In this article, Nature tells us that mathematicians have set up "equations that predict how fabric will fold." This theory of drapery could help fashion designers build clothes that hang straight. It also will allow computer animators to "model more realistically how clothes hang and move."
L. Maha Mahadevan of Harvard University in Cambridge, Massachusetts, and colleagues have come up with equations that predict the number and shape of folds in a draped sheet. The equations could be applied to anything from skirts to curtains.
The folds of draped fabrics have captivated artists and designers for hundreds of years, but only recently have scientists begun to understand what controls all those pleats and wrinkles.
Mahadevan has studied wrinkles and crumples for a while. Be sure to read "Folding, Wrinkling, and Crumpling," illustrated by a chiaroscuro by Leonardo da Vinci (ca. 1500) showing that a crumpled sheet exhibits deformations that are strongly localized around peaks and ridges (Credit: L. Maha Mahadevan).
You also can take a look at "New Wrinkle On Fighting Crow's Feet," which says that plastic garbage bags, human flesh, and the skins of apples all have wrinkles. Here is a photo showing these "crow's feet" wrinkles that appear around people's eyes as they age (Credit: L. Maha Mahadevan).
Now, let's go back to Nature and at other benefits from this mathematical theory of drapery.
Mahadevan’s equations could also allow clothing companies to give online shoppers a personalized, virtual view of how a garment will look on them -- something they are keen to do as web-based retailing gathers pace. "We don't have a formula for this stuff," says fabric modeller David Breen of Drexel University in Philadelphia. A mathematical model of drapery "could make things a lot easier", he says.
Part of the problem, says Breen, is that cloth is so stiff in the two dimensions that make up the plane of the cloth, but very floppy when it comes to folding in the third dimension. What also makes the maths so hard is that when a sheet of thin fabric crumples, nearly all the deformation gets concentrated into a single point or line, where the fabric kinks sharply. Determining where that point or line will fall is a tricky proposition.
This research work has been published by the Proceedings of the National Academy of Sciences on February 6, 2004 under the name "The elements of draping." Here is the abstract.
We consider the gravity-induced draping of a 3D 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 that arise because of the competition between elasticity and gravity. We analyze some of the simpler 3D 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. Our results are consistent with commonplace observations of drapes and complement large-scale computations of draping by providing benchmarks. They also yield a qualitative guide to fashion design and virtual reality animation.
Sources: Philip Ball, Nature, February 4, 2004; and various websites
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