The branch with the furthest reach

How should a given amount of material be moulded into a cantilevered beam clamped
at one end, so that it will have the furthest horizontal reach? Here, we formulate and solve this
variational problem for the optimal variation of the cross-section area of a heavy cantilevered beam
with a given volume V , Young’s modulus E, and density ρ, subject to gravity g. We find that
the cross-sectional area should vary according a universal profile that is independent of material
parameters, with both the length and maximum reach-out distance of the branch that scale as
(EV /ρg)
, with a universal self-similar shape at the tip with the area of cross-section a ∼ s
, s
being the distance from the tip, consistent with earlier observations of tree branches, but with a
different local interpretation than given before. A simple experimental realization of our optimal
beam shows that our result compares favorably with that of our observations. Our results for the
optimal design of slender structures with the longest reach are valid for cross-sections of arbitrary
shape that can be solid or hollow and thus relevant for a range of natural and engineered systems.

A pendulum in a flowing soap film

We consider the dynamics of a pendulum made of a rigid ring attached to an elastic
filament immersed in a flowing soap film. The system shows an oscillatory instability
whose onset is a function of the flow speed, length of the supporting string, the
ring mass, and ring radius. We characterize this system and show that there are
different regimes where the frequency is dependent or independent of the pendulum
length depending on the relative magnitude of the added-mass. Although the system
is an infinite-dimensional, we can explain many of our results in terms of a one
degree-of-freedom system corresponding to a forced pendulum. Indeed, using the
vorticity measured via particle imaging velocimetry allows us to make the model
quantitative, and a comparison with our experimental results shows we can capture
the basic phenomenology of this system.

Swarming, swirling and stasis in sequestered bristle-bots

The collective ability of organisms to move coherently
in space and time is ubiquitous in any group of
autonomous agents that can move and sense each
other and the environment. Here, we investigate
the origin of collective motion and its loss using
macroscopic self-propelled bristle-bots, simple
automata made from a toothbrush and powered
by an onboard cell phone vibrator-motor, that can
sense each other through shape-dependent local
interactions, and can also sense the environment nonlocally via the effects of confinement and substrate
topography. We show that when bristle-bots are
confined to a limited arena with a soft boundary,
increasing the density drives a transition from a
disordered and uncoordinated motion to organized
collective motion either as a swirling cluster or a
collective dynamical stasis. This transition is regulated
by a single parameter, the relative magnitude of
spinning and walking in a single automaton. We
explain this using quantitative experiments and
simulations that emphasize the role of the agent
shape, environment and confinement via boundaries.
Our study shows how the behavioural repertoire of
these physically interacting automatons controlled
by one parameter translates into the mechanical
intelligence of swarms.

Gyrification from constrained cortical expansion

The exterior of the mammalian brain—the cerebral cortex—has
a conserved layered structure whose thickness varies little across
species. However, selection pressures over evolutionary time scales
have led to cortices that have a large surface area to volume ratio in
some organisms, with the result that the brain is strongly convoluted into sulci and gyri. Here we show that the gyrification can
arise as a nonlinear consequence of a simple mechanical instability
driven by tangential expansion of the gray matter constrained by
the white matter. A physical mimic of the process using a layered
swelling gel captures the essence of the mechanism, and numerical
simulations of the brain treated as a soft solid lead to the formation
of cusped sulci and smooth gyri similar to those in the brain. The
resulting gyrification patterns are a function of relative cortical expansion and relative thickness (compared with brain size), and are
consistent with observations of a wide range of brains, ranging
from smooth to highly convoluted. Furthermore, this dependence
on two simple geometric parameters that characterize the brain
also allows us to qualitatively explain how variations in these
parameters lead to anatomical anomalies in such situations as polymicrogyria, pachygyria, and lissencephalia

Aging in complex interdependency networks

Although species longevity is subject to a diverse range of evolutionary forces, the mortality curves of a wide
variety of organisms are rather similar. Here we argue that qualitative and quantitative features of aging can be
reproduced by a simple model based on the interdependence of fault-prone agents on one other. In addition to
fitting our theory to the empiric mortality curves of six very different organisms, we establish the dependence
of lifetime and aging rate on initial conditions, damage and repair rate, and system size. We compare the size
distributions of disease and death and see that they have qualitatively different properties. We show that aging
patterns are independent of the details of interdependence network structure, which suggests that aging is a
many-body effect, and that the qualitative and quantitative features of aging are not sensitively dependent on the
details of dependency structure or its formation.

Continuum dynamics of elastocapillary coalescence and arrest

The surface-tension–driven coalescence of wet hair, nano-pillars and supported lamellae immersed in an evaporating liquid is eventually arrested elastically. To characterize this at
a continuum level, we start from a discrete microscopic model of the process and derive a mesoscopic theory that couples the inhomogeneous dynamics of drying to the capillary forcing and
elastic bending of the lamellae. Numerical simulations of the resulting partial differential equation capture the primary unstable mode seen in experiments, and the dynamic coalescence of the
lamellae into dimers and quadrimers. Our theory also predicts the elastic arrest of the pattern
or the separation of lamellar bundles into their constituents as a function of the amount of liquid
left at the end of the process.

Optimal control of plates using incompatible strains

A flat plate will bend into a curved shell if it experiences an inhomogeneous
growth field or if constrained appropriately at a boundary. While the forward
problem associated with this process is well studied, the inverse problem of
designing the boundary conditions or growth fields to achieve a particular
shape is much less understood. We use ideas from variational optimization
theory to formulate a well posed version of this inverse problem to determine
the optimal growth field or boundary condition that will give rise to an
arbitrary target shape, optimizing for both closeness to the target shape
and for smoothness of the growth field. We solve the resulting system of
PDE numerically using finite element methods with examples for both the
fully non-symmetric case as well as for simplified one-dimensional and
axisymmetric geometries. We also show that the system can also be solved
semi-analytically by positing an ansatz for the deformation and growth
fields in a circular disk with given thickness profile, leading to paraboloidal,
cylindrical and saddle-shaped target shapes, and show how a soft mode can
arise from a non-axisymmetric deformation of a structure with axisymmetric
material properties.

Biomimetic 4D printing

Shape-morphing systems can be found in many areas, including
smart textiles1
, autonomous robotics2
, biomedical devices3
drug delivery4 and tissue engineering5
. The natural analogues
of such systems are exemplified by nastic plant motions,
where a variety of organs such as tendrils, bracts, leaves and
flowers respond to environmental stimuli (such as humidity,
light or touch) by varying internal turgor, which leads to
dynamic conformations governed by the tissue composition
and microstructural anisotropy of cell walls6–10. Inspired by
these botanical systems, we printed composite hydrogel architectures that are encoded with localized, anisotropic swelling
behaviour controlled by the alignment of cellulose fibrils along
prescribed four-dimensional printing pathways. When combined with a minimal theoretical framework that allows us to
solve the inverse problem of designing the alignment patterns
for prescribed target shapes, we can programmably fabricate
plant-inspired architectures that change shape on immersion
in water, yielding complex three-dimensional morphologies.

Programming curvature using origami tessellations

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can
be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom
collapsible structures—we show that scale-independent elementary geometric constructions and constrained optimization
algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or
varying curvature. Paper models confirm the feasibility of our calculations. We also assess the diculty of realizing these
geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we
characterize the trade-o between the accuracy to which the pattern conforms to the target surface, and the eort associated
with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of
absolute scale, as well as the quantification of the energetic and material cost of doing so.

Controllable biomimetic birdsong

Birdsong is the product of the controlled generation of sound embodied in a
neuromotor system. From a biophysical perspective, a natural question is
that of the difficulty of producing birdsong. To address this, we built a biomimetic syrinx consisting of a stretched simple rubber tube through which
air is blown, subject to localized mechanical squeezing with a linear actuator. A large static tension on the tube and small dynamic variations in the
localized squeezing allow us to control transitions between three states: a
quiescent state, a periodic state and a solitary wave state. The static load
brings the system close to threshold for spontaneous oscillations, while
small dynamic loads allow for rapid transitions between the states. We use
this to mimic a variety of birdsongs via the slow– fast modulated nonlinear
dynamics of the physical substrate, the syrinx, regulated by a simple controller. Finally, a minimal mathematical model of the system inspired by our
observations allows us to address the problem of song mimicry in an
excitable oscillator for tonal songs