# Strain compatibility and gradient elasticity in morphing origami
metamaterials

Hussein Nassar<EMAIL_ADDRESS>Arthur Lebée<EMAIL_ADDRESS>Emily
Werner  Department of Mechanical and Aerospace Engineering, University of
Missouri, Columbia, MO 65211, USA  Laboratoire Navier, École des Ponts,
Université Gustave Eiffel, CNRS, France

###### Abstract

The principles of origami design have proven useful in a number of
technological applications. Origami tessellations in particular constitute a
class of morphing metamaterials with unusual geometric and elastic properties.
Although inextensible in principle, fine creases allow origami metamaterials
to effectively deform non-isometrically. Determining the strains that are
compatible with coarse-grained origami kinematics as well as the corresponding
elasticity functionals is paramount to understanding and controlling the
morphing paths of origami metamaterials. Here, within a unified theory, we
solve this problem for a wide array of well-known origami tessellations
including the Miura-ori as well as its more formidable oblique, non-
developable and non-flat-foldable variants. We find that these patterns
exhibit two universal properties. On one hand, they all admit equal but
opposite in-plane and out-of-plane Poisson’s ratios. On the other hand, their
bending energy detaches from their in-plane strain and depends instead on the
strain gradient. The results are illustrated over a case study of the self-
equilibrium geometry of origami pillars.

###### keywords:

Origami , Compliant shells , Poisson coefficient , Gradient elasticity ,
Geometric nonlinearity

##  1 Introduction

Origami has long surpassed its artistic vocation. Its early uses in
architectural geometry date back to the 1920s [1, 2]. By now, origami
techniques have been applied to the design of deployable structures in
astronautics, robotics and biomedical engineering [3, 4, 5, 6]. Origami
tessellations in particular make up versatile morphing 2D metamaterials that
can access various low-energy high-curvature configurations [7, 8]. Although
inextensible in principle, the fine crease patterns of origami metamaterials
enable them to effectively deform non-isometrically. In this letter, we
determine the strains that are compatible with coarse-grained origami
kinematics. Strain compatibility permits to fully characterize the 3D-embedded
geometry when suitable boundary conditions are prescribed. In most cases
however, the geometry further solves an elastostatic equilibrium for which we
calculate the appropriate elasticity functional.

Morphing in origami metamaterials can be triggered in response to different
loading conditions such as bending, twisting and stretching all of which
couple in a manner unparalleled in other plates and shells; see Fig. 1. For
instance, standard pure bending kinematics of plates imply that the curvatures
in the bending plane and in the plane orthogonal to it, $\kappa_{1}$ and
$\kappa_{2}$, are in a proportion equal and opposite to the in-plane Poisson’s
ratio $\nu$, i.e., $\nu=-\kappa_{2}/\kappa_{1}$. Thus auxetic plates tend to
bend synclastically while anauxetic plates tend to bend anticlastically [9].
Origami metamaterials do not [10]. In fact, in what follows, we demonstrate
that they typically exhibit the exact opposite behavior, i.e.,
$\nu=+\kappa_{2}/\kappa_{1}$. Other loads that induce morphing, such as
“pinching”, are unique to origami [11]. Indeed, in plates and membranes, a
pinch usually produces highly localized and singular deformations. By
contrast, pinching an origami metamaterial brings on global and nearly-uniform
deformations [12, 13]. The consequence, as we shall prove, is that membrane
energy penalizes, not strains, but their gradients, i.e., the Christoffel
symbols [14, 15, 16].

Figure 1: Two views of an origami-folded hyperboloid: A periodic “Mars”
pattern is laser-etched into a PET sheet then hand-folded, circularly bent and
fixated with adhesive tape. The pattern is auxetic but bends into a saddle.
The letter explains the observed self-equilibrium geometry. Figure 2: Zigzag
sums. (a) A zigzag sum in a periodic reference configuration: translating one
zigzag along another sweeps a polyhedral surface corresponding to a crease
pattern complete with a natural assignment of mountain and valley folds. (b) A
super-cell in an abstract reference state: indices $(ij)$ are discrete
coordinates for the vertices; the central node is at
$\bm{\xi}\equiv\bm{\xi}^{(00)}$. (c) A super-cell in a physical folded state:
vectors $(\mathbf{u},\mathbf{v},\mathbf{s},\mathbf{t})$ are
$\bm{\xi}$-dependent non-dimensional vectors aligned with the creases; they
fully characterize fold-only configurations as well as the tangent vectors
$\mathbf{x}_{1}$ and $\mathbf{x}_{2}$. (d) A super-cell following an
infinitesimal bending $\delta\mathbf{x}$ of the panels: note how the
parallelograms are slightly distorted now. (e) A global view of a finitely
deformed zigzag sum $(\Sigma)$: each super cell is designated by its central
node’s coordinates $\bm{\xi}=(\xi_{1},\xi_{2})$ (red dot) and is obtained by
the process illustrated in panels (b-d). (f) An orthogonal zigzag sum:
initially orthogonal zigzag directions remain orthogonal during folding
($g_{12}\equiv 0$). (g) An oblique zigzag sum: initially non-orthogonal zigzag
directions are sheared during folding; even then, $g_{12}$ remains constant.

Previous contributions have succeeded in describing several aspects of origami
mechanics, including calculations of Poisson’s ratios and bending moduli [17,
18, 19]. Hereafter, we derive an effective medium theory complete with its
field equations of compatibility and equilibrium. The theory naturally
accounts for the observations described above and shows how the different
deformation modes couple kinematically and energetically. The subject of the
theory is a class of origami patterns that we call “zigzag sums”. These are
generated by translating one zigzag line along another so as to sweep a
periodic polyhedral surface $\Sigma$ whose edges and panels we interpret as an
origami crease pattern; see Fig. 2a. Equivalently, $\Sigma$ is a periodic
crease pattern with a unit cell composed of four parallelograms. Four
quadrilaterals per unit cell is as simple as origami tessellations can be
without being trivial; moreover, here, the quads are parallelograms.
Nonetheless, zigzag sums span various well-known patterns (e.g., the Miura-
ori, the eggbox, “Sakoda’s staircase” and “Barreto’s Mars” patterns [20]) and
encompass their oblique, non-developable and non-flat-foldable variants by
maintaining two basic requirements: periodicity and parallelism. Meanwhile,
the zigzag generators provide a handful of geometric seeds that can readily
tune their deformation paths as well as the underlying elastic behavior.

Figure 3: Convergence analysis. (a) Convergence speed: the maximum error in
equation (7) decreases like $r$ or, equivalently, like the reciprocal of the
number of unit cells per row. The plot also shows that the maximum angular
deflection $r\delta^{(ij)}$ decreases in the same fashion confirming the fold-
to-bend orders of magnitude adopted in (1) and that, in the limit $r\to 0$,
all panels are flat. The inset shows the unit cell adopted for this example.
(b,c) Two iterates: snapshots of the way in which the zigzag sum converges to
a smooth surface taken for $64$ and $128$ unit cells per row. Note that the
zigzag sum is finitely and non-uniformly stretched and bent. Illustrated
surfaces are constructed by “propagating” inextensibility constraints away
from a given boundary; see [21] for details.

##  2 Geometric mechanics of zigzag sums

The deformations of a zigzag sum $\Sigma$ are the outcome of two competing
mechanisms [22]. The first is the folding and unfolding of $\Sigma$ along its
edges. The second is the isometric bending of its panels. Signature origami
kinematics emerges when the bending stiffness of the panels is significantly
larger than the folding stiffness of the edges. Accordingly, we can assume
that individual panels bend infinitesimally; $\Sigma$, on the other hand, can
still embrace finitely curved surfaces by accumulating small deflections
across many unit cells. Formally, we characterize the deformations of $\Sigma$
in terms of a continuous parametrization $\mathbf{x}=\mathbf{x}(\bm{\xi})$ of
its mid-plane, where $\mathbf{x}$ is the current position of vertex
$\bm{\xi}$, with $\bm{\xi}=(\xi_{1},\xi_{2})$ being the effective curvilinear
coordinate along the zigzags directions; see Fig. 2b-e. Consider then a non-
elementary unit cell, i.e., a supercell, centered about vertex $\bm{\xi}$ and
containing the neighboring vertices $\bm{\xi}^{(ij)}$ placed at
$\mathbf{x}^{(ij)}$. Then, the adopted origami kinematics imply that position

$\mathbf{x}^{(ij)}=\mathbf{X}^{(ij)}+r^{2}\delta\mathbf{x}^{(ij)}$ (1)

is attained by a fold-only motion $\bm{\xi}^{(ij)}\mapsto\mathbf{X}^{(ij)}$
(Fig. 2b-c) followed by a bending-induced perturbation
$r^{2}\delta\mathbf{x}^{(ij)}$ that is small compared to the characteristic
size of the panels denoted $r$ (Fig. 2c-d). In the limit of a tessellation
where $\Sigma$ contains a large number of unit cells, i.e., for $r\to 0$, the
plane locally tangent to $\Sigma$ is spanned by two vectors

$\begin{split}\mathbf{x}_{1}\equiv\frac{\partial\mathbf{x}}{\partial\xi_{1}}&=\lim_{r\to
0}\frac{\mathbf{x}^{(10)}-\mathbf{x}^{(\bar{1}0)}}{r}=\mathbf{u}-\mathbf{v},\\\
\mathbf{x}_{2}\equiv\frac{\partial\mathbf{x}}{\partial\xi_{2}}&=\lim_{r\to
0}\frac{\mathbf{x}^{(01)}-\mathbf{x}^{(0\bar{1})}}{r}=\mathbf{s}-\mathbf{t},\end{split}$
(2)

where $\mathbf{u}$, $\mathbf{v}$, $\mathbf{s}$ and $\mathbf{t}$ are the crease
vectors meeting at vertex $\bm{\xi}$ and normalized with respect to $r$. The
apparent in-plane deformations of $\Sigma$ can therefore be quantified in
terms of an effective metric tensor $\mathbf{g}$ of components

$g_{\mu\nu}=\left\langle{\mathbf{x}_{\mu},\mathbf{x}_{\nu}}\right\rangle=\begin{bmatrix}\left\lVert\mathbf{u}-\mathbf{v}\right\rVert^{2}&\left\langle{\mathbf{u}-\mathbf{v},\mathbf{s}-\mathbf{t}}\right\rangle\\\
\left\langle{\mathbf{u}-\mathbf{v},\mathbf{s}-\mathbf{t}}\right\rangle&\left\lVert\mathbf{s}-\mathbf{t}\right\rVert^{2}\end{bmatrix}.$
(3)

Note that bending-related contributions disappear in the limit $r\to 0$ and
can be disregarded for now.

It is noteworthy that
$g_{12}=\left\langle{\mathbf{u},\mathbf{s}}\right\rangle+\left\langle{\mathbf{v},\mathbf{t}}\right\rangle-\left\langle{\mathbf{u},\mathbf{t}}\right\rangle-\left\langle{\mathbf{v},\mathbf{s}}\right\rangle$
is a motion constant (i.e., $\mathrm{d}g_{12}\equiv 0$) as it is equal to a
combination of constant lengths and angles. Thus, if $g_{12}=0$ so that the
zigzags $\xi_{1}$\- and $\xi_{2}$-contours are initially orthogonal, then
orthogonality is maintained in any subsequent motion (Fig. 2f). However, in
oblique tessellations, with
$g_{12}=\left\lVert\mathbf{x}_{1}\right\rVert\left\lVert\mathbf{x}_{2}\right\rVert\cos\theta\neq
0$, stretching couples to shearing according to

$\tan\theta\,\mathrm{d}\theta=\frac{\mathrm{d}\\!\left\lVert\mathbf{x}_{1}\right\rVert}{\left\lVert\mathbf{x}_{1}\right\rVert}+\frac{\mathrm{d}\\!\left\lVert\mathbf{x}_{2}\right\rVert}{\left\lVert\mathbf{x}_{2}\right\rVert}=\frac{\mathrm{d}g}{2g_{11}g_{22}}$
(4)

with $g=\det\mathbf{g}$. Rather intuitively then, shearing, understood as
aligning the zigzags ($\theta\to 0$ or $\pi$), always goes to reduce the
pattern’s effective area (Fig 2g). As for the stretch ratios in directions $1$
and $2$, they are related to one another through a geometric Poisson’s ratio
$\nu\equiv-g_{11}/g_{22}\times\mathrm{d}g_{22}/\mathrm{d}g_{11}$. A general
expression for $\nu$ is pursued in the Supplemental Material [23, App. A].

Most important is the observation that the fold-only motion
$\bm{\xi}^{(ij)}\mapsto\mathbf{X}^{(ij)}$ maintains crease parallelism and
periodicity: this is truly where the fact that $\Sigma$ is made out of
parallelograms is crucial. Hence, folding alone produces no curvature and, in
that regard, it is bending that takes over now. This is most apparent in the
expressions of the second-order derivatives which, in the limit $r\to 0$, read

$\begin{split}\mathbf{x}_{11}&=\delta\mathbf{x}^{(20)}-2\delta\mathbf{x}+\delta\mathbf{x}^{(\bar{2}0)},\\\
\mathbf{x}_{22}&=\delta\mathbf{x}^{(02)}-2\delta\mathbf{x}+\delta\mathbf{x}^{(0\bar{2})},\\\
\mathbf{x}_{12}&=\delta\mathbf{x}^{(11)}-\delta\mathbf{x}^{(\bar{1}1)}-\delta\mathbf{x}^{(1\bar{1})}+\delta\mathbf{x}^{(\bar{1}\bar{1})},\end{split}$
(5)

where
$\mathbf{x}_{\mu\nu}\equiv\partial^{2}\mathbf{x}/\partial\xi_{\mu}\partial\xi_{\nu}$.
Therefore, the $\mathbf{x}_{\mu\nu}$ at $\bm{\xi}$ are linear functions of the
bending DOFs of the super cell centered at $\bm{\xi}$. That being said, not
all $\delta\mathbf{x}^{(ij)}$ are compatible with the inextensibility
constraints. Ultimately, it is possible to show that
$(\mathbf{x}_{11},\mathbf{x}_{22},\mathbf{x}_{12})$ belong to a linear
subspace of $\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}$ spanned
by 4 DOFs, each attributed to a planarity defect $(\delta^{(ij)})$ of one of
the four panels of the central unit cell (Fig 3a) [23, App. B]. By the rank-
nullity theorem, the $\mathbf{x}_{\mu\nu}$ satisfy $9-4=5$ linear
compatibility equations. Straightforward calculations show that the first two
of these equations are $g_{12,1}=g_{12,2}=0$, i.e., they re-produce
$\mathrm{d}g_{12}=0$. The other three relations are far more interesting; they
combine into the vector identity [23, App. B]

$\mathbf{x}_{22}=\frac{\left\langle{\mathbf{u},\mathbf{t}\wedge\mathbf{s}}\right\rangle\left\langle{\mathbf{v},\mathbf{t}\wedge\mathbf{s}}\right\rangle}{\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{v}}\right\rangle\left\langle{\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle}\mathbf{x}_{11}.$
(6)

The tangent components of $\mathbf{x}_{22}$ and $\mathbf{x}_{11}$ describe the
strain gradients in the mid-plane of $\Sigma$ in direction $\mathbf{x}_{1}$
and $\mathbf{x}_{2}$ whereas the normal components describe the normal
curvatures $\kappa_{1}$ and $\kappa_{2}$ of the mid-plane in the same
directions. The above identity states that these respective quantities
necessarily occur in equal proportions. Specifically, the in-plane and out-of-
plane Poisson’s ratios, $\nu$ and $-\kappa_{2}/\kappa_{1}$, of a zigzag sum
$\Sigma$ are equal and opposite and are given by the current geometry of the
folds as per

$\nu=\frac{\kappa_{2}}{\kappa_{1}}=\frac{g_{11}}{g_{22}}\frac{\left\langle{\mathbf{u},\mathbf{t}\wedge\mathbf{s}}\right\rangle\left\langle{\mathbf{v},\mathbf{t}\wedge\mathbf{s}}\right\rangle}{\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{v}}\right\rangle\left\langle{\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle},$
(7)

with
$\kappa_{\mu}=\left\langle{\mathbf{x}_{\mu\mu},\mathbf{n}}\right\rangle/g_{\mu\mu}$
(no sum),
$\mathbf{n}=\mathbf{x}_{1}\wedge\mathbf{x}_{2}/\left\lVert\mathbf{x}_{1}\wedge\mathbf{x}_{2}\right\rVert$
being the unit normal [23, App. C].

Identity (7) is of an asymptotic nature: in the limit $r\to 0$ of
infinitesimal unit cells, the Poisson’s ratios of a zigzag sum $\Sigma$
converge to one another. In other, more practical, terms, the equality is
accurate within an error proportional to $r$. See Fig 3 for a numerical
illustration of the convergence process and the relevant error bounds.
Identity (7) is also universal: it holds whether $\Sigma$ is developable,
flat-foldable, symmetric, rectangular, or not. Therein, the mixed products
(e.g., $\left\langle{\mathbf{u},\mathbf{t}\wedge\mathbf{s}}\right\rangle$)
have their signs determined by the mountain-valley assignment of the concerned
folds. This allows us to deduce the rather nice corollary: if a zigzag sum
$\Sigma$ has an odd number of mountain folds meeting at a vertex then: $(i)$
it is necessarily auxetic; and $(ii)$ it necessarily bends into a saddle. In
particular, all developable zigzag sums are auxetic and bend into saddles.
Note that these results are only valid for nondegenerate configurations, i.e.,
where all folds are partially folded. Indeed, when a fold is flat, be it open
or closed, mountain and valley assignments coalesce and the Poisson’s ratio
could become null, infinite or multivalued [19]. Note also that the proven
identity features the elongations and curvatures in two specific material
directions $\xi_{1}$\- and $\xi_{2}$-contours, be them orthogonal or not; in
particular, the identity bears no immediate consequences on the Poisson’s
coefficient defined for two general orthogonal directions.

Figure 4: “Mars”-folded origami pillars are one-sheeted hyperboloids. (a) A
unit cell of the “Mars” pattern: the supplementary opposite angles ensure
developability and flat-foldability; an equilateral pattern is further
rectangular. (b) Action of the first DOF on the shape of the pillar: given a
small enough aspect ratio $L_{2}/L_{1}$, the pillar can be inverted inside-
out; here, the pattern has $N=100$ unit cells per meridian, and $8$ cells per
parallel, with a flat-unfolded aspect ratio $L_{2}/L_{1}=8\%$. (c) Action of
the second DOF: the pattern flattens as the opening angle
$\angle(\mathbf{u},\mathbf{v})$ at a given parallel is increased; here, the
pattern has $N=100$ unit cells per meridian and $40$ unit cells per parallel
with $L_{2}/L_{1}=40\%$. (d) Bending energy: plot of normalized total bending
energy as a function of the opening angle $\angle(\mathbf{u},\mathbf{v})$ at
the equator for origami pillars centered about the equator with
$L_{2}/L_{1}=40\%$; the discretely summed energy for $N=100$ is plotted
against the continuum limit $N\to\infty$; data points $(i-iv)$ are the ones
illustrated on panel (c); greyed out domain cannot be accessed with the
current aspect ratio; the continuum theory satisfactorily predicts both energy
and kinematical admissibility. Adopted values: $\alpha=\pi/3$, $\beta=\pi/2$.

##  3 Self-equilibrium of origami pillars

The equality of the Poisson’s ratios greatly constrains the 3D geometries
accessible to a zigzag sum $\Sigma$ [24, 25, 21, 26]. Consider, for
tractability, the “Mars” tessellation of Fig. 4a. The tessellation lacks
mirror symmetry but is otherwise developable, flat-foldable and equilateral.
We find that compatible metrics and curvatures satisfy [23, App. D]

$\begin{gathered}g_{12}=0,\quad
g_{11}(4-g_{22})=4(\cos\alpha+\cos\beta)^{2},\\\
\frac{\kappa_{2}}{\kappa_{1}}=\nu=-4\frac{(\cos\alpha+\cos\beta)^{2}}{g_{11}g_{22}}.\end{gathered}$
(8)

This is in fact a system of non-linear PDEs weighing on the admissible
configurations $\mathbf{x}$. For instance, we explore the configurations of a
rectangular domain of flat dimensions $L_{1}\times L_{2}$ that are folded and
wrapped around an axis of symmetry into an origami pillar [20]. These pillars
are surfaces of revolution with a Cartesian parametrization of the form

$\mathbf{x}(\bm{\xi})=(\rho(\xi_{2})\cos(\omega\xi_{1}),\rho(\xi_{2})\sin(\omega\xi_{1}),z(\xi_{2})).$
(9)

Here, the axis of symmetry is the $z$-axis, $\xi_{1}$-contours are meridian
lines and $\xi_{2}$-contours are parallel lines. Equations (8) then simplify
into a system of ODEs governing $z$ and $\rho$. Its solutions constitute a
2-DOF family of one-sheeted hyperboloids (Fig. 4b,c) [23, App. E]. The range
of motion of each pillar is bounded by the maximally folded and unfolded
states of the pattern. Thus, for increasing aspect ratio $L_{2}/L_{1}$, the
pillar will have a smaller range of motion, until it jams, or even becomes
impossible to form without tearing.

Among all accessible pillars, the ones that are in self-equilibrium exhibit
minimum levels of strain energy $\psi$. Strain energy is composed of two
contributions: crease folding and panel bending. Crease folding energy takes
the form of a classical, however nonlinear, membrane energy. In what follows,
we neglect this contribution and focus on the less-explored influence of panel
bending [27]. For “Mars”-folded origami pillars, centered about the equator,
strain energy is exemplified on Fig. 4d. It appears that unfolding the
pattern, by increasing the angle $\angle(\mathbf{u},\mathbf{v})$, reduces
bending energy. The unfolding flattens the pillar and reduces both curvatures
$\kappa_{1}$ and $\kappa_{2}$. That trend continues up to a point where energy
is minimum (near state $(iv)$) and beyond which any further unfolding, or
flattening, of the pattern actually causes a dramatic increase in bending
energy. This suggests that bending energy $\psi$ depends on some other
deformation measures besides curvatures. As a matter of fact, the bending
energy density $b$, where $\psi=\int_{\Omega}b\mathrm{d}\bm{\xi}$, $\Omega$
being the reference domain of the pattern, is a quadratic form of the
planarity defects $\delta^{(ij)}$ of the panels. For their part, the defects
are linear forms of the full, both out-of-plane and in-plane, components of
the parametrization’s second derivatives $\mathbf{x}_{\mu\nu}$. The out-of-
plane components
$\Gamma_{3\mu\nu}\equiv\left\langle{\mathbf{n},\mathbf{x}_{\mu\nu}}\right\rangle$
are the coefficients of the second fundamental form; they quantify the
curvatures and torsion of the embraced surface. The in-plane components
$\Gamma_{\sigma\mu\nu}\equiv\left\langle{\mathbf{x}_{\sigma},\mathbf{x}_{\mu\nu}}\right\rangle$
are the Christoffel symbols; they quantify, not the strains, but their
gradient, i.e., terms of the form $g_{\mu\nu,\sigma}$. Accordingly,
$b=b(\bm{\Gamma};\mathbf{g})$ is a metric-dependent quadratic form of the
curvatures, torsion, and strain gradient. Detailed derivations of the
expression of $b$ leading to Fig. 4d can be found in the Supplemental Material
[23, App. F, G].

Back to the pillars, unfolding the pattern reduces both curvatures and strain-
gradients. However, as the geometric rigidities associated with the curvatures
(i.e., $\partial^{2}b/\partial\Gamma_{3\mu\nu}$) remain bounded, the ones
associated with the strain gradient (i.e.,
$\partial^{2}b/\partial\Gamma_{\sigma\mu\nu}$) diverge for states that are
close to being flat. Such states occur near the outer rims of sufficiently
unfolded pillars; their presence further signals that the pillar has reached
the boundary of the kinematically admissible domain, hence the energy blow-up
observed on Fig. 4d. Strain-gradient energy further dominates the response of
any plane, non-uniform, state such as the “ring” observed midway through Fig.
4b. Indeed, uniformly folded plane states do not engage panel bending and
therefore have zero energy. By contrast, non-uniformly folded states, with
gradients of folding angles, cannot be achieved without panel bending. When
the state is plane, it has zero curvatures and torsion, and energy becomes
function of the strain gradient alone [12]. More generally, it is noteworthy
that $\psi$ does not penalize strains or fold angles, however large, so long
as they are uniform. In fact, $\psi$ does not even refer to a specific natural
state or any specific natural fold angles in reference to which strains should
be measured [16]. Instead, it refers to a higher-order strain measure, namely
$\bm{\Gamma}$, which quantifies not how much the pattern is folded, but rather
how far it is from being uniformly folded.

##  4 Conclusion

In conclusion, we proposed a continuum theory of origami-folded shells where
the crease pattern is a zigzag sum. The theory accounts for geometric
nonlinearities and accurately predicts the states that are accessible under
isometric folding thanks to a universal identity relating in-plane
deformations and out-of-plane curvatures. The study further demonstrates that
the energy density of origami-folded shells can heavily depend on the in-plane
strain-gradient in a way that is unparalleled in classical shell theories. It
should be possible to extend the proposed theory, in one form or another, to
more general tessellations that possess periodic folding motions. By contrast,
alternative approaches are likely to be needed to tackle tessellations that do
not fold periodically (e.g., Huffman grids and Yoshimura-like patterns [28]).
Finally, it would be of interest to investigate how the proposed theory can
inform shape programming and morphing planning of origami and kirigami
structures [29, 30, 31, 32, 33, 34, 35, 36, 37] for robotics and 4D printing
applications.

## Acknowledgements

HN acknowledges support by the NSF under CAREER award No. CMMI-2045881. AL
acknowledges the support of the French Agence Nationale de la Recherche (ANR),
under grant No. ANR-17-CE08-0039 (project ArchiMatHOS). EW acknowledges
support by the NASA Missouri Space Grant Consortium.

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##  Appendix A The in-plane Poisson’s ratio: General case

The metric $\mathbf{g}$ of a zigzag sum has the components
$g_{11}=\left\lVert\mathbf{u}-\mathbf{v}\right\rVert^{2}$,
$g_{22}=\left\lVert\mathbf{s}-\mathbf{t}\right\rVert^{2}$ and
$g_{12}=\left\langle{\mathbf{u}-\mathbf{v},\mathbf{s}-\mathbf{t}}\right\rangle$.
Note that the scalar products
$\left\langle{\mathbf{u},\mathbf{s}}\right\rangle$,
$\left\langle{\mathbf{u},\mathbf{t}}\right\rangle$,
$\left\langle{\mathbf{v},\mathbf{s}}\right\rangle$ and
$\left\langle{\mathbf{v},\mathbf{t}}\right\rangle$ are combinations of lengths
and angles that are isometrically preserved; they are motion constants and so
is $g_{12}$. It is somewhat more challenging to derive a relationship between
$g_{11}$ and $g_{22}$; this is done next. First, note that $\mathbf{s}$ and
$\mathbf{t}$ can be decomposed into

$\begin{split}\mathbf{s}&=\left\langle{\mathbf{s},\mathbf{u}}\right\rangle\mathbf{u}^{*}+\left\langle{\mathbf{s},\mathbf{v}}\right\rangle\mathbf{v}^{*}+s_{w}\mathbf{w},\\\
\mathbf{t}&=\left\langle{\mathbf{t},\mathbf{u}}\right\rangle\mathbf{u}^{*}+\left\langle{\mathbf{t},\mathbf{v}}\right\rangle\mathbf{v}^{*}+t_{w}\mathbf{w},\end{split}$
(10)

with

$\mathbf{u}^{*}=\frac{\mathbf{v}\wedge\mathbf{w}}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert},\quad\mathbf{v}^{*}=\frac{\mathbf{w}\wedge\mathbf{u}}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert},\quad\mathbf{w}=\frac{\mathbf{u}\wedge\mathbf{v}}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert}.$
(11)

Components $s_{w}$ and $t_{w}$ are easily determined by considering the
magnitudes of $\mathbf{s}$ and $\mathbf{t}$. Indeed, we have

$\begin{split}s_{w}^{2}=s^{2}-\frac{\left\langle{\mathbf{s},\mathbf{u}}\right\rangle^{2}v^{2}+\left\langle{\mathbf{s},\mathbf{v}}\right\rangle^{2}u^{2}-2\left\langle{\mathbf{s},\mathbf{u}}\right\rangle\left\langle{\mathbf{s},\mathbf{v}}\right\rangle\left\langle{\mathbf{u},\mathbf{v}}\right\rangle}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert^{2}},\\\
t_{w}^{2}=t^{2}-\frac{\left\langle{\mathbf{t},\mathbf{u}}\right\rangle^{2}v^{2}+\left\langle{\mathbf{t},\mathbf{v}}\right\rangle^{2}u^{2}-2\left\langle{\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{t},\mathbf{v}}\right\rangle\left\langle{\mathbf{u},\mathbf{v}}\right\rangle}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert^{2}},\end{split}$
(12)

with $u=\left\lVert\mathbf{u}\right\rVert$, and so on. Therein, note that the
only variables are

$\left\langle{\mathbf{u},\mathbf{v}}\right\rangle=\frac{u^{2}+v^{2}-g_{11}}{2},\quad\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert^{2}=u^{2}v^{2}-\left\langle{\mathbf{u},\mathbf{v}}\right\rangle^{2},$
(13)

and are both functions of $g_{11}$. Last, we write

$g_{22}=\frac{\left\langle{\mathbf{s}-\mathbf{t},\mathbf{u}}\right\rangle^{2}v^{2}+\left\langle{\mathbf{s}-\mathbf{t},\mathbf{v}}\right\rangle^{2}u^{2}}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert^{2}}\\\
-\frac{2\left\langle{\mathbf{s}-\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{s}-\mathbf{t},\mathbf{v}}\right\rangle\left\langle{\mathbf{u},\mathbf{v}}\right\rangle}{\left\lVert\mathbf{u}\wedge\mathbf{v}\right\rVert^{2}}\\\
+(s_{w}-t_{w})^{2},$ (14)

and accordingly deduce $g_{22}$ as a function of $g_{11}$, as well as $\nu$ as
a function of $g_{11}$. It is worth mentioning that $g_{11}$ and $g_{22}$ have
upper and lower bounds corresponding to some creases being maximally folded or
unfolded. Beyond these bounds the pattern would penetrate itself or break
apart. These situations can be avoided by enforcing $s_{w}^{2}>0$ and
$t_{w}^{2}>0$.

##  Appendix B The admissible second-order derivatives

Consider the vertices of a super cell placed at
$\mathbf{x}^{(ij)}=\mathbf{X}^{(ij)}+r^{2}\delta\mathbf{x}^{(ij)}$ where the
$\mathbf{X}^{(ij)}$ describe a pre-folded state and the
$r^{2}\delta\mathbf{x}^{(ij)}$ are bending-induced perturbations. Similarly,
the central vertex $\bm{\xi}$ is at $\mathbf{X}$ subsequent to the fold-only
motion and is perturbed by $r^{2}\delta\mathbf{x}$ subsequent to the
infinitesimal bending of the panels. Note that the second-order derivatives
$\mathbf{x}_{\mu\nu}$ are invariant by composition with linear motions
including rigid body motions and periodic stretching and contraction.
Accordingly, it is possible, without loss of generality, to default the
displacements of the central four creases to zero. In other words, we set

$\delta\mathbf{x}=\delta\mathbf{x}^{(10)}=\delta\mathbf{x}^{(01)}=\delta\mathbf{x}^{(\bar{1}0)}=\delta\mathbf{x}^{(0\bar{1})}=\mathbf{0}.$
(15)

Furthermore, by linearity, the contributions of the remaining displacements
can be investigated independently then superposed.

So let $\delta\mathbf{x}^{(\bar{1}1)}$, $\delta\mathbf{x}^{(\bar{1}\bar{1})}$,
$\delta\mathbf{x}^{(1\bar{1})}$, $\delta\mathbf{x}^{(\bar{2}0)}$ and
$\delta\mathbf{x}^{(0\bar{2})}$ all be null for now and consider a nonzero
displacement $\delta\mathbf{x}^{(11)}$: the latter must be orthogonal to
$\mathbf{u}$ so as to preserve the length of
$\mathbf{x}^{(11)}-\mathbf{x}^{(01)}$ as well as to $\mathbf{s}$ so as to
preserve the length of $\mathbf{x}^{(11)}-\mathbf{x}^{(10)}$, to leading
order. Hence,

$\delta\mathbf{x}^{(11)}=\delta^{(11)}\mathbf{u}\wedge\mathbf{s}$ (16)

for some planarity defect $\delta^{(11)}$. Similarly, we have

$\delta\mathbf{x}^{(02)}=\delta^{+}\mathbf{v}\wedge\mathbf{t},\quad\delta\mathbf{x}^{(20)}=\delta^{-}\mathbf{v}\wedge\mathbf{t}.$
(17)

Expressions for $\delta^{\pm}$ are easily obtained by observing that the
lengths of $\mathbf{x}^{(11)}-\mathbf{x}^{(02)}$ and
$\mathbf{x}^{(11)}-\mathbf{x}^{(20)}$ are preserved. Indeed, infinitesimal
inextensibility implies

$\begin{split}\left\langle{\mathbf{t}+\mathbf{u},\delta^{(11)}\mathbf{u}\wedge\mathbf{s}-\delta^{+}\mathbf{v}\wedge\mathbf{t}}\right\rangle&=0,\\\
\left\langle{\mathbf{v}+\mathbf{s},\delta^{(11)}\mathbf{u}\wedge\mathbf{s}-\delta^{-}\mathbf{v}\wedge\mathbf{t}}\right\rangle&=0,\end{split}$
(18)

namely,

$\frac{\delta^{+}}{\delta^{-}}=\frac{\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{t}}\right\rangle\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle}.$
(19)

As for the $\mathbf{x}_{\mu\nu}$, they are given by

$\begin{split}\mathbf{x}_{11}&={\delta\mathbf{x}^{(20)}}={\delta^{-}}\mathbf{v}\wedge\mathbf{t},\\\
\mathbf{x}_{22}&={\delta\mathbf{x}^{(02)}}={\delta^{+}}\mathbf{v}\wedge\mathbf{t},\\\
\mathbf{x}_{12}&={\delta\mathbf{x}^{(11)}}=\delta^{(11)}\mathbf{u}\wedge\mathbf{s}.\end{split}$
(20)

In particular,

$\mathbf{x}_{22}=\frac{\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{t}}\right\rangle\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle}\mathbf{x}_{11}.$
(21)

The contributions of $\delta\mathbf{x}^{(\bar{1}1)}$,
$\delta\mathbf{x}^{(\bar{1}\bar{1})}$ and $\delta\mathbf{x}^{(1\bar{1})}$
(i.e., $\delta^{(\bar{1}1)}$, $\delta^{(\bar{1}\bar{1})}$ and
$\delta^{(1\bar{1})}$, respectively) can be readily calculated in the same
fashion and shown to yield exactly the same constraint on $\mathbf{x}_{22}$
and $\mathbf{x}_{11}$. This can be predicted by appreciating the symmetric way
in which the proportionality coefficient depends upon the crease vectors. In
conclusion, the above constraint holds for any combination of admissible
infinitesimal displacements of the super cell.

Last, note that

$\begin{split}\left\langle{\mathbf{x}_{12},\mathbf{x}_{1}}\right\rangle&=\left\langle{\delta^{(11)}\mathbf{u}\wedge\mathbf{s},\mathbf{u}-\mathbf{v}}\right\rangle\\\
&=-\delta^{(11)}\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle\\\
&=-\delta^{-}\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle\\\
&=-\delta^{-}\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}-\mathbf{t}}\right\rangle\\\
&=-\left\langle{\mathbf{x}_{11},\mathbf{x}_{2}}\right\rangle.\end{split}$ (22)

That is:
$\partial_{1}\\!\left\langle{\mathbf{x}_{1},\mathbf{x}_{2}}\right\rangle=0$.
In the same manner, one shows that
$\partial_{2}\\!\left\langle{\mathbf{x}_{1},\mathbf{x}_{2}}\right\rangle=0$ in
order to recover $\mathrm{d}g_{12}=0$ as claimed in the main text.

##  Appendix C The in- and out-of-plane Poisson’s coefficients are equal and
opposite

Projecting the proportionality constraint of $\mathbf{x}_{11}$ and
$\mathbf{x}_{22}$ along the normal $\mathbf{n}$ yields

$\kappa_{2}g_{22}=\frac{\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{t}}\right\rangle\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle}\kappa_{1}g_{11}$
(23)

by the definition of the normal curvatures in directions 1 and 2. Projecting
along $\mathbf{x}_{1}$ yields

$\left\langle{\mathbf{x}_{22},\mathbf{x}_{1}}\right\rangle=\frac{\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{t}}\right\rangle\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle}\left\langle{\mathbf{x}_{11},\mathbf{x}_{1}}\right\rangle.$
(24)

But

$\begin{split}\left\langle{\mathbf{x}_{22},\mathbf{x}_{1}}\right\rangle&=\partial_{2}\\!\left\langle{\mathbf{x}_{2},\mathbf{x}_{1}}\right\rangle-\left\langle{\mathbf{x}_{2},\mathbf{x}_{12}}\right\rangle\\\
&=\partial_{2}\\!\left\langle{\mathbf{x}_{2},\mathbf{x}_{1}}\right\rangle-\frac{1}{2}\partial_{1}\\!\left\langle{\mathbf{x}_{2},\mathbf{x}_{2}}\right\rangle\\\
&=-\frac{1}{2}\partial_{1}g_{22},\end{split}$ (25)

since $\mathrm{d}g_{12}=0$. Also,
$\left\langle{\mathbf{x}_{11},\mathbf{x}_{1}}\right\rangle=\partial_{1}g_{11}/2$.
Projecting over $\mathbf{x}_{2}$ yields similar relations so that overall

$-\frac{\mathrm{d}g_{22}}{2}=\frac{\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{t}}\right\rangle\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle}\frac{\mathrm{d}g_{11}}{2}.$
(26)

Hence, the in-plane Poisson’s coefficient is

$\nu=-\frac{\mathrm{d}g_{22}/g_{22}}{\mathrm{d}g_{11}/g_{11}}=\frac{g_{11}}{g_{22}}\frac{\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{t}}\right\rangle\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v}\wedge\mathbf{t},\mathbf{u}}\right\rangle\left\langle{\mathbf{u}\wedge\mathbf{s},\mathbf{v}}\right\rangle}=\frac{\kappa_{2}}{\kappa_{1}}$
(27)

and matches, up to a sign, the out-of-plane Poisson’s coefficient
$-\kappa_{2}/\kappa_{1}$.

##  Appendix D The Poisson’s ratios of the “Mars” pattern

The “Mars” pattern is a developable flat-foldable zigzag sum. Let us suppose
that the pattern is equilateral so that $u=v=s=t$. Now we have

$\left\langle{\mathbf{u},\mathbf{s}}\right\rangle=-\left\langle{\mathbf{v},\mathbf{t}}\right\rangle=\cos\alpha,\quad\left\langle{\mathbf{u},\mathbf{t}}\right\rangle=-\left\langle{\mathbf{v},\mathbf{s}}\right\rangle=\cos\beta.$
(28)

We are at liberty to adopt a mountain-valley assignment convention: we let
creases $\mathbf{s}$, $\mathbf{t}$ and $\mathbf{v}$ be mountains and let
crease $\mathbf{u}$ be a valley. Then, inspection of (12) reveals that
$s_{w}=-t_{w}$. This allows to greatly simplify (14) ultimately into

$g_{22}=4-4\frac{(\cos\alpha+\cos\beta)^{2}}{g_{11}}.$ (29)

The Poisson’s coefficient is deduced by differentiation. It reads

$\nu=-4\frac{(\cos\alpha+\cos\beta)^{2}}{g_{11}g_{22}}$ (30)

and is automatically equal to $\kappa_{2}/\kappa_{1}$.

Recall that $g_{11}$ and $g_{22}$ are bounded. In the maximally unfolded
state, $\cos(\mathbf{u},\mathbf{v})=-\cos(\alpha-\beta)$ so that, in any case,
$g_{11}$ remains smaller than $2+2\cos(\alpha-\beta)$. The other bounds are
similarly deduced and, overall, we have

$\begin{split}4\cos^{2}\left(\frac{\alpha+\beta}{2}\right)&\leq g_{11}\leq
4\cos^{2}\left(\frac{\alpha-\beta}{2}\right),\\\
4\sin^{2}\left(\frac{\alpha-\beta}{2}\right)&\leq g_{22}\leq
4\sin^{2}\left(\frac{\alpha+\beta}{2}\right).\end{split}$ (31)

##  Appendix E “Mars”-folded hyperboloids

Consider an axisymmetric origami pillar folded out of an equilateral “Mars”
pattern with a parametrization $\mathbf{x}\in\mathbb{R}^{3}$ of the form

$\mathbf{x}=(\rho(\xi_{2})\cos(q\xi_{1}),\rho(\xi_{2})\sin(q\xi_{1}),z(\xi_{2})).$
(32)

We denote $L_{1}$ and $L_{2}$ the dimensions of the origami pattern in its
flat, maximally unfolded, reference state. We then let
$\xi_{1}\in[-S_{1}/2,S_{1}/2]$ and $\xi_{2}\in[-S_{2}/2,S_{2}/2]$ with

$S_{1}=\frac{L_{1}}{2\cos((\alpha-\beta)/2)},\quad
S_{2}=\frac{L_{2}}{2\sin((\alpha+\beta)/2)}.$ (33)

The pillar is obtained by folding the pattern then glueing together the ends
$\xi_{1}=0$ and $\xi_{1}=S_{1}$. In other words, we let $q=2\pi/S_{1}$.

With these notations, it straightforward to see that

$\begin{split}\mathbf{x}_{1}&=(-q\rho(\xi_{2})\sin(q\xi_{1}),q\rho(\xi_{2})\cos(q\xi_{1}),0),\\\
\mathbf{x}_{2}&=(\rho^{\prime}(\xi_{2})\cos(q\xi_{1}),\rho^{\prime}(\xi_{2})\sin(q\xi_{1}),z^{\prime}(\xi_{2})),\\\
\mathbf{x}_{11}&=(-q^{2}\rho(\xi_{2})\cos(q\xi_{1}),-q^{2}\rho(\xi_{2})\sin(q\xi_{1}),0),\\\
\mathbf{x}_{22}&=(\rho^{\prime\prime}(\xi_{2})\cos(q\xi_{1}),\rho^{\prime\prime}(\xi_{2})\sin(q\xi_{1}),z^{\prime\prime}(\xi_{2})),\end{split}$
(34)

with $\rho^{\prime}\equiv\mathrm{d}\rho/\mathrm{d}\xi_{2}$,
$\rho^{\prime\prime}\equiv\mathrm{d}\rho^{\prime}/\mathrm{d}\xi_{2}$ and so
on.

Then,

$g_{11}=q^{2}\rho^{2}\quad\text{and}\quad g_{22}=\rho^{\prime 2}+z^{\prime 2}$
(35)

depend on one another through the in-plane kinematical constraint

$\rho^{\prime 2}+z^{\prime
2}=4-4\frac{(\cos\alpha+\cos\beta)^{2}}{q^{2}\rho^{2}}.$ (36)

The out-of-plane constraint, i.e., $\mathbf{x}_{11}\parallel\mathbf{x}_{22}$,
immediately yields

$z^{\prime\prime}=0.$ (37)

Together, these two ODEs can be integrated into

$z^{\prime}=\text{cst},\quad\rho=\sqrt{4c^{2}(\xi_{2}-\xi_{o})^{2}+\frac{(\cos\alpha+\cos\beta)^{2}}{q^{2}c^{2}}},$
(38)

with $c^{2}=1-z^{\prime 2}/4$. Hence, the origami pillar embraces a
hyperboloid. The pattern does not cover the whole hyperboloid however and is
limited to the band spanned by $\xi_{2}\in[-S_{2}/2,S_{2}/2]$. Overall then,
and up to rigid body motions, the folded pillar has two degrees of freedom.
The first is $c$: it describes the shape of the hyperboloid; as it goes from 1
to 0, the hyperboloid changes from a “yoyo” into a cylinder. The second is
$\xi_{o}$: it specifies how the band of length $L_{2}$ that the pattern covers
is centered on or offset away from the equator.

Not all values $(c,\xi_{o})$ are admissible however since candidate solutions
must further satisfy the upper and lower bounds weighing on the metric, namely
(31). In the present case, these read

$4\cos^{2}((\alpha+\beta)/2)\leq q^{2}\rho^{2}\leq
4\cos^{2}((\alpha-\beta)/2).$ (39)

For a centered band ($\xi_{o}=0$) in particular, these bounds reduce to

$\begin{split}4\cos^{2}\left(\frac{\alpha+\beta}{2}\right)&\leq\frac{(\cos\alpha+\cos\beta)^{2}}{c^{2}},\\\
c^{2}q^{2}S_{2}^{2}+\frac{(\cos\alpha+\cos\beta)^{2}}{c^{2}}&\leq
4\cos^{2}\left(\frac{\alpha-\beta}{2}\right),\end{split}$ (40)

and place a maximum bound on the aspect ratio $S_{2}/S_{1}=qS_{2}/(2\pi)$ of
origami pillars that can be formed. In the main text, the angle
$\angle(\mathbf{u},\mathbf{v})$ at the equator is used rather than $c$. These
two are in a one-to-one correspondence

$2-2\cos(\mathbf{u},\mathbf{v})=g_{11}=\frac{(\cos\alpha+\cos\beta)^{2}}{c^{2}}.$
(41)

##  Appendix F Bending energy – General case

We have found that the admissible $\mathbf{x}_{\mu\nu}$ satisfy five linear
constraints: three describe the proportionality of $\mathbf{x}_{11}$ and
$\mathbf{x}_{22}$, and two reduce to $\mathrm{d}g_{12}=0$. Thus, the
admissible $\mathbf{x}_{\mu\nu}$ belong to a four-dimensional linear subspace
of $(\mathbb{R}^{3})^{3}$ space. Each dimension is spanned by a DOF that we
attribute to one planarity defect $\delta^{(ij)}$,
$(i,j)\in\\{1,\bar{1}\\}^{2}$ such that

$\begin{split}\delta\mathbf{x}^{(11)}&=\delta^{(11)}\mathbf{u}\wedge\mathbf{s},\\\
\delta\mathbf{x}^{(\bar{1}1)}&=\delta^{(\bar{1}1)}\mathbf{s}\wedge\mathbf{v},\\\
\delta\mathbf{x}^{(\bar{1}\bar{1})}&=\delta^{(\bar{1}\bar{1})}\mathbf{v}\wedge\mathbf{t},\\\
\delta\mathbf{x}^{(1\bar{1})}&=\delta^{(1\bar{1})}\mathbf{t}\wedge\mathbf{u}.\end{split}$
(42)

Here too, we began by defaulting the same displacements as in (15) to zero.
Next, we derive expressions for the $\delta^{(ij)}$ in function of the
$\mathbf{x}_{\mu\nu}$.

Generalizing equation (20), it is possible to derive full expressions for the
$\mathbf{x}_{\mu\nu}$ in terms of the $\delta^{(ij)}$. We find,

$\begin{split}\mathbf{x}_{11}=&\frac{\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{v}}\right\rangle}{\left\langle{\mathbf{t},\mathbf{s}\wedge\mathbf{v}}\right\rangle}\delta^{(11)}\mathbf{v}\wedge\mathbf{t}+\frac{\left\langle{\mathbf{v},\mathbf{u}\wedge\mathbf{s}}\right\rangle}{\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{s}}\right\rangle}\delta^{(\bar{1}1)}\mathbf{t}\wedge\mathbf{u}\\\
+&\frac{\left\langle{\mathbf{v},\mathbf{t}\wedge\mathbf{u}}\right\rangle}{\left\langle{\mathbf{s},\mathbf{t}\wedge\mathbf{u}}\right\rangle}\delta^{(\bar{1}\bar{1})}\mathbf{u}\wedge\mathbf{s}+\frac{\left\langle{\mathbf{u},\mathbf{v}\wedge\mathbf{t}}\right\rangle}{\left\langle{\mathbf{s},\mathbf{v}\wedge\mathbf{t}}\right\rangle}\delta^{(1\bar{1})}\mathbf{s}\wedge\mathbf{v}.\end{split}$
(43)

As for $\mathbf{x}_{12}$, it is readily available and reads

$\mathbf{x}_{12}=\delta^{(11)}\mathbf{u}\wedge\mathbf{s}-\delta^{(\bar{1}1)}\mathbf{s}\wedge\mathbf{v}+\delta^{(\bar{1}\bar{1})}\mathbf{v}\wedge\mathbf{t}-\delta^{(1\bar{1})}\mathbf{t}\wedge\mathbf{u}.$
(44)

These two vector equations provide a system (with two redundant equations),
which can be solved for the $\delta^{(ij)}$. For instance, projecting
$\mathbf{x}_{11}$ over $\mathbf{u}$ and $\mathbf{x}_{12}$ over $\mathbf{v}$
leads to a $2\times 2$ system

$\begin{split}\left\langle{\mathbf{x}_{11},\mathbf{u}}\right\rangle&=\frac{\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{v}}\right\rangle\left\langle{\mathbf{u},\mathbf{v}\wedge\mathbf{t}}\right\rangle}{\left\langle{\mathbf{t},\mathbf{s}\wedge\mathbf{v}}\right\rangle}(\delta^{(11)}+\delta^{(1\bar{1})}),\\\
\left\langle{\mathbf{x}_{12},\mathbf{v}}\right\rangle&=\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{v}}\right\rangle\delta^{(11)}-\left\langle{\mathbf{u},\mathbf{v}\wedge\mathbf{t}}\right\rangle\delta^{(1\bar{1})},\end{split}$
(45)

which can be solved for $\delta^{(11)}$ and $\delta^{(1\bar{1})}$ to give

$\begin{split}\delta^{(11)}&=\frac{\left\langle{\mathbf{x}_{11},\mathbf{u}}\right\rangle\left\langle{\mathbf{t},\mathbf{s}\wedge\mathbf{v}}\right\rangle+\left\langle{\mathbf{x}_{12},\mathbf{v}}\right\rangle\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{v}}\right\rangle}{\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{v}}\right\rangle\left\langle{\mathbf{t}-\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle},\\\
\delta^{(1\bar{1})}&=\frac{\left\langle{\mathbf{x}_{11},\mathbf{u}}\right\rangle\left\langle{\mathbf{t},\mathbf{s}\wedge\mathbf{v}}\right\rangle-\left\langle{\mathbf{x}_{12},\mathbf{v}}\right\rangle\left\langle{\mathbf{u},\mathbf{v}\wedge\mathbf{t}}\right\rangle}{\left\langle{\mathbf{u},\mathbf{v}\wedge\mathbf{t}}\right\rangle\left\langle{\mathbf{t}-\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle}.\end{split}$
(46)

Similar considerations lead to

$\begin{split}\left\langle{\mathbf{x}_{11},\mathbf{v}}\right\rangle&=\frac{\left\langle{\mathbf{v},\mathbf{u}\wedge\mathbf{s}}\right\rangle\left\langle{\mathbf{v},\mathbf{t}\wedge\mathbf{u}}\right\rangle}{\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{s}}\right\rangle}(\delta^{(\bar{1}1)}+\delta^{(\bar{1}\bar{1})}),\\\
\left\langle{\mathbf{x}_{12},\mathbf{u}}\right\rangle&=-\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{v}}\right\rangle\delta^{(\bar{1}1)}+\left\langle{\mathbf{u},\mathbf{v}\wedge\mathbf{t}}\right\rangle\delta^{(\bar{1}\bar{1})},\end{split}$
(47)

and to

$\begin{split}\delta^{(\bar{1}1)}&=\frac{\left\langle{\mathbf{x}_{11},\mathbf{v}}\right\rangle\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{s}}\right\rangle-\left\langle{\mathbf{x}_{12},\mathbf{u}}\right\rangle\left\langle{\mathbf{v},\mathbf{u}\wedge\mathbf{s}}\right\rangle}{\left\langle{\mathbf{v},\mathbf{u}\wedge\mathbf{s}}\right\rangle\left\langle{\mathbf{t}-\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle},\\\
\delta^{(\bar{1}\bar{1})}&=\frac{\left\langle{\mathbf{x}_{11},\mathbf{v}}\right\rangle\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{s}}\right\rangle+\left\langle{\mathbf{x}_{12},\mathbf{u}}\right\rangle\left\langle{\mathbf{v},\mathbf{t}\wedge\mathbf{u}}\right\rangle}{\left\langle{\mathbf{v},\mathbf{t}\wedge\mathbf{u}}\right\rangle\left\langle{\mathbf{t}-\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle}.\end{split}$
(48)

Most importantly, the planarity defects $\delta^{(ij)}$ are linear forms of
the full, in- and out-of-plane, components of the parametrization’s second
derivatives.

Now each panel in the unit cell, contributes a term
$g^{(ij)}(r^{2}\delta^{(ij)})$, $(i,j)\in\\{1,\bar{1}\\}^{2}$, to the bending
energy density of the tessellation equal to the bending energy of the relevant
panel for a given planarity defect $r^{2}\delta^{(ij)}$. The potential
$g^{(ij)}$ can be nonlinear in principle, but given that the planarity defects
are of order $O(r^{2})$, it is reasonable to linearize it in the vicinity of
$r\to 0$, that is while assuming, at the same time, that the plane is the
natural state of the panels. In conclusion, the bending energy density of the
tessellation takes the form

$b=\sum_{(i,j)\in\\{1,\bar{1}\\}^{2}}\frac{1}{2}D^{(ij)}(\delta^{(ij)})^{2}$
(49)

where the $D^{(ij)}$ are the panels flexural rigidities normalized with
respect to the area of a reference unit cell and where the $\delta^{(ij)}$
have been shown to be configuration-dependent (i.e.,
$(\mathbf{u},\mathbf{v},\mathbf{s},\mathbf{t})$-dependent) linear forms of the
second derivatives $\mathbf{x}_{\mu\nu}$. Equivalently, bending energy density
$b=b(\Gamma_{i\mu\nu};g_{\alpha\beta})$ can be written as a metric-dependent
quadratic form of the Christoffel symbols and of the curvatures. This is done
next in the particular case of the “Mars” pattern.

Note that while we linearized $b$ into a quadratic form in the limit $r\to 0$
(i.e., for infinitesimal panel bending), we make no a priori assumptions
regarding the macroscopic curvatures (i.e., the $\mathbf{x}_{\mu\nu}$). These
are free to take finite values. Last, the flexural rigidities $D^{(ij)}$ are
material constants. Here, we suppose for simplicity that they are all equal to
$D$.

##  Appendix G Bending energy – The “Mars” pattern

Similar considerations to the ones that led to equation (12) show that the
current configuration of the folds
$(\mathbf{u},\mathbf{v},\mathbf{s},\mathbf{t})$ can be determined from the
tangent vectors $(\mathbf{x}_{1},\mathbf{x}_{2})$. Indeed, we have, in the
case of an equilateral “Mars” pattern,

$\begin{split}\mathbf{u}&=\frac{1}{2}\mathbf{x}_{1}+\frac{\cos\alpha-\cos\beta}{g_{22}}\mathbf{x}_{2}-u_{n}\mathbf{n},\\\
\mathbf{v}&=-\frac{1}{2}\mathbf{x}_{1}+\frac{\cos\alpha-\cos\beta}{g_{22}}\mathbf{x}_{2}-u_{n}\mathbf{n},\\\
\mathbf{s}&=\frac{\cos\alpha+\cos\beta}{g_{11}}\mathbf{x}_{1}+\frac{1}{2}\mathbf{x}_{2},\\\
\mathbf{t}&=\frac{\cos\alpha+\cos\beta}{g_{11}}\mathbf{x}_{1}-\frac{1}{2}\mathbf{x}_{2},\end{split}$
(50)

with

$u_{n}=\sqrt{1-\frac{g_{11}}{4}-\frac{(\cos\alpha-\cos\beta)^{2}}{g_{22}}}$
(51)

being a known function of the metric tensor. Thus, the coefficients appearing
in the expression of $b$ can be written in terms of the metric tensor as well

$\begin{split}\left\langle{\mathbf{s},\mathbf{u}\wedge\mathbf{v}}\right\rangle&=-\left\langle{\mathbf{t},\mathbf{u}\wedge\mathbf{v}}\right\rangle=u_{n}\sqrt{g}/2,\\\
\left\langle{\mathbf{u},\mathbf{s}\wedge\mathbf{t}}\right\rangle&=\left\langle{\mathbf{v},\mathbf{s}\wedge\mathbf{t}}\right\rangle=-\frac{\cos\alpha+\cos\beta}{g_{11}}\sqrt{g}u_{n},\end{split}$
(52)

with $g=\det\mathbf{g}=g_{11}g_{22}$. As for the remaining terms involving the
$\mathbf{x}_{\mu\nu}$, they can be expanded into combinations of the
Christoffel symbols
$\Gamma_{\sigma\mu\nu}\equiv\left\langle{\mathbf{x}_{\mu\nu},\mathbf{x}_{\sigma}}\right\rangle$
and of the coefficients of the second fundamental form
$\Gamma_{3\mu\nu}\equiv\left\langle{\mathbf{x}_{\mu\nu},\mathbf{n}}\right\rangle$.
Namely, we have

$\begin{split}\left\langle{\mathbf{x}_{11},\mathbf{u}}\right\rangle&=\frac{1}{2}\Gamma_{111}+\frac{\cos\alpha-\cos\beta}{g_{22}}\Gamma_{211}-u_{n}\Gamma_{311},\\\
\left\langle{\mathbf{x}_{11},\mathbf{v}}\right\rangle&=-\frac{1}{2}\Gamma_{111}+\frac{\cos\alpha-\cos\beta}{g_{22}}\Gamma_{211}-u_{n}\Gamma_{311},\\\
\left\langle{\mathbf{x}_{12},\mathbf{u}}\right\rangle&=\frac{1}{2}\Gamma_{112}+\frac{\cos\alpha-\cos\beta}{g_{22}}\Gamma_{212}-u_{n}\Gamma_{312},\\\
\left\langle{\mathbf{x}_{12},\mathbf{v}}\right\rangle&=-\frac{1}{2}\Gamma_{112}+\frac{\cos\alpha-\cos\beta}{g_{22}}\Gamma_{212}-u_{n}\Gamma_{312}.\end{split}$
(53)

Recall that, in the present case where the curvilinear coordinates are
rectangular, the Christoffel symbols read

$\displaystyle\Gamma_{111}$ $\displaystyle=\frac{1}{2}g_{11,1},\quad$
$\displaystyle\Gamma_{211}$ $\displaystyle=-\frac{1}{2}g_{11,2},$ (54)
$\displaystyle\Gamma_{112}$ $\displaystyle=\frac{1}{2}g_{11,2},\quad$
$\displaystyle\Gamma_{212}$ $\displaystyle=\frac{1}{2}g_{22,1},$
$\displaystyle\Gamma_{122}$ $\displaystyle=-\frac{1}{2}g_{22,1},\quad$
$\displaystyle\Gamma_{222}$ $\displaystyle=\frac{1}{2}g_{22,2}.$

In conclusion, the total strain energy of a “Mars” pattern is

$\psi(\mathbf{x})=\int_{\Omega}b(\bm{\Gamma};\mathbf{g})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}$
(55)

where the bending strain energy density is

$b(\bm{\Gamma};\mathbf{g})=\frac{D}{2}\left((\delta^{(11)})^{2}+(\delta^{(\bar{1}1)})^{2}+(\delta^{(1\bar{1})})^{2}+(\delta^{(\bar{1}\bar{1})})^{2}\right)$
(56)

with the planarity defects $\delta^{(ij)}$ being the metric-dependent linear
forms given in (46) and (48). Therein, note how the rigidity coefficients in
front of the in-plane strain-gradient are proportional to the term $1/u_{n}$
which diverges when $g_{11}$ and $g_{22}$ approach their maximum and minimum
values, i.e., at the boundary of the domain of kinematical admissibility.
Meanwhile, the rigidity coefficients weighing the curvatures remain bounded.


# The recessionary pressures of generative AI: A threat to wellbeing

Jo-An Occhipintia,b,c,* Ante Prodana,b,c,d William Hynesb,e,f Roy Greeng
Sharan Burrowh Harris A. Eyreb,i,j,k Adam Skinnera Goran Ujdura,b,c John
Buchanana,b Ian B. Hickiea,b Mark Heffernand,m Christine Songa,b Marcel
Tannern,o

###### Abstract

Generative Artificial Intelligence (AI) stands as a transformative force that
presents a paradox; it offers unprecedented opportunities for productivity
growth while potentially posing significant threats to economic stability and
societal wellbeing. Many consider generative AI as akin to previous
technological advancements, using historical precedent to argue that fears of
widespread job displacement are unfounded, while others contend that
generative AI’s unique capacity to undertake non-routine cognitive tasks sets
it apart from other forms of automation capital and presents a threat to the
quality and availability of work that underpin stable societies. This paper
explores the conditions under which both may be true. We posit the existence
of an AI-capital-to-labour ratio threshold beyond which a self-reinforcing
cycle of recessionary pressures could be triggered, exacerbating social
disparities, reducing social cohesion, heightening tensions, and requiring
sustained government intervention to maintain stability. To prevent this, the
paper underscores the urgent need for proactive policy responses, making
recommendations to reduce these risks through robust regulatory frameworks and
a new social contract characterised by progressive social and economic
policies. This approach aims to ensure a sustainable, inclusive, and resilient
economic future where human contribution to the economy is integrated with
generative AI to enhance the Mental Wealth of nations.

###### keywords:

artificial intelligence, labour market, recession, economic policy, wellbeing

††journal: …

[first]Brain and Mind Centre, University of Sydney, Camperdown, Australia
Mental Wealth Initiative, University of Sydney, Camperdown, Australia Computer
Simulation & Advanced Research Technologies (CSART), Sydney, Australia

School of Computer, Data and Mathematical Sciences, Western Sydney University,
Sydney, Australia

The World Bank, Washington D.C., USA Santa Fe Institute, Santa Fe, New Mexico,
USA

University of Technology Sydney, Broadway, Sydney, Australia

Visiting Professor in Practice, London School of Economics Grantham Institute,
London, UK

Brain Capital Alliance, San Francisco, California, USA Baker Institute for
Public Policy, Rice University, Houston, Texas, USA Meadows Mental Health
Policy Institute, Dallas, Texas, USA

Business School, University of Sydney, Sydney, Australia

Dynamic Operations, Sydney, Australia

Swiss Academies of Arts and Sciences, Bern, Switzerland Swiss Tropical and
Public Health Institute & University of Basel, Switzerland

## “Why then, can one desire too much of a good thing?”

— Shakespeare, 1623

Stable, quality employment is the bedrock of societal strength, providing not
just economic stability but also a source of shared purpose, connectedness,
and psychological fulfillment. Work is the steady pulse that sustains
individual lives, families, the vitality of communities, and the prosperity of
nations. Exclusion from stable, high-quality work can precipitate poor
psychological wellbeing with its most severe consequence being the potential
for deaths of despair.[1, 2] Guarding against economic downturns and the
associated risks to employment is therefore a shared concern for economists,
governments, business leaders, labour organisations, and mental health
researchers. Enhancing productivity has long been a key focus, not only for
preventing economic downturns, but to improve standard of living, enhance
competitiveness, foster innovation, and create a more resilient labour market.
However, since the 2008 global financial crisis, there has been a global
slowdown in productivity growth affecting about 70% of advanced and developing
economies.[3] In this context, the launch of generative Artificial
Intelligence (AI), particularly of large language models like GPT-3 by OpenAI
in November 2022 opened up the possibility of a new frontier in generalised
productivity growth. Alongside this prospect has been debate on whether AI-
driven productivity growth represents an opportunity or a threat to social and
economic wellbeing. This paper explores the question: in realising the full
potential for AI-driven productivity growth, could too much of a good thing be
damaging? We argue that generative AI holds the capacity to profoundly reshape
labour market dynamics and paradoxically, if left to market dynamics,
undermine the very economic growth it aims to achieve.

Many consider generative AI as akin to previous technological advancements
such as the internet or computers and emphasize this development as simply a
continuity of the historical pattern of technology-driven progress. They argue
that, like past innovations, generative AI is another form of capital that can
augment human capabilities to enhance labour productivity, efficiency,
economic growth, and hence societal wellbeing. Their perspective contends that
fears of widespread job displacement are unfounded, using historical precedent
and analyses to demonstrate that technological advancements typically resulted
in the creation of new types of jobs, accompanied by a gradual adjustment in
the tasks performed by workers, the requisite skill levels, and the share of
income earned, rather than an aggregate reduction in the amount of work
available.[4, 5] Certainly, the post-pandemic reductions in unemployment rates
and increases in skills shortages in industrialised nations are giving
credence to this perspective.[6] An additional fundamental assumption that
underpins this optimistic business-as-usual scenario is that the new jobs that
are inevitably created as a result of this technological disruption will
require human input, and at a level to offset aggregate job loss. The World
Economic Forum’s Future of Jobs Report, 2023, estimates that the largest
drivers of job growth (to offset job displacement) will likely be in the areas
of big data analytics, climate change and environmental management
technologies, encryption and cybersecurity, education, agricultural
professionals, and e-commerce and trade.[7] While this analysis has some
merit, it is too limited in scope and reveals a fundamental misunderstanding
of the capabilities of generative AI.

Generative AI is not simply automation, it is a form of intelligence rapidly
evolving to acquire experiential learning, creativity, adaptability, strategic
thinking, and the ability to generate knowledge surpassing human capacities.
GPT-4 already has capability to ‘solve novel and difficult tasks that span
mathematics, coding, vision, medicine, law, psychology and more, without
needing any special prompting.[8] On professional and academic exams it
exhibits human-level performance scoring in the top 10% of test takers in a
simulated version of the Multistate Bar Examination.[8, 9] More important to
productivity and innovation than human performance in any given area, is the
ability to perform across multiple disciplines (e.g. engineering, medicine and
law) where human intelligence is constrained due to the sheer amount of
information it has to manipulate. OpenAI are developing GPTs that are
enhancing productivity not only in generalised everyday tasks but also in
professional assignments without the necessity of programming skills.
Estimates from the Brookings Institution indicate that, in the coming decade,
around 60% of job tasks in the United States alone are at medium to high risk
of being replaced by AI. [10] Progress is being made at unprecedented speeds.
On average, across a broad range of task categories, a 25% performance
improvement has been achieved with GPT-4 (released March 2023) compared to
GPT-3.5 released months earlier.[11] While timelines remain uncertain,
substantial investments are being made (anticipated to surpass US$300 billion
in 2026 [12]) to progress capability, reliability, and implementation of
generative AI, including the potential for self-advancement with minimal
future human input. Those that understand the capability and trajectory of
generative AI see the current surge in jobs for those with specialised AI
skills as containing the seeds for their own obsolescence.[13, 14] The
pioneers of this technology are now openly acknowledging that generative AI is
fundamentally a labour replacing tool.[15]

To appreciate the likely impact of generative AI on the nature of work and the
economy it is helpful to first consider historic trends in four occupational
categories; routine manual, non-routine manual, routine cognitive, non-routine
cognitive (see Appendix Table 1 of Borland & Coelli, 2023).[4] Figure 1
illustrates a typical example of these trends based on the Australian context
but similar across other advanced nations. The share of employment consisting
primarily of routine manual and routine cognitive tasks have declined with
increasing automation, particularly with the rise in IT-enabled technologies
from the 1990s.[4] However, offsetting this decline has been a 12.3 percentage
point increase in the share of employment in non-routine cognitive occupations
between 1986 to 2022 demonstrating the creation of new higher skilled, higher
paid jobs where the skills required are unique to labour and cannot be
replaced by capital.[4] The departure from this historical trend comes with
the advent of a form of capital (generative AI) that already has the capacity
to handle non-routine cognitive tasks. Unlike earlier automation technologies
that were primarily suited for routine and repetitive manual and cognitive
tasks, generative AI can analyse complex data, generate creative content, and
most recently has demonstrated the capacity to perform other non-routine
functions at or above human level performance.[8]

Figure 1: Share of employment by type of occupation, Australia, 1986 – 2022
(August) (reprinted with permission. Source: Borland, J. & Coelli, M. The
Australian labour market and IT-enabled technological change. Working Paper
No. 01/23. Melbourne Institute of Applied Economic & Social Research,
Melbourne, Australia, 2023).

Non-routine cognitive occupations in this classification include managers,
professionals, and technicians.[4] A broad range of professional and technical
occupations have been identified as facing medium to high exposure to
displacement by AI in the coming decade including accountancy, finance,
banking, legal, human resources, teaching, data analyst, creative writing,
marketing and communications, journalism, and medical diagnostic roles to name
a few.[16] While it is unlikely that AI will eliminate labour in affected
occupations entirely, the need for human input will be dramatically reduced,
even for new non-routine cognitive jobs and sectors that may be created as a
result of this technological disruption.[17] Occupations characterised by
routine cognitive tasks (such as sales and office/administration roles), and
some aspects of manual categories are also likely to be increasingly affected
by generative AI. While the deployment of generative AI may indeed create new
high-skilled occupations where labour is preferred to capital such as those
related to AI ethics, data privacy, and algorithm oversight, opportunities in
these new occupations will not be sufficient to offset the large-scale
displacement of workers across the cognitive occupations. With contraction of
the labour market, particularly in the cognitive occupational categories,
increases in unemployment, underemployment, non-participation in the labour
market, and displacement to lower paid manual jobs (including protective
services, food/cleaning, care services, labourers etc) should be expected in
the absence of regulation, placing downward pressure on demand for goods and
services in the economy.

## Recessionary pressures

The amount of downward pressure on demand will depend on the extent of the
shift in the ratio of AI-capital to labour (i.e. the extent of job
displacement) that is permitted. Replacement of existing capital with AI-
capital or smaller increases in the aggregate ratio of AI-capital to labour
across the economy are likely to simply augment labour productivity, lowering
prices, increasing consumers’ real incomes, increasing demand for goods and
services, and creating employment opportunities. However, a threshold likely
exists beyond which the substitution of labour for AI-capital (capital
deepening) will trigger the following self-reinforcing loop (in the absence of
government intervention) depicted visually by a causal loop diagram in Figure
2:

  * 1.

Decreasing availability of higher paid high- and middle-skilled jobs will
displace workers from a broad range of professions. This will increase
competition for lower-skilled jobs, lowering wage rates (wage compression),
reducing job quality, and likely expanding the insecure and exploitive gig
economy.

  * 2.

Average household income will fall, reducing demand for goods and services,
thereby exacerbating labour market contraction, and placing further downward
pressure on demand.

  * 3.

Even among workers who retain high-skilled jobs, rising job insecurity will
undermine consumer confidence leading to reduced discretionary spending. This
shift in consumption patterns would further exacerbate recessionary pressures.

  * 4.

A broad range of businesses would face challenges in maintaining
profitability, investment would decline, and financial markets are likely to
experience greater volatility.

  * 5.

Labour share of income (already in decline since the mid-1980s and attributed
largely to the impact of technology)[18] will decrease further as owners of
AI-capital secure a growing portion of national income.

  * 6.

Increasing income inequality, job scarcity, and associated population
displacement, coupled with a contraction in brain capital and diminishing
public trust in the economic system, is likely to intensify polarization and
political unrest, posing additional challenges to social and economic
stability.

Generative AI enables an aggregate decoupling of economic productivity from
labour and its virtualised infrastructure can scale quickly (in days instead
of years). While this would theoretically permit rapid and continuous
productivity growth, in practice substantial increases in the AI-capital-to-
labour ratio needed to achieve that continuous productivity growth would
likely fundamentally change labour market dynamics, undermine economic
stability, and be detrimental to individual and social wellbeing. Figure 3
provides a stylised graph of the theorized relationship between the AI-
capital-to-labour ratio, productivity, and demand, suggesting the potential
presence of a threshold beyond which demand falls, and highlighting a
‘productivity overhang’ (the productivity that cannot be realized due to
bounded demand for goods and services).

Figure 2: A causal loop diagram of the recessionary pressures of unrestrained
AI-capital displacement of labour. Figure 3: The theorised AI-capital-to-
labour ratio threshold beyond which demand growth cannot be sustained without
intervention.

## Additional considerations

Social prosperity: In addition to contributing to recessionary pressures, more
substantial increases in the AI-capital-to-labour ratio will precipitate a
broad range of threats to social prosperity. With middle- to high-income
cognitive occupational categories constituting more than 60% of the total
labour market (in the Australian context), shrinking of job opportunities in
these categories will catalyze a middle-class contraction as well as an
erosion of brain capital. Social mobility will decline, graduate opportunities
and pathways for career development will become scarce, creating extreme
competition in affected professions. These impacts will further exacerbate the
youth mental health crisis. As seen recently among young people in China,
extreme competition and declining economic opportunities can feed despondency,
reduce engagement with higher education, and increase a sense of
hopelessness.[19] Similar youth cohort scaring effects due to diminished
economic opportunities were seen as a result of austerity measures following
the 2008 global financial crisis.[20, 21] As generative AI becomes proficient
in replicating a broader range of cognitive tasks, the value of degrees and
certifications associated with certain cognitive skills and professions is
likely to diminish, posing significant risks to the traditional structures and
functions of the tertiary education sector.

The incentive to replace labour with AI-capital will be high. A 2017 analysis
across 800 occupations estimated that ‘almost half the activities people are
paid almost $16 trillion in wages to do in the global economy have the
potential to be automated by adapting currently demonstrated technology…’ This
is likely an underestimate given latest generative AI advances. To remain
nationally and globally competitive, companies across industries will seek to
reduce costs, maintain, or increase productivity, and remain relevant through
the adoption of this transformative technology. Without intervention, the risk
of significant and multifaceted disruptions, which cannot be ‘naturally’
corrected by market forces, are profound.

Traditional responses: It is important to consider that early in the onset of
declining demand a range of responses may be initiated to try to head off
further deterioration. However, traditional government and central bank
interventions, such as fiscal and monetary policies, might be less effective
as AI-induced job displacement is understood to be more than a temporary
market disruption. For instance, fiscal measures like increased government
spending or tax incentives may have limited impact if AI-induced job
displacement leads to a change in consumer behaviour, with individuals
choosing to save in anticipation of challenging times ahead. Similarly,
monetary policy measures, such as reduced interest rates or quantitative
easing, could be less effective if businesses are reluctant to invest due to
concerns about longer term demand suppression arising from AI-induced job
displacement. These potential limitations underscore the need for early
recognition and mitigation of the risks and mechanisms to prevent a self-
perpetuating cycle of recessionary pressures that could undermine economic
stability, democracy, and social wellbeing.

Implications for low-to-middle-income countries (LMICs): There are additional
implications of generative-AI-related job displacement for LMICs both in terms
of their exposure to risks and capacity for governments to respond. The
implications will vary based on factors such as national income, access to AI-
capital and supporting infrastructure, economic structure, industry
composition, regulatory capacity, social safety nets and their degree of
integration with the global economy. While generative AI might initially be
inaccessible or unaffordable in LMIC contexts due to the cost of computing
power and data to train large language models in a way that ensures privacy
and data sovereignty, the technology is improving at a rapid pace. As the high
costs of training AI models decreases, deploying this technology in LMICs
could become affordable, thus facilitating its spread.[22] The prospect of
widespread accessibility of generative-AI offers substantial opportunities for
LMICs, particularly through existing mobile phone technology. Among these
opportunities, generative AI could help support entrepreneurship and mitigate
skills shortages across various areas including healthcare, education,
software development, data analysis and decision making, and enable climate-
resilient infrastructure development and engineering, thereby driving overall
development.[23, 24]

Countering these potential benefits are additional risks. The economies of
India, the Philippines, China, Malaysia, Vietnam, and countries in Eastern
Europe and Africa, depend on offshored non-routine cognitive and routine
cognitive tasks like IT services, customer support, and accounting (Business
Process Outsourcing).[25, 26] These jobs, previously considered safe from
automation due to their cognitive nature, are now under threat as generative
AI evolves to perform complex tasks at or above human level performance.
Traditionally, high-income countries have offshored these tasks to LMICs to
take advantage of lower labour costs. However, it is likely to become more
economically viable for these tasks to be ‘reshored’ locally. This could lead
to significant job losses in LMICs. Additionally, economic instability in
high-income countries due to generative-AI-related job displacement could
result in reduced foreign investment and aid to LMICs, as well as decreased
demand for their exports, negatively impacting their economies. Between- and
within-country inequality could also be exacerbated. As AI-capital becomes a
dominant factor of production, the returns could accrue largely to the owners
of AI-capital, who will likely be concentrated in high-income countries or
among the wealthy elites in LMICs. Globally, the challenge lies in leveraging
the potential benefits of generative AI while mitigating risks as countries
transition to the Age of Intelligence.

## Recommendations

While we are at an early stage of diffusion, the rapid advancement of
generative AI demands an immediate and proactive response. The nature of this
response, however, hinges on our collective vision for the future. Will we
seek to uphold the status quo, prioritizing policies from the failed forty-
year experiment with free-market principles?[27] Or do we seek to forge a new
path, one where human contribution to the economy is harmoniously integrated
with AI, regulated, and valued in terms beyond the capacity to generate
profit? This latter vision suggests the need for a fundamental re-evaluation
of our current economic and social structures. To realise this, potential
government responses might include:

  1. 1.

A fiscal policy response. This may include an Automation Tax[28, 29] aimed at
disincentivising the over-leveraging of AI for productivity gains, and as a
means to support programs to ameliorate the externalities associated with
increased automation. While various perspectives exist regarding the
implementation of an Automation Tax alongside adjustments to other capital and
labour taxes and incentives, a common proposal suggests allocating the
generated revenue to programs supporting displaced workers, financing
alternative vocational education and training initiatives, or contributing to
social welfare programs. The Automation Tax seeks to strike a balance between
technological advancements and social responsibility, acknowledging the
potential economic impacts of job displacement and income inequality.[28]
Policymakers must carefully design the tax framework to achieve its intended
goal of disincentivising mass job displacement and ensuring equitable
distribution of benefits without stifling the potential of generative AI to
revitalise sluggish productivity growth and improve the quality of work.

  2. 2.

Regulation. This involves implementing robust mechanisms for dynamic
monitoring of labour displacement, gaining a comprehensive understanding of
where the threshold for the AI-capital-to-labour ratio lies, and instituting a
regulatory framework to ensure that the ratio is maintained above the critical
threshold. Systems modelling, a method of simulating the dynamics of complex
systems, including feedback loops,[30, 31] is necessary to determine where
economy-wide thresholds lie for different countries and to understand the
sensitivity of these thresholds to variations in the AI-capital-to-labour
ratio across different sectors. For example, in the Australian context, the
mining sector employs 2.1% of total workforce while professional, scientific,
and technical services employ 9% in 2023.[32] Therefore, labour displacement
in the latter sector is likely to have more significant implications for
shifting the AI-capital-to-labour ratio towards the recessionary threshold.

  3. 3.

Progressive social policies. Embracing the AI-revolution and accommodating
some increased labour displacement will require the design of a new social
contract to prevent economic harm, ensure equitable distribution of
productivity gains and maintain a central focus on the dignity of decent work
at the heart of societal wellbeing. Progressive social policies including Job
Guarantee Programs[33, 34] or a Participation wage[35, 36] focusing on
engaging populations in socially beneficial projects (social production) such
as civic infrastructure development, environmental conservation, or improving
community cohesion may offer some benefit. So too may initiatives to support
entrepreneurship, affordable housing, and cost of living to improve standard
of living for workers in lower-paid roles or the unemployed. However, while
such strategies may help counter recessionary pressures, alone they are
unlikely to be sufficient to prevent a significant contraction of the middle
class needed to sustain a thriving economy unless wages provided for social
production were above existing, relatively low, award rates. Mechanisms to
engineer real wage growth and improved job quality will therefore be needed in
sectors that are likely to remain labour intensive such as those in the care
economy; healthcare, education, childcare, and aged care, along with other
social, protective, and personal services.

  4. 4.

Frameworks for data-knowledge ownership: Mechanisms that ensure democratic
control and fair remuneration for both individual data and the intellectual
contributions made through interactions with Large Language Models (LLMs) are
critical for distributive justice, preventing exploitation and an exacerbation
of inequality. Human interactions contribute significantly to the learning and
evolution of LLMs, and individuals should have the ability to consent to the
use of their data and process knowledge, determine how it is shared, and be
fairly compensated for the value generated. This could be accomplished through
frameworks such as data cooperatives or trusts,[37, 38] ensuring that the
benefits of AI evolution are equitably shared. This approach not only
recognizes the value of an individual’s data but also rewards the cognitive
processes and innovative thinking that contribute to the development and
refinement of AI systems.

Generative AI, distinguished by its unique capacity to handle both routine and
non-routine cognitive tasks will increasingly challenge conventional
assumptions about job susceptibility to displacement. As this technology
evolves at a rapid pace, the prevailing belief that market forces will self-
correct after a period of disruption, leading to the creation of sufficient
new jobs, appears overly optimistic. This assumption overlooks the threat to
economic stability and social wellbeing. Drawing on a systems perspective,
this paper illustrates the implications of generative AI for productivity,
demand, and economic stability. It reveals a potential paradox where excessive
productivity growth driven by AI’s displacement of labour may lead to
recessionary pressures. It is imperative we grasp a realistic understanding of
AI’s potential to shift job market structure in ways that will exacerbate
social disparities, leading to reduced social mobility, cohesion and
heightened social tension. This highlights the critical need for proactive
policy responses and the consideration of a new social contract to ensure a
sustainable, inclusive, and resilient economic future in this era of rapid
technological advancement.

##

This paper solely reflects the views, opinions, arguments of its authors and
does not necessarily represent the perspectives of the organizations that
authors are associate with.

### Statement of potential competing interests:

Authors AP, WH, RG, SB, AS, GU, JB, MH, CS, and MT declare they have no
conflicts of interest relevant to this work. Author JO is both Head of Systems
Modelling, Simulation & Data Science, and Co-Director of the Mental Wealth
Initiative at the University of Sydney’s Brain and Mind Centre. She is also
Managing Director of Computer Simulation & Advanced Research Technologies
(CSART) and acts as Advisor to the Brain Capital Alliance. Author HE is a
consultant to PRODEO LLC (an executive services group for brain health
technologies), the Meadows Mental Health Policy Institute and the Euro-
Mediterranean Economists Association. In the past, he has received consulting
income from Delix Therapeutics, Neo Auvra and Johnson and Johnson. IBH is the
Co-Director, Health and Policy at the Brain and Mind Centre (BMC) University
of Sydney. The BMC operates an early-intervention youth services at Camperdown
under contract to headspace. He is the Chief Scientific Advisor to, and a 3.2%
equity shareholder in, InnoWell Pty Ltd which aims to transform mental health
services through the use of innovative technologies.

### Funding and acknowledgements:

This work was undertaken under the Mental Wealth Initiative supported by seed
funding and philanthropic gifts provided to the Brain and Mind Centre,
University of Sydney.

### Author contribution:

Manuscript concept and drafting: JO & AP; Critical revision of manuscript and
contribution of important intellectual content: all authors.

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