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Mathematical Drapery
The
couturier
drapes the
three-dimensional
human body
with a two-dimensional
fabric, working
hard to subvert
the relentless
force of
gravity to
her cause
using a combination
of cuts,
folds and
tucks to
transform
a featureless
textile into
a piece of
art. Indeed
the depiction
of drapery,
in the form
of a carelessly
thrown shawl
on one's
knee is an
important
theme in
renaissance
art, both
in sculpture
and in sketching.
Modern art
has found
another expression
for the aesthetics
of drapery
in the carefully
orchestrated
wrapping
of an entire
building.
Inspired
by these
observations,
we consider
the gravity-induced
draping of
a three-dimensional
object with
a naturally
flat, isotropic
elastic sheet.
As the size
of the sheet
increases,
we observe
the appearance
of new folded
structures
of increasing
complexity
which arise
due to the
competition
between elasticity
and gravity.
We analyze
some of the
simpler 3-dimensional
structures
by determining
their shape,
and analyzing
their response
and stability,
and show
that these
structures
can easily
switch between
a number
of metastable
configurations.
For more
complex draperies,
we derive
scaling laws
for the appearance
and disappearance
of new length
scales.
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