
allows for opposite halves of slightly different pattern angle to collapse
together.

Given these new observations, it is necessary to increase the number of valid
microstates we consider for each module. We will designate half-deployed
barrels with new symbols, indicating whether the deployed half faces the
stiffer (+) or softer (-) ends of the bellows. Since a single defect concerns
two barrels, we should consider 54 additional microstates for a 4-barrels
bellows, 48 with a single defect, and 6 with two defects (the counting process
is detailed in appendix B). In order to study them, we built a second stack of
four modules with pattern angles
$\phi_{1}=[81^{\circ},79^{\circ},77^{\circ},75^{\circ}]$, where the three
stiffest modules were each made with $\phi_{1}$ $1^{\circ}$ smaller compared
to the first stack, making them slightly softer. We only managed to reach
configurations with a single defect that maintain a small net twist. The
potential two-defects configurations were not stable, and the system
immediately erased one defect. Moreover, we only managed to observe 39 out of
the 48 microstates with a single defect with manual actuation before the
sample became too damaged to continue. All the configurations and their
corresponding observed transitions are transcribed in the directed graph in
fig. 8 (b) where we differentiate between the reachable microstates (darker
color) and the GoE microstates (lighter color).

Following recent work on the modeling of amorphous solid with directed graphs
[50], we will focus our following analysis on the strongly connected
components (SCCs) of the transition graph. The SCCs are the collections of
subgraphs where every configuration is reachable from every other
configuration. The largest SCC turns out to be the main graph and – by
definition – includes only the reachable microstates. It is identical to the
graph presented in the previous subsection. Therefore, there is no transition
from a reachable microstate to a defective configuration, making them GoE
microstates. We define $k$-GoE loops as the SCC of the graph formed by $k$ GoE
microstates with identical defects. For instance, 1-GoE loops, which we will
also name _isolated_ GoE microstate, are single GoE microstate that can not be
reached from any other configuration, even defectives ones, through quasi-
static transitions. Our study managed to identify eight isolated GoE
microstates, five 2-GoE loops, and two 4-GoE loops.

###  II.4 Mechanics and hierarchical structure of the GoE loops

The mechanical origin of the GoE loop can be understood as follows. For any
$2^{l}$-GoE loops, with $l$ a positive integer, we notice the defect does not
involve the $l$ softer barrels. Therefore, it remains kinetically stuck as
long as we keep the external loading small enough. If this range is enough to
actuate the $l$ softer barrels, they will generate a Preisach graph for $l$
independent modules, the $2^{l}$-GoE loops we observe. This logic holds for
the 4-GoE and 2-GoE loops in the reported graph. Following this reasoning, for
a bellows with $N$ barrels and a single defect, we expect to obtain a total
number $n_{l}$ of $2^{l}$-GoE loops such that

$\displaystyle n_{l}=\left(N-l-1\right)\,2^{N-l-1}$ (5)

with $l$ a positive integer and $l<N$. Here the term $N-l-1$ counts the number
of locations at which the other defective barrel can be placed, while the
factor $2^{N-l-1}$ accounts for the possibility of the exchange of the two
defects and the assignments of states $0$ and $1$ to the $N-l-2$ inactive
barrels. Since for $l=1$ and $N=4$ we have $n_{1}=8$, this implies that there
are three additional 2-GoE loops that we did not record in the diagram.

A careful analysis of the graph in fig. 8 (c) shows a clear hierarchical
structure with nested loops consistent with return-point-memory [48]. Any
transition from a $2^{l}$-GoE loop to another loop only occurs when the $l$
softer barrels are either completely deployed or collapse. The next barrel to
be actuated is the $l+1$ softer barrel, which presents a defect. This leads to
two possible outcomes:

  * •

_Erasure_ : The defect disappears, and the system shifts to the main cycle.
This removal is necessary when the total number of deployed or folded modules
becomes $N$ and the system transition to the absorbing microstates 0000 or
1111.

  * •

_Shift_ : The defect moves to a stiffer module. This allows additional soft
barrels to be freed and contributes to a larger loop. In most cases, the
defect moves to the next softest barrel. If this barrel were also defective,
the system might eliminate the fault and transition to the main cycle.

While this analysis successfully predicts most of the transitions observed,
some remain inconsistent, such as ${\tt 00-+}\rightarrow{\tt 1-+0}$ for
instance. We argue that these discrepancies come from the aging of the creases
during our testing, leading to modifications of the configuration stability.

In our investigation of a stack of modules, we did not observe the last type
of transition: the spontaneous _appearance_ of a defect. This transition does
not appear because there is insufficient coupling between modules to drive it.
Instead, we observe that the constraints are inadequate to favor the formation
of defective configurations over geometrically preferred states. By
introducing a correlation between barrels, we can produce systems for which
the formation of defects is beneficial. An attempt to achieve this is provided
in the appendix C, where we look at a different cutting pattern that provides
inter-unit coupling. In that case, we find that some defective configurations
are reachable states. This observation confirms that tweaking the correlation
between barrels is a promising avenue for controlling the transition diagram,
but the technicalities are beyond the scope of this study.

##  III Discussion

In this study, we presented the multiple memory capabilities of cylindrical
origami bellows. We based our analysis on a specific subset of the pattern’s
configuration space for Kresling cylindrical origamis only using pairs of unit
cells, here named barrels. Our choice guarantees that the folded structures
hold three properties: (1) it does not have a global twist under loading, (2)
it has two stable configurations, and (3) the pattern’s geometry is governed
by a single parameter $\phi_{1}$. We investigated the mechanical properties of
the simplest bellows that we call _modules_ and which is composed of only one
barrel. The probing reveals a monotonic relation between the parameter
$\phi_{1}$ and the threshold forces $F_{\pm}$ of transition between the stable
microstates. Moreover, it confirms that a geometrical model based on flat
faces correctly predicts the height of the deployed structure.

We exploited all these properties to design a long origami with mechanical
memory by stacking modules with different $\phi_{1}$, and considering the
state of each module, deployed or folded, as one bit of memory. The system
requires three distinct features in order to work as effective memory: (1)
Each stable microstate has to remain stable in time. This is the case for our
stack as long as the external loading stays small. (2) The microstate of the
system can be actively changed through external action. The internal
equilibrium assures the force imposed at the extremity of the stack is applied
on all the modules. Furthermore, as we assume them to be independent, and
given their mechanical properties, we always obtain a Preisach system where
every existing stable configuration is reachable with quasi-static actuation.
This property holds for any number of modules as long as their patterns are
all different. (3) The microstate of the system can be deduced from non-
destructive measurements. From a minute deployment-folding cycle, we showed
how the configuration is identifiable with the height and the stiffness of the
origami at rest.

More remarkably, the hidden degrees of freedom due to the material elasticity
[49] allow for new stable configurations that a zero global twisting should
forbid. These defective configurations create a more convoluted diagram with
Garden of Eden states and irreversible transitions. Our experimental study
shows a hierarchical structure with the graph’s strongly connected components
induced by the relative mechanical response of the modules. Notably, the size
of any defective loop depends on the stiffness of the softest defective bit of
memory which irreversibly increases with the highest load applied to the
system during shifting events. While the model for our system remains simple,
we recover behavior typically observed in amorphous materials like the memory
of the largest input [3, 4] or irreversible plastic events with internal
rearrangement due to external stress [51].

From an engineering standpoint, the memory capabilities of the presented
cylindrical origamis, in addition to its capacities to tune its mechanical
properties, make it an interesting reprogrammable metamaterial in line with
recent work on very similar origami structures [31]. In this paper, we only
considered systems composed of four barrels. In principle, larger systems with
distinct reachable states are obtainable since degeneracy only appears if two
or more barrels have identical pattern angles $\\{\phi_{1},\phi_{2}\\}$. In
practice, the material we use and our fabrication method limit the scalability
to only a few barrels before a precise reading becomes impossible. Developing
a method to reduce measurement error is essential to make the origami bellows
a tractable memory system. We believe that the observed hysteresis comes from
the non-elastic response of the creases, which manifests as changes of their
rest angle. Consequently, we expect new production schemes that reduce creases
stiffness to diminish the experimental error and facilitate state reading.
Moreover, long deployed bellows might prefer to buckle out of the central axis
during compression instead of folding barrels, a behavior already exploited to
create origami octopus-like bendable arms [30]. We also expect longer bellows
to generate stable configurations with more defects. The internal strain in
defective configurations is related to the internal twist per unit length. By
increasing the number of barrels, the internal strain generated by any single
defect is reduced, which should support the introduction of additional folding
defects.

We conclude by pointing out that our works follow a very recent surge of
interest for the Preisach model and its generalizations in order to explain
the complex transition diagram of multistable systems by introducing
interactions between hysterons [52, 53, 54, 55]. While our barrel-based
approach considers independent hysterons, the existence of defective
configuration highlight that each barrel is separable into two mirror halves,
which are both bistable. Consequently, another way to analyze our system is to
divide it and consider each half-barrel as a hysteron. The condition of no-
rotation imposed at the ends creates correlations between hysterons and favors
transitions where hysterons of opposite chirality transition together. We
leave a thorough investigation of how these correlations precisely relate to
the folding pattern and crease properties for future works.

## Acknowledgements

T.J. thanks Pedro Reis for interesting discussions. Research by K.E.D and M.M.
was supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958. M.M. was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Projektnummer 398962893, the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer
211504053 - SFB 1060, and by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-
ID 390685813.

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##  Appendix A Response of the stack near equilibrium

We probed the elastic response to small deformation around equilibrium of each
reachable state for the stacks of four modules with
$\phi_{1}=[82^{\circ},80^{\circ},78^{\circ},75^{\circ}]$ presented in section
II.2. To do so, we started from a loading $F\approx-0.5$ N and carried one
cycle of extension-compression as shown in fig. 9. We used linear regression
on a small region around $F=0$, approximately between $-0.1$ and $0.1$ N,
during both extension and compression to set the two limit values for the
stiffness $k^{eq}$ used in fig. 7.

Figure 9: Force-displacement response of the stack of 4 modules studied in
section II.2 near the rest position for the configuration 0000. The dotted
lines represent the limit values of $H_{z}^{eq}$ we used in fig. 7

##  Appendix B Number of configuration with $k$ defects for $N$ barrels

Employing the convention taken in the paper, we consider a defect as a pair of
${\tt+}$ and ${\tt-}$ barrels. As a result, a defect always concerns two
barrels and necessarily $N\geq 2k$, with $N$ the total number of barrels.
Starting from this postulate, counting the number of configurations
$n^{N}_{k}$ with $k$ defects for a system with $N$ barrels is straightforward.
First, we need to account for all the combinations of barrels concerned by the
defects leading to $\binom{N}{2k}$ possibilities. Then, for all the defectives
barrels, half must be in the state ${\tt+}$ while the other half must be in
the state ${\tt+}$, leading to $\binom{2k}{2}$ possibilities. Finally, each
non-defective barrel has two state, ${\tt 0}$ or ${\tt 1}$, generating
$2^{N-2k}$ possibilities. As we multiply all these contributions, the total
number of configurations with $k$ defects is

$\displaystyle
n^{N}_{k}=\binom{N}{2k}\binom{2k}{k}2^{N-2k}=\frac{N!}{k!^{2}(N-2k)!}2^{N-2k}.$
(6)

The system we studied has $N=4$ barrels and may have up to $k=2$ defects.

The total number of configurations should simply be the sum of configurations
realizable with $k$ defects:

$\displaystyle n$ $\displaystyle=\sum\limits_{k=0}^{2}n_{k}^{4}$
$\displaystyle=16+48+6$ $\displaystyle=70$

However, since we did not observe any configurations with two defects, we
consider a total of $n_{0}^{4}+n_{1}^{4}=64$ configurations.

##  Appendix C Case of a long bellows with coupled barrels

While developing our production schemes, we also looked alternative solutions
for generating multi-barrel bellows: a single long cutting pattern rather than
preparing individual modules and connecting them afterward. We extended the
design of the branches for the star-like pattern to include multiple barrels.
However, we cut each branch separately to compromise with the limited size of
the original A4 transparent sheets. The typical pattern of a single branch is
shown in fig. 10 (a). Once again, we used double-sided tape and pre-cut tabs
to generate a single coherent structure. To follow as closely as possible our
previous analysis, we folded an origami bellows with five branches and four
barrels, with pattern angles
$\phi_{1}=[81^{\circ},79^{\circ},77^{\circ},75^{\circ}]$. As for the stack of
modules presented in the main paper, we tested the transition diagram for the
system through quasi-static actuation. The result is shown in fig. 10 (b).

At first glance, the diagram seems identical to the Preisach graph presented
in section II.2 and follows the same hierarchical structure of transitions.
Yet, some reachable states are defective configurations. Furthermore, we
observe transitions already defined in the context of GoE microstate
transition with _shifting_ and _erasure_ of the defect. But we also observe
transitions that we name _appearance_ where one degree of defect is added to
the system. These three types of transitions round up the necessary framework
to analyze the transformation of defective structures.

This specific diagram is actually easy to understand empirically. We observe
the appearance of a defect only when the third barrel, corresponding to
$\phi_{1}=77^{\circ}$ , transition from 1 to 0. If the configuration allows
it, i.e. if at least another barrel is deployed, the third barrel prefers to
transition to the - state and create a defect. Finally, the system remains
with a defect until either it has to be folded entirely (0000) or the three
softer barrels have to be deployed (0111 or 1111). Meanwhile, the defect
shifts and the other bellows deploy and fold according to the mechanical rules
established in section II.4.

The appearance of defects can be attributed to two elements. First, it might
simply come from an error during production. Then, the analysis of the
transition diagram is an interesting tool to understand where the irregularity
lies. For instance, a misplacement of the double-sided tape could lead to one-
half of the third barrel having difficulty folding. A second possibility comes
from the difference in the cutting pattern. Contrary to the modular approach,
the barrels in a multi-barrel design are linked by a single crease with an
angle that depends on the state of the two sections it connects. The angle
ranges from $\pi$, if both connecting sections are deployed, to 0, if they are
folded. An interesting case emerges when only one side is folded. The crease’s
stiffness may change the stability of the structure, effectively changing the
mechanical properties of each module depending on the states of its neighbors.
As a result, a destabilization of the non-defective relative to the defective
configurations facilitates the formation of defects and explains why we manage
to reach them quasi-statically. We leave an investigation of the relation
between properties of the connecting crease and change of the transition
diagram to future studies.

Figure 10: (a) Cutting pattern for a single branch of a 4-barrels stack with
$\ell_{1}=32$ mm, $\phi_{1}=[81^{\circ},79^{\circ},77^{\circ},75^{\circ}]$ and
the resulting stack for $n=5$ after folding. The drawings are available online
[42]. Below is a photo of the produces bellows for $n=5$ in the 1111
configurations. (b) Observed transition diagram. The black (red) circles
represent normal (defective) configuration. A dotted arrow correspond to the
_appearance_ of a defect, a dashed arrow its _shift_ and a doted-dashed arrow
its _erasure_.