Cracking the Mystery of the Egg Shape
Not all eggs are shaped like a chicken’s—now we know why!
A stem’s ‘sense of self’ contributes to shape
Mathematical framework explains diverse plant stem forms
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 di culty 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.
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.
Replicating folding of a fetal human brain
The distinctive troughs and crests of the human brain are actually not present in most animal brains; highly folded brains are seen only in a handful of species, including some primates, dolphins, elephants and pigs. In humans, folding begins in fetal brains around the 20th week of gestation and is completed only when the child […]
Designing a pop-up future
Simple origami fold may hold the key to designing pop-up furniture, medical devices and scientific tools
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.
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
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.
Prof. L. MahadevanPierce Hall 322
Harvard University
29 Oxford Street
Cambridge, MA 02138
lmahadev@g.harvard.edu
(617) 496-9599
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